Abstracts

On the Cauchy problem for first-order elliptic systems

Abdunabi Abdukarimov Tashkent State Architectural Construction University

In this note we present a formula for recovering the solution of systems of first-order differential equations of elliptic type of a special class in a domain on the basis of their values on the boundary.

Let A_{l×n}(x) denote the class of matrices D(x^{T})(x^{T} = (x_{1}, ..., x_{m})^{T} the transposed column vector of x ) with constant coefficients in C^{1} for which

D^{∗}(x^{T})D(x′) = ϵ((| x |^{2} +λ)U^{0},

where D^{∗}(x^{T}) is the Hermitian conjugate of the matrix D^{∗}(x^{T}) and ϵ(x) is a diagonal matrix , U^{0} is the unit vector. We denote by G_{ρ} a domain in R^{m} whose boundary consists of part of the - surface of the cone

y_{1}^{2} + ... + y_{m−1}^{2} = τy_{m}^{2}, (τ = tan π/2ρ)

and a smooth surface S lying inside the cone.

We consider the system of differential equations ( 1 )

(1)  D( \frac{∂}{∂x})u(x) = 0,

where the characteristic matrix

D(x^{T}) ∈ A_{l×n}(x).

In this note we consider the Cauchy problem in the formulation of M.M. Lavrent’ev. Suppose u(x) ∈ C^{1}(G_{ρ}) ∩ C(G_{ρ}) and u(x) satisfies (1) , and let u_{δ}(x) be continuous approximation to u(x) on S , i . e . ,

max | u(x) − u_{δ}(x) |< δ, 0 < δ < 1.

We want recover u ( x ) in G_{ρ}.

Theorem . Suppose u ( x ) is a solution of ( 1 ) and

| u(y) |= M, y ∈ T = ∂G_{ρ} \ S,

where M is a given positive number . Set

u_{σ}(x) = ∫_{S} N_{σ}(x, y)u(y)ds_{y}, x ∈ G_{ρ}.

Then

| u(x) − u_{σ}(x) |≤ MC(x)C(σ)exp(−σγ^{ρ}), x ∈ G_{ρ},

where

C(x) = C_{ρ} ∫_{S} \frac{ds}{r^{m−1}} ,

C(σ) = \{ \begin{matrix}σ^{m} for m = 2k + 1, k ≥ 1, \\ σ^{m−1} for m = 2k, k ≥ 2. \end{matrix}

Corollari . The limit equality

lim_{σ→∞} u(x) = u(x), x ∈ G_{ρ},

is satisfied uniformly in each compact set in G_{ρ}.

Localization in Schrödinger Dynamics with Random Potentials

Michael Aizenman Princeton University

A new method is introduced for the study of the Anderson localization in operators with random potential in the continuum (R^{d}). The results include exponential decay for the transition amplitudes throughout the localization regime, which can in principle be established through finite volume constructive criteria. The analysis utilizes analytic and probabilistic tools, and extends to the continuum the fractional moment method which was previously developed for discrete systems. (Work done in collaboration with A. Elgart, N. Naboko, J. Schenker, and G. Stolz.)

Wave focusing on the line

Tuncay Aktosun Mississippi State University

We consider “wave focusing” for the plasma-wave equation on the line with a potential that is integrable and has a finite first moment. We explicitly construct those solutions that are causal, incoming from either the left or right, consisting of a plane wavefront and a trailing tail behind in such a way that the tail totally disappears at a desired instant. Some applications are discussed.

Asymptotic expression of negative eigenvalue number of an operator differential equation with even order according to small parameter having weighted function in eigenvalue

Inci Albayrak Yildiz Technical University Mathematical Engineering Davutpasa Istanbul Turkey

Let H be seperable Hilbert space and while ρ(x) > 0 is a continuous function in [0, ∞). Let H_{1} a Hilbert space of H-valued measurable function f(x), (0 ≤ x < ∞) that satisfies ∫_{0}^{∞}ρ(x)∥f(x)∥^{2}dx < ∞ condition.

Let L_{h}, an operator generated by differential expression

ℓ_{h}(y) = \frac{1}{ρ(x)} \{(−1)^{n}hy^{(2n)} − Q(x)y\}, (0 ≤ x < ∞)

and boundary conditions

y^{(j)}(0) = 0, j = 0, 1, ..., n − 1

in H_{1} space. Q(x) in ℓ_{h}(y) is an operator function which satisfied Q^{∗}(x) = Q(x) ≥ 0, Q(x) ∈ σ_{∞}, lim_{x→∞} ∥Q(x)∥ = 0 and h > 0

is small parameter. Under this conditions, negative spectrum of L_{h} operator consists of eigenvalues. Let N_{h}(ϵ) denoted numbers of negative eigenvalues of L_{h} that smaller than −ϵ, (ϵ > 0). When h → 0

N_{ϵ}(h) ∼ \frac{1}{π \sqrt[2n]{h}} ∑_{j} ∫_{α_{j} (x)>ϵ} ρ(x) \sqrt[2n]{α_{j}(x) − ϵ}dx

is obtained. Here, α_{1}(x) ≥ α_{2}(x) ≥ ... are eigenvalues of Q(x). While n = 1, asymptotic expression of N_{h}(ϵ) is found in [1].

REFERENCE

1. M.Bayramoğlu,I.Albayrak, Second International Symposium on Mathematical and Computational Applications, Baku, September 1-3,1999.

Joint work with Oya Baykal and Mehmet Bayramoğlu.

Heat Kernel Asymptotics of Zaremba Boundary Value Problem

Ivan Avramidi New Mexico Tech

The Zaremba boundary-value problem is a boundary value problem for Laplace-type second-order partial differential operators acting on smooth sections of a vector bundle over a smooth compact Riemannian manifold with smooth boundary but with non-smooth (singular) boundary conditions, which include Dirichlet conditions on one part of the boundary and Neumann ones on another part of the boundary. We study the heat kernel asymptotics of Zaremba boundary value problem. The construction of the global parametrix of the heat equation is described and some first non-trivial coefficients of the heat kernel asymptotic expansion are computed explicitly.

Spline Approximations of the 3D Navier-Stokes Equations

Gerard Awanou University of Georgia

We consider numerical approximations of the 3D Navier-Stokes equations in velocity-pressure formulation. The pressure is eliminated from the equations by using a space of velocity fields which are divergence free. The later is discretized by means of splines of arbitrary degree and arbitrary smoothness. Energy arguments are used to derive the discrete equations satisfied by the equations. The pressure term is computed by solving a Poisson problem with Neumann boundary.

A note on existence and stability of periodic and almost periodic solutions of quasilinear equations with ”maxima”

Nataliya Bantsur Institute of Mathematics, NAS of Ukraine

Let us consider differential equation with ”maxima”

y′(t) = −δy(t) + p max_{s∈[t−h,t]} y(x) + f(t)  (1)

with initial conditions

y(t) = φ(t), t ∈ [−h, 0],

where f(t) ∈ C(R^{n}), f(t + T) = f(t), y(t) and φ(t) are continuous functions with values in R^{n}, p and δ are some positive constants.

We investigate problems relation to existence and stability of periodic and almost periodic solutions of Eq.(1). Note, that analogocal problems for some other types of differential equations with ”maxima” were studied in [1-3].

The sufficient conditions for existence, uniqueness and stability of periodic (almost periodic) solutions of Eq.(1) were obtained.

References:

1. Bantsur N.R., Trofimchuk E.P. Existence and stability of the periodic and almost periodic solutions of quasilinear system with maxima // Ukr. Math. J. - 1998. - 50, N 6. - P. 747-754.

2. Liz E., Trofimchuk S. Existence ans stability of almost periodic solutions for quasilinear delay systems and the Halanay inequality // J. Math. Anal. and Appl. 248, 625-644 (2000).

3. Pinto M., Trofimchuk S. Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima // Preceedings of the Royal Society of Edinburgh, 130A, 1103-1118 (2000).

The Spectrum of Differential Operators with Almost Constant Coefficients

Horst Behncke University of Osnabruck

The absolutely continuous spectrum of differential operators of order 2n of the form

Ly = w^{−1} ∑_{k=0}^{n}(−1)^{n}(p_{k}y^{(k)})^{(k)}

on L^{2}((0, ∞), w) is determined. We assume w, p_{n} > 0 and that the real valued coefficients admit a decomposition

p_{k} = p_{k1} + p_{k2} + p_{k3} + p_{k4}  k = 0, …, n
w = w_{1} + w_{2} p_{n3} = p_{n4} = 0

with

[p_{k3}, q_{k}, p′_{k,2}, p′′_{k1}γ^{−1}, p_{k,1}^{′2}w^{−1}γ^{2k−1}], γ^{2k}w^{−1}w′_{2}w^{−1}, w′′_{1}w^{−1}, w_{1}^{′2}w^{−1}γ^{−1} ∈ L_{1}
q_{k}γ^{2k}w^{−1}, p′_{k,1}γ^{2k−1}w^{−1}, w′_{1}w^{−1}γ^{−1} → 0 x → ∞

where q_{k} = ∫_{∞}^{t} p_{k4} and γ = (wp_{n}^{−1})^{1/2n}. Moreover assume (p_{k,1} + p_{k,2})γ^{2k}w^{−1} → c_{k}.

Then the absolutely continuous part of the spectrum of any selfadjoint extension of L is unitarily equivalent to that of the operator multiplication with ∑_{k=0}^{n} c_{k}x^{2k} on L^{2}(0, ∞).

This theorem extends all known results for such operators. The proof is based on a variant of the Liouville-Kummer transformation and asymptotic integration.

Various extensions of this result are also considered.

Stability of a general linear mechanical system with a white noise in parameters

Boris Belinskiy University of Tennessee at Chattanooga

This is joint work with Peter Caithamer.

We discuss the behavior, for large values of time, of a linear mechanical system with a white noise in its parameters. The initial data may be random as well but are independent of white noise. It is well known that a free linear nonstochastic mechanical system with viscous damping is stable, i.e., its energy approaches zero as time increases. We calculate the expected energy and check that this behavior takes place for the stochastic system under consideration in the case when the initial data are random but the parameters are not. When the parameters contain a random noise the expected energy may be infinite, approach zero, remain bounded, or increase with no bound. This regime is similar to but more interesting than the known regime for the solutions of differential equations with time dependent periodic coefficients that describes the behavior of a mechanical system with characteristics which are periodic functions of time. We give necessary and sufficient conditions for stability in terms of the structure of roots of an auxiliary equation. We also present results of numerical simulations that confirm our results on stability. We explain why our approach may not be applied to some other models.

The Lane-Emden Equation revisited

Rafael Benguria P. Universidad Católica de Chile

Several results concerning existence and nonexistence of radial (but not necessarily positive) solutions to the nonlinear boundary value problem

−Δu = λu + |u|^{p−1}u,

on a (geodesic) ball of R^{N} or S^{3}, with Dirichlet boundary condition will be presented. In particular, a whole classification of radial solutions in the Euclidean case will be presented (depending on the possible values of λ and the exponent p). The dependence on the underlying space will be illustrated through a discussion of the BVP on S^{3}.

Supported by FONDECYT (Chile) 199-0427.

Paley-Wiener methods in inverse spectral theory

Christer Bennewitz Dept of Mathematics, Lund University

A new method for obtaining uniqueness theorems in inverse spectral theory is described. It is based on a generalization of the classical Paley-Wiener theorem for the Fourier transform to the generalized Fourier transforms occuring in the spectral theory of differential operators. Several situations where this method has been successful will be described.

D-S projection methods for discrete Schrodinger operators

Lyonell Boulton Universidad Simon Bolivar, Caracas, Venezuela

Let H be the discrete Schrodinger operator Hu(n) := u(n − 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l^{2}(Z^{+}) where the potential v is real and v(n) → 0 as n → ∞. Let P be the orthogonal projection onto a closed linear subspace Λ ⊂ l^{2}(Z^{+}). In a recent paper E.B. Davies defines the second order spectrum Spec_{2}(H, Λ) of H relative to Λ as the set of z ∈ C such that the restriction to Λ of the operator P (H − z)^{2}P is not invertible within the space Λ. We study properties of Spec_{2}(H, Λ) when Λ is large but finite dimensional. Our particular interest is the conection between this set, the spectrum and the “resonances” of H. In our main result we provide sharp bounds in terms of the potential v for the asymptotic behaviour of Spec_{2}(H, Λ) as Λ increases towards l^{2}(Z^{+}).

Interpolation and Inverse Spectral Theory

Amin Boumenir State University of West Georgia

We are interested in finding necessary and sufficient conditions for interpolation at irregular points to hold. Given a sequence of real numbers {μ_{n}}_{n≥0} , can we find a space containing a sequence {S_{n}(μ)}_{n≥0} such that the values {F (μ_{n})}_{n≥0} define a unique function by the following sampling expansion formula, F (μ) = ∑_{n≥0} F (μ_{n})S_{n}(μ)? We show that the inverse spectral problem, and more precisey the Gelfand Levitan theory, can be used to construct sampling type theorems from the knowledge of the sampling points only. Thus one can recover a function from the knowedge of its given values on a sequence of points. This improves Kramer’s theorem as it reveals all possible distributions of the sampling points together with a construction of the sampling functions.

Computing the spectrum of the one-dimensional p-Laplacian

B Malcolm Brown Cardiff University

The talk will focus on some recent work to compute the eigenvalues and Fucik spectrum of the one-dimensional p-Laplacian. This is joint work with Wolfgang Reichel of Basel.

Some recent results, applications, and open problems concerning Opial’s inequality

Richard Brown University of Alabama, Tuscaloosa

We sketch some recent developments concerning the classic inequality

∫_{0}^{1} |yy′| ≤ (1/4) ∫_{0}^{1} |y′|^{2}
y(0) = y(1) = 0

discovered by Zdzislaw Opial in 1960. There are significant applications of this inequality to differential equations, some of which we discuss. Also there are several interesting extensions when the derivative is replaced by an operator and/or weights are considered. Although there has been much progress due to such writers as Boyd, Agarwal, Bloom, Anastassiou, Pecaric, and others, many open problems remain.

Walks in Rigid Environments: Continuous Limits

Leonid Bunimovich Georgia Tech

We consider the models where a particle hopes at integer times to one of the neighboring vertices of a lattice. The scatterers are randomly distrubuted along the vertices of the lattice and their type prescribe the particle where to jump. This class of systems describes deterministic walks in random environments (or multidimensional, i.e. with many tapes, Turing machines). A general class of these models has been recently shown to be completely solvable in 1D. We consider continuous limits of these systems and demonstrate that their behavior is quite different from probabilistic models like a classical biased random walk. Deterministic dynamics ensures that in a continuous limit fluctuations disappear and thus only Euler-type limit exist.

On the Cauchy problem for a dynamical Euler’s elastica

Almut Burchard University of Virginia, Department of Mathematics

We consider the dynamics for a thin, closed loop inextensible Euler’s elastica moving in three dimensions. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives, and a second order elliptic equation for its tension. Local existence and uniqueness of solutions are established for initial data in suitable Sobolev spaces.

The proof uses a Hasimoto transformation to rewrite the equations in convenient coordinates for the space and time derivatives of the tangent vector. A feature of this elastica is that it exhibits time-dependent monodromy – a frame parallel-transported along the elastica is in general quasi-periodic, resulting in time-dependent boundary conditions for the coordinates. We address this complication by a gauge transformation, after which a contraction mapping argument can be applied.

(Joint work with L. E. Thomas.)

Qualitative Analysis of Three Dimensional System Differential Equations of Special Forms.

Tolibjon Buriev Samarkand State University

Pulse Waves in Elastic Tubes

Radu Cascaval University of Missouri - Columbia

I will describe a simplified one-dimensional model for the pulse wave propagation through the blood vessels, considered as fluid-filled tubes with elastic walls. Our model takes into account the elasticity of the wall as well as the tapering effect. The influence of fluid viscosity on the wave deformation will be briefly discussed. The spatial dynamics in this model is governed by a variable coefficient dispersive equation with initial conditions given at the inflow. An existence theory for the associated evolution equation, based on an extension of the semilinear Hille-Yosida theory, will be detailed.

On a Dirac-Fock type models for crystals

Isabelle Catto CNRS and Univerity of Paris IX-Dauphine

In this talk, we will propose a relativistic model of Dirac-Fock type for crystals, which is the analogue of the Dirac-Fock model for molecules in this periodic setting. This is a work in collaboration with Eric Paturel.

Multidimensional Transonic Shocks and Free Boundary Problems for the Euler Equations for Potential Fluids

Gui-Qiang Chen Northwestern University

We will first discuss recent results on the existence and stability of multidimensional transonic shocks (hyperbolicelliptic shocks) for the Euler equations for steady potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second-order, nonlinear elliptichyperbolic equation of mixed type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of smooth flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We will introduce a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow; and the free boundary is also stable under the steady perturbation of the hyperbolic phase. We will also present some further results in this direction, including the results on the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks in unbounded domains (joint work with Mikhail Feldman).

Infrared renormalization and infraparticle states in QED

Thomas Chen Courant Institute, NYU

We consider a freely propagating, non-relativistic, charged particle interacting with the quantized, ultraviolet regularized electromagnetic field. Introducing an infrared regularization σ_{0}, we study H_{p}(σ_{0}), the restriction of the corresponding hamiltonian to the fiber Hilbert space H_{p} associated to the conserved total momentum p. For |p| sufficiently small, we show that E_{0}(|p|, σ_{0}) =infspec(H_{p}(σ_{0})) is an eigenvalue, and that E_{0}(|p|, σ_{0}) − \frac{|p|^{2}}{2} has small first and second derivatives w.r.t. |p|, uniformly in σ_{0}. Furthermore, the corresponding eigenvector is an element of H_{p}(σ_{0}) for all σ_{0} > 0, but not in the limit σ_{0} → 0. The proof is based on the Bach-Fröhlich-Sigal operator-theoretic renormalization group.

Hopf’s last hope: Spatiotemporal chaos in terms of unstable recurrent patterns

Predrag Cvitanovic Georgia Tech

If everthing in a turbulent systems is in constant flux, how is it that humans are able to distinguish different kinds of turbulence? Hopf’s answer was that dynamics drives a given spatially extended system through a repertoire of unstable patterns; as we watch a given “turbulent” system evolve, every so often we catch a glimpse of a familiar pattern. For any finite spatial resolution, the system follows approximately for a finite time a pattern belonging to a finite alphabet of admissible patterns, and the long term dynamics can be thought of as a walk through the space of such patterns, just as chaotic dynamics with a low dimensional attractor can be thought of as a succession of nearly periodic (but unstable) motions.

Hopf’s proposal is in its spirit very different from most ideas that animate current turbulence research. It is not the Kolmogorov’s 1941 homogeneous turbulence with no coherent structures fixing the length scale, here all the action is in specific coherent structures. It is emphatically not universal; spatiotemporally periodic solutions are specific to the particular set of equations and boundary conditions. And it is not probabilistic; everything is fixed by the deterministic dynamics with no probabilistic assumptions on the velocity distributions or external stochastic forcing.

I will describe a modest implementation of Hopf’s program on a 1-dimensional spatially extended Kuramoto-Sivashinsky system, a PDE that describes interfacial instabilities such as unstable flame fronts.

Quantum Dynamical Bounds for One-Dimensional Quasicrystals

David Damanik California Institute of Technology

We consider discrete one-dimensional Schrödinger operators with Sturmian potentials and discuss quantum dynamical upper and lower bounds for these models.

Coexistence of spectra in rank-one perturbation problems

Rafael del Rio Castillo UNAM

I shall report on joint work with A. Poltoratski and S. Fuentes on the behavior of spectra of selfadjoint operators when rank one perturbations are considered. It is well known that a.c. spectra is stable and singular spectra is in general unstable under these kind of perturbations. We study the phenomenon of coexistence of differents types of spectra for families of perturbations which depend on a parameter. Almost anything can happen.

The theory of orthogonal polynomials and spectral analysis of Schrödinger operators.

Serguei Denissov California Institute of Technology

Some new results in the spectral analysis of Schrödinger operators will be presented. Deep relations to the theory of polynomials orthogonal on the unit circle will be discussed.

Derivation of the nonlinear Schrödinger equation with Coulomb potential

Laszlo Erdos GeorgiaTech

We consider the time evolution of N bosonic particles interacting via a mean field Coulomb potential. Suppose the initial state is a product wavefunction. We show that at any finite time the correlation functions factorize in the limit N → ∞. Furthermore, the limiting one particle density matrix satisfies the nonlinear Hartree equation. The key ingredients are the uniqueness of the BBGKY hierarchy for the correlation functions and a new apriori estimate for the many-body Schrödinger equations.

Qualitative Analysis of Three Dimensional System Differential Equations of Special Forms.

Vafokul Ergashev Samarkand State University

On the Zero modes of Pauli and Dirac operators

W. Desmond Evans Cardiff University, Wales, UK

The talk will present some recent results obtained jointly with A. Balinsky which prove that, within an appropriate class, magnetic fields which give rise to zero modes are rare. An estimate of the number of zero modes is also given.

Generalized Schrödinger operators of the graph type

Pavel Exner Czech Academy of Sciences

We discuss generalized Schrödinger operators of the form −Δ − αδ(x − Γ) in L^{2}(ℝ^{2}), where Γ is a planar graph. Various spectral properties of these operators can be derived by combining minimax estimates with the generalized Birman-Schwinger principle. In particular we give conditions under which the geometry gives rise to a non-void discrete spectrum for a noncompact Γ. We also discuss the relations between the asymptotic properties of such operators as α → ∞ and the spectral behavior of a suitable operator on L^{2}(Γ) in the situation when Γ an infinite curve or a loop, as well as emergence of persistent currents in such an asymptotics.

Time-reversal mirror techniques in the regime of separation of scales

Jean-Pierre Fouque North Carolina State University

A time-reversal mirror is, roughly speaking, a device able to memorize an ultrasonic signal and to resend a part of it back into the same medium in the reverse direction of time. We analyse the refocalization effect in the context of layered random media. This is done in the regime of high frequencies and ultra-short correlation lengths of the medium. Mathematically this translates into a system of PDE’s with random coefficients depending on a small parameter representing the separation of scales. Diffusion-approximation results are used to perform an asymptotic analysis of this system.

The total magnetic moment of large atoms in strong magnetic fields

Soeren Fournais Lab. de Math., Univ. Paris-Sud

We look at the magnetisation of neutral atoms in large magnetic fields. Asymptotic formulas for the energy exist with high precision in different regions depending on how large the nuclear charge Z is compared to the magnetic field strength B. All these formulas take the splitting of the kinetic energy into Landau levels as the principal feature and then treat the electric potential as a perturbation. We prove that these approximate formulas predict correctly (to highest order) the total magnetic moment of the atom. The proof of this fact relies on a ”virial theorem” for Coulomb systems in a constant magnetic field.

On the spectrum of second order elliptic differential operators with periodic coefficients

Leonid Friedlander University of Arizona

Under an additional symmetry assumption, we prove that the spectrum of an elliptic second order differential operator with epriodic coefficients in R^{n} is purely absolutely continuous.

A dyadic model for non-Newtonian fluid equations

Susan Friedlander University of Illinois, Chicago

We present a dyadic model for the equations for a non-Newtonian fluid introduced by Ladyzhenskaya. We prove a bound for the hausdorff dimension of the singular set at the first time of blow-up. The result interpolates between the solvability results of Ladyzhenskaya and the bound for the dimension of the singular set of Caffarelli, Kohn, and Nirenberg.

This is joint work with Natasa Pavlovic.

On the singular Cauchy type integrals with a continuous density for function of many variables

Abdurasul Gaziev Department of Mathematics Samarkand State University , University Boulevard 15, 703004 , Samarkand , Uzbekistan

The singular integral operators are met in the solution of many important problems of applied character; particularly, in solution of boundary problems of analytical functions, linear and non-linear singular integral equations, various problems of the mathematical physics, theory of elasticity, queuing problems, etc.

At the present time the properties of the singular Cauchy type integral in various functional spaces for onedimensional case are rather well studied. At the same time the properties of Cauchy type integral for multi-dimensional case (on a continuity up io the boundary of a semi- cylindrical domain; on a behavior, at the boundary of the semicylindrical domain, of limiting values of Cauchy type integral that are expressed by Sokhotskii type formulas via repeated singular integral ) are not well investigated.

In the present paper we consider a multiple Cauchy type integral

F (z) = \frac{1}{(2πi)^{n}} ∫_{Δ} \frac{f(τ)}{τ − z} dτ,  (1)

where z = (z_{1}, z_{2}, ..., z_{n}), τ = (τ_{1}, τ_{2}, ..., τ_{n}), dτ = dτ_{1}dτ_{2}..., dτ_{n},

τ − z = ∏_{k=1}^{n}(τ_{k} − z_{k}), Δ = γ^{1} × γ^{2} × ⋯ × γ^{k}

γ^{k} is a closed Jordon rectifiaable curve (c. j. r. c) on the complex plane z_{k} (k = \overline{1, n}), f ∈ C_{Δ}, where C_{Δ} is a space of continuous functions on the spanning set Δ.

To study the properties of the limiting values of integral (1) the partial and mixed moduli of continuity of function f, and a new characteristic

θ(δ) (θ^{k}(t_{k}, δ) = ∫_{γ_{δ}^{k}(t_{k})} |dτ_{k}|, θ_{(δ)}^{k} = sup_{t_{k}∈γ^{k}} θ^{k}(t_{k}, δ), γ_{δ}^{k}(t_{k}) = {τ ∈ γ^{k};

|τ − t_{k}| ≤ δ, δ ∈ (0, d_{k}), d_{k} = sup(|τ_{k} − t_{k}|))

of the curve γ^{k} were chosen.

With the help of these characteristics we obtained inequalities analogous to those of Zygmund type, which connect the partial and mixed moduli of the continuity of an image and pre-image of the limiting value of integral (1) for any (c. j. r. c) γ.

With the help of the obtained inequalities we constructed the Banach space, which is invariant with respect to the limiting value of integral (1), which is spread to a class of curves, satisfying the condition θ^{k}(δ) ∼ δ (k = \overline{1, n}). This class is significantly wider than that of piecewise smooth curve (the presence of cusps is supposed).

Spectral analysis of a class of singular differential operators

Daphne Gilbert Dublin Institute of Technology

Lower order terms in Szego type asymptotic formulas

Dimitri Gioev Department of Mathematics, University of Pennsylvania

We present two results. The first is a three term Szego type asymptotic formula for a zeroth order pseudodifferential operator (PsDO) on a Zoll manifold of an arbitrary dimension. An example is a Szego type asymptotics for multiplication by a smooth function on a sphere of any dimension. The second result is a four term Szego asymptotics for a PsDO on the unit circle. The proofs utilize a combinatorial formula which generalizes the formula of G.A.Hunt and F.J.Dyson given in an article by M.Kac, where the usual Strong Szego Limit Theorem is provided with a simple combinatorial proof.

Predictability and Solution Error Models for Flow in Porous Media

James Glimm Stony Brook University

Prediction with the quantification of uncertainty is needed to take advantage of the opportunities created by modern simulation. As more of the stages of scientific inquiry are computationally based, there is an increased need to automate some of the decision processes associated with the computation.

Error models for numerical solutions (as well as for observations) are the basic ingredient for uncertainty quantification. It is the comparison of solutions to observations which refines our models of porous media (through history matching). The probability of error in either generates the probability for accuracy of some model of the media. This probability, together with a forward simulation (and its error with probability) generates confidence intervals for prediction.

This talk will develop a general stochastic framework for prediction, based on error models for numerical solutions of partial differential equations, and illustrated by problems of turbulent mixing and flow in porous media. Recent results of the speaker and collaborators will be presented.

The n-dimensional Heat Equation with Nonlinear Generalized Wentzel Boundary Conditions

Gisele Goldstein U of Memphis

Of concern is the Cauchy problem for the heat equation on D ⊂ ℝ^{n}

u_{t} = ∇ · (α(x)∇u) x ∈ D, t > 0
u(x, 0) = f(x) x ∈ ∂D

with nonlinear generalized Wentzel boundary conditions

∇ · (α(x)∇u) + b(x) \frac{∂u}{∂n} ∈ c(x)β(x, u) for x ∈ ∂D, t > 0.

The usual Wentzel boundary condition corresponds to b(x) = c(x) = 0. Here we assume that α ∈ C^{1}(\overline{D}), α > 0 on \overline{D}, b, c ∈ C(∂D), b ≥ 0, c ≤ 0 and β is a maximal monotone graph in ℝ^{2} containing the origin. We present an L^{p} theory for this problem, which gives rise to a nonlinear semigroup on a Banach space X^{p}, 1 ≤ p < ∞ and on C(\overline{D}) in the case p = ∞. Hence, we obtain a solution to the initial boundary value problem. One novel feature of this work is that X^{p} is not L^{p}(D) but rather an L^{p} space based on functions on D and on ∂D, with a measure on ∂D based on the boundary conditions. Questions of regularity and smoothing of the solutions are also obtained via Nash-Moser iteration.

This is joint work with Angelo Favini, Jerry Goldstein, and Silvia Romanelli.

Instantaneous blow-up for heat equations

Jerry Goldstein U of Memphis

Of concern is the existence of nontrivial positive solutions of du/dt = Lu + V (x)u, where L is a second order symmetric elliptic operator and the potential function V has singularities. In many cases there are sharp borderlines between good and bad behavior. The main cases are: x in R^{n} and L=Laplacian [Baras-Goldstein], the analogous problem on the Heisenberg group [Goldstein-Zhang], L uniformly elliptic but with bad coefficients [Goldstein-Zhang], and V highly oscillatory [Goldstein-Kombe-Zhang].

Asymptotic Completeness for Compton Scattering

Marcel Griesemer University of Alabama at Birmingham

Scattering in a model of a massive quantum-mechanical particle, an “electron”, interacting with massless, relativistic bosons, “photons”, is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of “photons” by the “electron”; but “electron” pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed “electron” is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one–electron states with propagation estimates for the “electron” and the “photons”.

Joint work with J. Froehlich and B. Schlein.

Spectral theory of Schrödinger Operators with periodic magnetic field

Michael J Gruber The University of Arizona

Schrödinger operators with periodic electric fields are at the heart of solid state physics, their spectrum (band structure) is well known. The case of periodic magnetic fields (non-zero magnetic flux) poses new challenges since the operator is not periodic any more; indeed it allows for new spectral phenomena: infinitely degenerate point spectra and Cantor spectra are possible. In this talk we describe criteria on the operator that decide about the appearance of band structure, and we give more detailed information on the spectrum (spectral type, degeneracy).

Effective dynamics of magnetic vortices

Stephen Gustafson ETH-Zuerich

Ginzburg-Landau equations are nonlinear PDEs arising in many areas (superconductors, superfluids, particle physics). A basic qualitative feature of their solutions is the presence (and dynamics) of localized, topological structures called vortices. We show that solutions (with appropriate initial data) are well-described for long times as a collection of moving vortices, and we identify the dynamical system which describes (to leading order in the vortex separation) the motion of the vortex centres. Once certain spectral properties of an associated linearized operator are established, the argument is quite general. This is joint work with I.M. Sigal.

A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates

George Hagedorn Virginia Tech

We present joint work with Alain Joye on the construction of an exponentially accurate time–dependent Born– Oppenheimer approximation for molecular quantum mechanics.

We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to ϵ^{−4}, where ϵ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time–dependent Schrödinger equation that agree with exact normalized solutions up to errors whose norms are bounded by C exp ( −γ/ϵ^{2} ), for some C and γ > 0.

One non-relativistic particle coupled to a photon field

Christian Hainzl LMU-Muenchen

We investigate the ground state energy of a charged particle coupled to a photon field. First, we regard the self-energy of a “free” electron, which we describe by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of the coupling constant α, the leading order term is represented by 8πα(Λ − ln[1 + Λ]).

Secondly, we treat the self-energy of a charged boson and provide a different proof for recovering the next to leading order term in α, which has already been obtained in Hainzl-Vougalter-Vugalter.

Thirdly, we estimate from above the binding energy of a charged boson in the field of a nucleus. The first order radiative correction turns out to behave like ln[1 + Λ] for large values of the ultraviolet-cutoff parameter Λ.

Gap estimates for Schrödinger operators depending on curvature

Evans Harrell Georgia Tech

Schrödinger operators depending on curvature, of the form

−∇^{2} + q(κ)

on L^{2}(M), where M is a compact surface or submanifold of R^{n} arise in the study of thin structures, among other topics in physics. Typically, q is a quadratic function of the principal curvatures. The author will present some estimates of the low-lying eigenvalues and the gaps between them. In some cases these estimates are shown to be optimal.

On the discrete spectrum of Schrödinger operators with strong magnetic fields of compact support.

Rainer Hempel TU Braunschweig

We discuss the discrete eigenvalues of magnetic Schrödinger operators in R^{2}

H(λa⃗) = (−i∇ − λa⃗)^{2} + V (x), λ ≥ 0,

where V is a bounded potential and a⃗ a magnetic vector potential of class C^{1} in R^{2} which is such that the associated magnetic field B = curl a⃗ has compact support consisting in a finite number of components. The interesting case is when the flux ∫ B dx is non-zero. As for the unperturbed operator H = −Δ + V, we consider the periodic case as well as the case where H has compact resolvent, like the harmonic oscillator.

More precisely, we study the (signed) flow of spectral multiplicity across a fixed energy level E ∈ R \ σ(H), where σ(H) is the spectrum of H. It is shown that there exists a sequence of couplings λ_{k} → ∞ such that the total signed spectral flow for H(λ_{k}a⃗) corresponds to the spectral flow for the family of operators H + λχ_{Ω}, as λ → ∞, where Ω = {B(x) ≠ 0}.

On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

Lotfi Hermi University of Arizona

We produce a new proof and extend recent results by Harrell and Stubbe for the discrete spectrum of a selfadjoint operator. An abstract approach is used to produce the theorems. We also analyze the strength of the various inequalities that ensue, and display various applications to physical and geometric problems. The results contain all the known classical results (Payne-Polya-Weinberger, Hile-Protter, H.C. Yang).

This is joint work with Mark S. Ashbaugh

Bifurcation of relative equilibria in symmetric Hamiltonian systems at singular momentum values

Antonio Hernandez IIMAS-UNAM

We will recall the notions of momentum map and reduction of Hamiltonian systems with symmetry, along with known results in bifurcation of relative equilibria at points with non-generic or singular value of the momentum map. We will summarize and illustrate some of our results in bifurcation of relative equilibria in the context of cotangent bundles.

Delocalization for Schrödinger operators

Andreas M. Hinz Technical University Munich

The phenomenon of localization, i.e. the appearance of intervals of dense point spectrum associated with exponentially decaying eigenfunctions, has become a popular topic in the spectral theory of Schrödinger operators. The antithetical event, namely isolated eigenvalues with long range eigenfunctions, is not so well-known. There are examples for the one-dimensional case which provide insight into the limitations of some theorems relating the behavior of eigensolutions at infinity to the spectrum. Such ideas have also been employed to show that Dirac operators with potentials tending to infinity may have holes in the essential spectrum. We will summarize these results and indicate possible approaches for higher dimensions.

Global Continuity of the Integrated Density of States

Peter Hislop University of Kentucky

We prove that the integrated density of states for random Schrödinger operators with Anderson-type potentials on L^{2}(R^{d}), for d ≥ 1, is Hölder continuous at all energies. The single-site potential must be nonnegative and compactly supported, and the distribution of the random variable must be absolutely continuous with a bounded, compactly supported density. We also prove this result for random Anderson-type perturbations of the Landau Hamiltonian in two-dimensions under a rational flux condition. This is joint work with J. M. Combes and F. Klopp.

Trace distributions and heat invariants for Schrödinger operators

Michael Hitrik University of California at Berkeley

We provide a complete description of the asymptotics of regularized traces for functions of Schrödinger operators with long range potentials. The traces in question involve higher order terms in associated noncommutative Taylor expansions, and the coefficients in the asymptotics are expressed in terms of powers of the full operator applied to the Euclidean distance function. This is joint work with I. Polterovich.

A geometrical version of Hardy’s inequality

Maria Hoffmann-Ostenhof University of Vienna

This is joint work with T. Hoffmann-Ostenhof and A. Laptev. A Hardy’s type inequality is proven for domains in ℝ^{n}, where the distance to the boundary and the volume of the domain is involved. In particular a result is obtained which gives a positive answer to a question asked by Brezis and Marcus.

Eigenvalues in Symmetry subspaces

Thomas Hoffmann-Ostenhof Erwin Schrödinger Institute for mathematical Physics

This is joint work with B. Helffer, M. Hoffmann-Ostenhof and N. Nadirashvili. We consider 2-dimensional Schrödinger operators on bounded domains with suitable boundary conditions which commute with the actions of a group. For certain cases (dihedral group) we prove that the multiplicity of the groundstate eigenvalues in the symmetry subspaces associated to the irreducible representations of the group equals the degree of the corresponding irreducible representation. Furthermore for some cases it is possible to order these groundstate eigenvalues in the same way as the radial problem would imply.

Modelling of Nonlinear Dynamic Linked Elastic Structures

Mary Ann Horn Vanderbilt University

In recent years, the tools required to study elastic systems have been sufficiently developed to allow researchers to focus on linked elastic structures. For example, linear models of plate-beam configurations which involve geometric junction conditions are known to be wellposed and controllable. Extensions to nonlinear plate-beam systems are motivated by issues arising from large-amplitude periodic oscillations, but such systems have not been seen in the literature. A model comprised of a nonlinear von Kármán plate coupled with a nonlinear beam equation is developed. Dynamic junction conditions are imposed at the interface. Wellposedness is established by first considering a corresponding linear problem, then applying a perturbation theorem for nonlinear semigroups. Proof of regularity takes advantage of elliptic theory, as well as the regularity of the Airy’s stress function.

The maximal smoothing effect of dispersive equations

Toshihiko Hoshiro Himeji Institute of Technology

In this talk, we discuss some smoothing estimates of the initial value problem for dispersive equations. We consider the initial value problem of the form: idu/dt + P (D)u = f(t, x), and u(0, x) = u_{0}(x). We will show that, for the maximal smoothing in space-time, it is necessary and sufficient that the principal part of the symbol does not vanish on the unit sphere.

A diamagnetic inequality for semigroup differences

Dirk Hundertmark California Institute of Technology

Recently several groups (Nakamura, Doi et al, and Hupfer et al) showed that the integrated density of states for random Schrödinger operators with magnetic fields is independent of the boundary conditions used in its construction.

This result is well-known without magnetic fields. We will give a short proof of it in the magnetic case by reducing it to the non-magnetic case via a diamagnetic inequality for the difference of the Neumann and Dirichlet semigroups. (Joint work with Barry Simon.)

Birkhoff normal forms for Fourier integral operators

Alexei Iantchenko Malmö University

We consider the problem of constructing a microlocal logarithm and a normal form for an elliptic semi-classical Fourier integral operator near a fixed point of the corresponding canonical transformation. In [Ia] the canonical transformation was assumed to be of real hyperbolic type. In [IS] this assumption was relaxed considerably, to what we think are the natural conditions.

This is a joint work with J. Sjöstrand.

References.

[Ia] A. Iantchenko, La forme normale de Birkhoff pour un opérateur intégral de Fourier, Asymptotic Analysis, 17(1)(1998), 71–92.

[IS] A. Iantchenko and J. Sjöstrand, Birkhoff normal forms for Fourier integral operators II, preprint, 2001.

http://xxx.lanl.gov/abs/math.SP/0111134

Method of Supplementary Argument

Taalaibek Imanaliev Kyrgyz State National University

Implementation of the method of supplementary argument for the equations of the first order was considered in [1]. In [2] it was shown the implementing of this method for the equation of third order. In this article the method of supplementary argument is implemented for the systems of quasilinear equations. By the method of supplementary argument the problem is reduced to integral equation, in which auxiliary functions are defined from the system of equations without right parts with matched initial conditions. The systems of equations of

(2)  u_{t} (t, x) + g(t, x, u (t, x))u_{x} (t, x) = f(t, x, u (t, x)),

type with the initial condition

(3)  u(0, x) = φ(x), (t, x) ∈ R_{+} × R

are considered.

Theorem. Let vector-function φ (x) ∈ \overline{C} (R^{n}) ∩ Lip (L_{1}),

f(t, x, u) ∈ C^{(0,1,1)}(R_{+} × R × R) ∩ Lip(L(t) |_{x} , L(t) |_{u}),

g(t, x, u) ∈ C^{(0.1.1)}(R_{+} × R × R) ∩ Lip(L(t) |_{x} , L(t) |_{u}),

∫_{0}^{∞}L(s)ds(2 + L_{1}) ≤ α < 1.

Then the problem (1) - (2) has the unique solution u(t, x) ∈ C^{(1,1)}(R_{+} × R), which is presented as

(4)  u(t, x) = φ(x) − ∫_{0}^{t} g(t, p(s, t, x), v(s, t, x))ds + ∫_{0}^{t} f(s, p(s, t, x), v(s, t, x))ds,

where the vector-function ν(τ, t, x) and scalar function p(τ, t, x) is determined from the system

v_{t}(τ, t, x) + g(t, x, v(t, t, x))v_{x}(τ, t, x) = 0, v(t, t, x) = u(t, x),
p_{t}(τ, t, x) + g(t, x, v(t, t, x))p_{x}(τ, t, x) = 0, p(t, t, x) = x,

and it is the solution of the problem (1)-(2) with the same properties.

References

[1] Imanaliev, Murzabek, Pankov, Paul, Imanaliev, Taalaibek, Method of Supplementary Argument in Theory of Nonlinear Wavy Equations in Partial Derivatives, Doklady Academy Nauk, 1995, Vol.343, 5, pp. 596-598

[2] Imanaliev, Murzabek, Pankov, Paul, Imanaliev, Taalaibek, On the Theory of Differential Equations In Partial Derivatives of Korteweg-De Vries Type, Doklady Academiyi Nauk, 1995, Vol. 342, 1, pp. 17-19

Notes on the spontaneous decay

Ketill Ingolfsson Cmty College of Philadelphia

Suppose that two strongly continuous one-parameter semi-groups of contraction operators are given in the separable Hilbert space and in a specific subspace respectively such that the latter determines a correct exponential decay for all positive t. We develop an asymptotic series under strictly singular perturbation of the infinitesimal generator in the general case. A possible deviation from the exponential decay is then well defined through the remainder-term of the series-expansion. The numerical evaluation of the series will yield criteria by which one can determine if the deviations from the exponential decay can be confirmed by observations.

Stiffness Identification in Biological Tissue

Lin Ji Rensselaer Polytechnic Institute

By cross-correlating successive ultrasonic scans of the biological tissue that is excited by a transient source on the surface, recent experiments show that the wave displacement during the interval between the two scans can be measured everywhere in the tissue. Assumming we know the displacement history of the shear wave everywhere, we identify the stiffness change inside the tissue in terms of the Lamé constant, μ. Our method starts with the asymptotic expansion of geometrical optics. By tracing the amplitude change of the displacement along the gemoetrical rays as they travel into the medium, we are able to recover μ without directly taking derivatives of the displacement data.

Nonperturbative localization.

svetlana jitomirskaya UC Irvine

Study of fine spectral properties of quasiperiodic and similar discrete Schrodinger operators involves dealing with problems caused by small denominators, and until recently was only possible using perturbative methods, requiring certain small parameters and complicated KAM-type schemes. I will first review the nonperturbative methods for such study which lead to stronger results and are significantly simpler. Next, I will discuss recent work (joint with J. Bourgain) on nonperturbative localization for the quantum kicked rotor model.

Equations in involution and intermediate integrals for a scalar second-order hyperbolic PDE in the plane

Martin Juras North Dakota State University

We find necessary and sufficient conditions for a scalar second-order hyperbolic equation in the plane

(1)  F (x, y, u, u_{x}, u_{y}, u_{xx}, u_{xy}, u_{yy}) = 0

to admit an equation in involution of a given order. We also show that there are essentially no first integrals of a total vector field T, provided T is not one of the infinitely prolonged Monge characteristic vectors. We reformulate the classical notions of the general and complete intermediate integrals for a hyperbolic equation (1) in the terms of modern terminology and find characterizations of equations that admit intermediate integrals in terms of the generalized Laplace invariants.

Anti-eigenvalues and a conjecture of McKenna and Walter

Bernhard Kawohl Univ. of Cologne, Cologne, Germany

McKenna and Walter remarked in 1987 that the following linear problem needs more study:

(2)  \{ \begin{matrix}Δ^{2}v + b v & = & f & in Ω, \\ v = Δv & = & 0 & on ∂Ω, \end{matrix}

where Ω is a bounded domain in ℝ^{n} and f a given function:

… we should be able to estimate the value b_{c}(Ω) with the property that for 0 ≤ b < b_{c}(Ω) the inverse operator is positivity preserving (…). We conjecture that this constant b_{c}(Ω) is largest among all regions Ω with given volume when Ω is a ball.

In the lecture I present results on this conjecture and on related questions that were obtained jointly with G.Sweers from Delft. The parameter b can be interpreted as a spring constant, and u as a deformation of a hinged plate in a bed of springs under a load f.

Trace Formulae and Tri-diagonal Matrices

Rowan Killip University of Pennsylvania

Some applications of trace formulae to the spectral analysis of tri-diagonal matrices will be described. In particular, the speaker’s recent joint work with Barry Simon (math-ph/0112008). A major result of this work was the determination of the set of spectral measures that arise from Hilbert–Schmidt perturbations of the free Jacobi matrix. That is, the Jacobi matrix with ones on the sub/super-diagonals and zeros elsewhere.

On a spectral notion of integrability

oliver knill harvard university

We discuss a spectral notion of integrability for dynamical systems. The definition applies to a large class of systems including systems with higher-dimensional time. We will look at the relation with other notions of integrability on classes of finite or infinite dimensional Hamiltonian systems.

Asymmetric Fermi Surfaces for Magnetic Schrödinger Operators

Horst Knorrer ETH Zürich/Institute for Advanced Study

Consider Schrödinger operators with peridic magnetic field having zero flux through a fundamental cell of the period lattice. We show that, for a generic small magnetic field and a generic Fermi energy, the corresponding Fermi surface is convex and not invariant under inversion in any point

Multiplicity of Unstable Nontrivial Solutions in Semilinear Elliptic Indefinite Weight Boundary Value Problems

Bongsoo Ko Cheju National University

We discuss the existence and multiplicity of unstable nontrivial solutions about the following problems: −Δu = λg(x)f(u) in Ω, (1 − α) \frac{∂u}{∂n} + αu = 0 on ∂Ω. Here, Ω is a bounded open region in R^{n}, with the smooth boundary ∂Ω.

On attractors of nonlinear Hamiltonian equations

Alexander Komech Moscow State University

For generic nonlinear Hamiltonian equation, the attractor is the set of all stationary states. For generic equations with a symmetry group, the attractor is the set of all solitary waves. The problem and the results are inspired by the N.Bohr postulate on transitions to quantum stationary states and Schroedinger’s identification of the quantum stationary states to the solitary waves corresponding to the gauge symmetry group.

Perturbation of Spectral Projections and the Riccati Equation

Vadim Kostrykin Institut fuer Lasertechnik

Let A and V be bounded self-adjoint operators on a separable Hilbert space. Let P = E_{A}((−∞, λ]) and Q = E_{A+V} ((−∞, μ]), λ, μ ∈ ℝ be the spectral projections of A and A + V, respectively. Under the assumption that

sup spec(P (A + V)|_{RanP}) ≤ μ ≤ inf spec(P^{⊥}(A + V)|_{RanP^{⊥}})

we give a complete description of the relative geometry of the subspaces RanP and RanQ. We establish a necessary and sufficient condition on A and V for the pair (P, Q) to be Fredholm. We also obtain new existence and uniqueness results for the operator Riccati equation associated with the perturbation problem.

This talk is based on results of a joint work with K.A. Makarov and A.K. Motovilov.

nonlinear analogues of e^{i(kx−ωt)} for integrable systems

Mikhail Kovlayov university of alberta

The similarity between the inverse scattering method for solving nonlinear integrable systems and the Fourier transform method for solving linear PDEs has been known for a while. But do there exist nonlinear anlogues of e^{i(kx−ωt)}, the Fourier representation formula, amplitude modulation and other things naturally appearing as applications of the Fourier transform? The answer will be discussed in the talk.

On a Complete Scale of Isomorphisms for Elliptic and Parabolic Pseudodifferential Boundary-Value Problems

Alexander Kozhevnikov Dept. of Math., Univ. of Haifa, Israel

In monographs by J.-L.Lions and E.Magenes (1968) and by Ya.Roitberg (1996) theorems on complete scales of isomorphisms have been established which, roughly speaking, mean that operators generated by elliptic differential boundary-value problems are isomorghisms (Fredholm operators) between Sobolev-type spaces of functions with s and s-d derivatives, where d is the order of the the elliptic operator. The completeness of the scale means that s can be an arbitrary real number.

In a monogragh by S.Eidelman and N.Zhitarashu (1990) a theorem on a complete scale of isomorphisms has been obtained for parabolic differential initial boundary-value problems.

Due to the fact, that for elliptic and parabolic pseudodifferential initial boundary-value problems there exist parametrices belonging to the Boutet de Monvel algebra, a much shorter proof has been found as well as some applications of the results to the spectral theory.

An approach to solving boundary-value problems of elasticity in coordinate systems with incomplete separation of variables

Pavlo Krokhmal University of Florida

We present an approach to constructing exact analytical solutions to the displacement boundary-problem of elasticity for bodies that are described by surfaces in cyclidal coordinate systems with incomplete separation of variables.

We consider the displacement boundary-value problem of elasticity formulated in a simple sub-family of cyclidal coordinates that includes bi-spherical and toroidal coordinate systems.

One of the key points in our approach is the developed form of general solution for the elastostatics (Lamé) equation, where a solution is represented as a linear combination of a scalar and vector harmonic functions. These functions have to satisfy not only boundary conditions, but also additional differential constraint. It is shown that this general solution reduces the boundary-value problem to functional or algebraic equations of the simplest form.

As an example, exact analytical solution of the displacement boundary-value problem for a toroidal elastic body is presented. According to the approach, the original boundary-value problem is reduced to a series of infinite linear algebraic equations with three-diagonal matrices. An analytical technique for solving systems of diagonal form is developed.

General aspects of uniqueness of solution of generalized Cauchy-Riemann equations in double-connected spatial toroidal regions are considered, and the uniqueness of the obtained solution is proved.

Spectra of Neumann Laplacians in thin graph like domains

Peter Kuchment Texas A&M University, College Station

We discuss convergence of spectra of Neumann Laplace operators on thin domains to spectra of appropriate differential operators on graphs. The results are obtained jointly with H. Zeng.

Inverse scattering problem for quantum graphs

Pavel Kurasov Dept. of Mathematics, Lund Institute of Technology

The inverse scattering problem for the Schrödinger operator on a graph having several infinite branches is studied. Such Schrödinger operator is determined by potentials on the branches and certain boundary conditions at the vertices. It is shown that the knowledge of the scattering matrix in general is not enough to reconstruct the potential, the graph, the boundary conditions or even the topological structure of the graph. We discuss possible generalization to the case of quantum graphs of the Borg-Marchenko theorem concerning the uniqueness of the one-dimensional Schrödinger operator.

The Efimov effect of three-particle Schrödinger operators on a lattice. Asymptotics for the number of eigenvalues

Saidakhmat Lakaev Samarkand State University

The Efimov effect is one of the most remarkable results in the spectral analysis for three-particle Schrödinger operators on Euclidean space. Roughly speaking,the effect will be explained as follows:

If none of the three two-particle subsystem has bound states with negative energy, but at least two of them have a resonance state with zero energy, then the three-particle system has an infinite number of three-particle bound states with negative energy, accumulating at zero.

This effect first discovered by Efimov in [1]. The mathematically rigorous proof of the result has been presented(see for instance [2-3]).

We consider a system of three-particle,which move in three-dimensional lattice Z^{3} and interact through a pair short-range negative potentials. The Hamiltonian of this system has form

H = △_{i} − ∑_{i<j} V_{ij},

where △_{i} is the lattice Laplacian and V_{ij} in the coordinate representation is the multiplication operator by function V_{ij}(r_{i} − r_{j}), V_{ij} ≥ 0. We reduce the investigation of the spectral properties of the operator H to the analysis of the family of self-adjoint,bounded operators (the three-particle discrete Schrödinger operators ) {H(K), K ∈ T^{3}}, acting in the Hilbert space of square-integrable functions defined on the torus, applying the Fourier transform and the decomposition into the direct operator integrals.

We assume that for the two-particle discrete Schrödinger operators h_{α}(k), k ∈ T^{3}, α = 1, 2, 3 some natural assumptions are fulfilled and establish the following results:

Let N_{0}(z) be the number of eigenvalues of the three-particle Schrödinger operator H(0) lying on the left from point z < 0.

(i)The N_{0}(z) obeys asymptotics

lim_{z→0} \frac{N_{0}(z)}{| log |z||} = C_{0} > 0,

where the C_{0} > 0 depend only on the masses of the particles.

Let U_{0} ⊂ T^{3} be sufficiently small punctured neighborhood of the point 0 ∈ T^{3}.

(ii) For any K ∈ U_{0} the operator H(K) has only finite number eigenvalues lying on the left-hand side of the essential spectrum.

We also shall point to some important spectral properties of the operators H(K), K ∈ T^{3},which are analogously or different for both of the Schrödinger operators on Euclidean space and lattice.

[1]V. Efimov: Nucl.Phys. A.210 (1973),157-158. [2]D. R. Yafaev: Math.USSR-Sb. 23 (1974),535-559. [3] A. V. Sobolev: Comm.Math. Phys. 156 (1993), 127–168.

Negative spectrum of Schrödinger operators

Ari Laptev Royal Institute of Technology in Stockholm

We shall discuss different types of Lieb-Thirring and CLR inequalities for the negative spectrum of Schrödinger operators.

Spectrum of the linearized Euler operator

Yuri Latushkin University of Missouri-Columbia

This talk is scheduled at the Focus Session on fluid mechanics. We study spectral properties of the linearized Euler operator. For the generalized Kolmogorov flow we give the full description of the spectrum. Also, we prove spactral mapping propert that relates the spectrum of the linearized Euler operator and the spectrum of the group generated by this operator.

Complex Time Evolution of Simple Quantum Systems

Joel Lebowitz Rutgers

I will describe some exact results about the long time behavior of a simple quantum system, having both bound and continuum states, when subjected to a periodic external force. This turns out to be surprisingly complex.

Joint work with O. Costin, R. Costin and A. Rokhlenko.

Large deviations for ideal quantum systems

Marco Lenci Stevens Inst of Tech

We consider a general d-dimensional quantum system of non-interacting particles, with suitable statistics, in a very large (formally infinite) container. We prove that, in equilibrium, the fluctuations in the density of particles in a subdomain of the container are described by a large deviation function related to the pressure of the system. That is, untypical densities occur with a probability exponentially small in the volume of the subdomain, with the coefficient in the exponent given by the appropriate thermodynamic potential. Furthermore, small fluctuations satisfy the central limit theorem.

Cantorspectrum for one-dimensional quasicrystals

Daniel Lenz TU Chemnitz

The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue measure for a large class of aperiodic subshifts including all aperiodic subshifts arising from primitive substitutions.

Stability of a Model of Relativistic Quantum Electrodynamics

Elliott H. Lieb Princeton University

The relativistic “no pair” model of quantum electrodynamics uses the Dirac operator, D(A) for the electron dynamics together with the usual self-energy of the quantized ultraviolet cutoff electromagnetic field A — in the Coulomb gauge. There are no positrons because the electron wave functions are constrained to lie in the positive spectral subspace of some Dirac operator, D, but the model is defined for any number, N, of electrons, and hence describes a true many-body system. In addition to the electrons there are a number, K, of fixed nuclei with charges ≤ Z. If the fields are not quantized but are classical, it was shown earlier (with Siedentop and Solovej) that such a model is always unstable (the ground state energy E = −∞) if one uses the customary D(0) to define the electron space, but is stable (E > −const.(N + K)) if one uses D(A) itself (provided the fine structure constant α and Z are not too large). This result is extended to quantized ultraviolet cutoff electromagnetic fields here, and stability is proved for α = 1/137 and Z ≤ 42. This formulation of QED (using the spectrum of D(A) instead of D(0) to define an electron) is somewhat unusual because it means that the electron Hilbert space is inextricably linked to the photon Fock space. But such a linkage appears to better describe the real world of photons and electrons. (joint work with Michael Loss)

Concentration and Cavitation of Density in Invicid Compressible Fluid Flow

Hailiang Liu UCLA

I discuss recent work with Guiqiang Chen in which we identify and analyze the phenomena of concentration and cavitation in the vanishing pressure limit in inviscid compressible fluid flow. It is shown that any two-shock Riemann solution of the Euler equations for isentropic fluids tends to a δ-shock solution of the Euler equations for pressureless fluids and the intermediate density between the two shocks tends to a δ-mass that forms the δ-shock; by contrast, any two-rarefaction-wave Riemann solution for isentropic fluids tends to a two-contact-discontinuity solution for pressureless fluids and the intermediate state between the two rarefaction waves, even away from the vacuum, tends to a vacuum state. Some numerical results exhibiting the formation process of δ-shocks and vacuum states are presented.

Boltzmann equation and conservation laws in gas dynamics

Tai-Ping Liu Stanford University and Academia Sinica (Taiwan)

There has been important progresses in recent years in the theory of shock waves for conservation laws. We will discuss the recent efforts on the Navier-Stokes equations in gas dynamics and Boltzmann equation in kinetic theory. The relationship between these equations has been much studied in two directions: One is for the incompressible limits and the other is for compressible shockless flows. We will view the Boltzmann equation via the compressible Euler and Navier-Stokes equations. This viewpoint yields new setup and analytical possibilities, which we will illustrate with the stability problem for shock and diffusion waves for the Boltzmann equation.

A Bound on Binding Energies and Mass Renormalization in Models of Quantum Electrodynamics

Michael Loss Gergia Tech

We study three well known models of matter coupled to the ultraviolet cutoff, quantized radiation field and to the Coulomb potential of arbitrarily many nuclei. Two are nonrelativistic: the first uses the kinetic energy (p + eA(x))^{2} and the second uses the Pauli-Fierz energy (p + eA(x))^{2} + eσ · B(x). The third, no-pair model, is relativistic and replaces the kinetic energy with the Dirac operator D(A), but restricted to its positive spectral subspace, which is the “electron subspace”. In each case we are able to give an upper bound to the binding energy – as distinct from the less difficult ground state energy. This implies, for the first time we believe, an estimate, albeit a crude one, of the mass renormalization in these theories.

On a Subspace Perturbation Problem

Konstantin Makarov University of Missouri-Columbia

Let A and V be bounded self-adjoint operators on a separable Hilbert space. Assume that the spectrum of A consists of two disjoint parts Δ and Σ such that dist(Δ, Σ) > 0. Denote by {P (s)} the real analytic family of spectral projections of A + sV uniquely determined by the requirement that P (0) is the spectral projection of A associated with the set Δ. Under various assumptions on the perturbation V and mutual disposition of the sets Δ and Σ we obtain (optimal) sufficient conditions on the magnitude of the coupling constant s ∈ ℝ that ensure the strict inequality ∥P (s) − P (0)∥ < 1.

This is a joint work with V. Kostrykin and A. K. Motovilov.

On boundary value problems for one-dimensional Schrödinger operator

Alexander Makin Moscow State Academy of Instrument-Making and Informatics

We will consider nonselfadjoint boundary value problems for one-dimensional Schrodinger operator u′′ − q(x)u defined on the segment [0, 1]. We establish sufficient conditions of summability of spectral expansions which hold for many types of boundary conditions, for instance, Birkhoff-regular but not strongly regular, irregular, multipoint boundary conditions etc. A lot of interesting examples will be considered.

Invertibility of the functional operators with shift having singularity

Rasul Mardiev Department of Mathematics. Samarkand State University , University Boulevard 15,

The functional operators with shift in cases, when the shift is a diffeomorphism are well studed. If the shift is ergodic and has one point of singularity, the invariant measure is singular with respect to Lebesgue measure.Thearefore in this case situation very difference from a case of diffeomorphisms. By us are obtained a condition of invertibility of the functional operators with shift, when the shift has Break-type singularity point.

The website functions.wolfram.com

Oleg Marichev Wolfran Research Inc.

Currently the website www.functions.wolfram.com includes more than 37000 formulas for about 250 elementary and special functions. The structure and organization of the website will be discussed in detail. An overview is given how to manage, maintain, develop, and update this large mathematical knowledge base and how to deliver various formats types (Mathematica notebooks, html, MathML, and pdf). Future development directions will be outlined. New formulas like new connection between Sin[z] and Sqrt[1-Cos[z]ˆ 2] will be demonstrated.

The Navier-Stokes equation in distribution spaces

Anna L. Mazzucato Yale University and IMA

We discuss the Navier-Stokes equation in distribution spaces. We investigate existence and uniqueness by means of fixed-point methods and Littlewood-Paley theory. In particular, we consider the scale of Besov-Morrey spaces.

Torsional Waves in Functionally Graded Isotropic Elastic Rods

C. Maeve McCarthy Murray State University

The purpose of this work is to investigate the influence of material inhomogeneity on the propagation of torsional waves in linearly elastic isotropic rods. The special case of a circular rod with shear modulus depending on the radial coordinate only is examined and the influence of various cases of inhomogeneity on the fundamental frequencies of the rod is assessed. Although the associated Sturm-Liouville problem is singular, the spectrum can be shown to be discrete. Upper bounds on the frequencies are obtained by a Rayleigh-Ritz method. Lower bounds are obtained by an integral equation technique.

Analysis of a van der Pol Type Differential Equation Having a Nonpolynomial Elastic Term

Ronald Mickens Clark Atlanta University

The standard van der Pol differential equation provides a model for many important phenomena in the natural and engineering sciences. For this equation the force function is a linear function of the dependent variable; consequently, in the absence of nonlinear terms, the ODE reduces to a damped harmonic oscillator. We consider a modification for which the force function contains terms that are odd, but nonpolynomial in the dependent variable. We use a variety of techniques (energy arguments, phase-space methods, and perturbation procedures) to study the asymptotic behavior of solutions to these van der Pol type ODE’s.

The work reported here was supported in part by research grants from DOE and the MBRS-SCORE Program at Clark Atlanta University.

Factorization Problem for Operator-Valued Herglotz Functions: Geometric Approach

Alexander K. Motovilov University of Missouri-Columbia and JINR, Dubna, Russia

The spectral subspace perturbation problem for linear self-adjoint operators can be viewed as the factorization problem for a class of operator-valued Herglotz functions. Splitting off a linear factor from the Herglotz function in question ensures the representation of a perturbed spectral subspace as the graph of a linear operator which, in turn, solves the related operator Riccati equation. In the present work we obtain a series of new factorization results for operator-valued Herglotz functions based, in particular, on geometric consideration. Some of our factorization theorems are optimal.

This is a joint work with V. Kostrykin and K.A. Makarov.

Qualitative Analysis Of Two Dimensional System Differential Equations Of Special Form

Yahyo Muhtorov Samarkand state University

The whole system of two differential equations, right parts which consists of equation of m grade that corresponds to the equation of the searching function with gradual coefficients into which changeable enter to m grade (m.n. the natural numbers) We got enough conditions about absence of limit cycles. In dependence on legibility searching allocation of the special points of the system.

Sequences of Classical Hard-Core Collisions

Thomas Murphy University of Maryland, College Park

Knowledge of the collective dynamics of small groups of particles is needed in the calculation of the density dependence of the transport properties of dilute gases. For the particularly simple case of hard - core repulsion, this requires knowledge of the possible sequences of collisions among small groups of such particles in infinite space. New results for the one - dimensional case are presented in the context of the current state of knowledge for the one - dimensional case as well as for the case of more than one dimension.

From Schrödinger dynamics to the Euler equations

Bruno Nachtergaele University of California, Davis

We show that, in the hydrodynamic limit, the Heisenberg dynamics of the energy, momentum, and particle densities for fermions with short-range pair interactions converges to the compressible Euler equations with the pressure function given by quantum statistical mechanics. Our derivation is based on a quantum version of the entropy method and a suitable quantum virial theorem. We require a number of technical conditions that will be discussed in the talk.

Darboux-Lamé equation and isomonodoromic deformation on the torus

Mayumi Ohmiya Doshisha Universuty

For the n-th Lamé equation

\frac{d^{2}f}{dx^{2}} = (n(n + 1)℘(x, τ) − λ)f,

where ℘(x, τ) is Weierstrass elliptic function with the primitive periods 1 and τ, ℑτ > 0, and n is a natural number, one can construct the double Darboux transformation via some specified eigenfunction, which is the 1-parameter family of ordinary differential equations with the algebro geometric elliptic potentials. It is shown that such 1parameter family of ordinary differential equations is the isomonodromic family of Fuchsian equations on the torus ℂ/L, where L = ℤ ⊕ ℤτ is the lattice.

The Asymptotic Formula For The Eigenvalues Of The Second Order Partial Differential Equation With Operator Coefficient

Kevser Ozden Koklu Yildiz Technical University

In this study, it is shown that the spectrum is pure discrete and obtained the asymptotic expression for number eigenvalues of the operator generated with the differential expression

− ∑_{i=1}^{3} \frac{∂}{∂x_{i}} (p(x) \frac{∂u}{∂x_{i}} ) + Q(x)u

and the boundary condition

u|_{x_{3}=0} = 0

in E_{3}^{+} = {x = (x_{1}, x_{2}, x_{3}) : x_{3} ≥ 0}, a semi space of space E_{3}. Here p(x) is a continuously differentiable positive scalar function, Q(x) is a self-adjoint positive operator defined in separable Hilbert space for every x ∈ E_{3}^{+} whose inverse is completely continuous.

Joint work with Mehmet Bayramoglu and Oya Baykal.

An alternative proof of Pascu’s theorem on hot spots

Michael Pang University of Missouri-Columbia

Let Ω be a bounded C^{1,α} convex domain on the plane that has one axis of symmetry. Let ϕ be an eigenfunction corresponding to the smallest non-zero eigenvalue of the Neumann Laplacian on Ω. Suppose that ϕ is antisymmtric with respect to the axis of symmetry. M. Pascu recently proved that ϕ attains its maximum at the boundary of Ω. We give an alternative proof of Pascu theorem using elementary properties of reflected Brownian motions on planar domains.

Recent Developments in the Theory of the Periodic Euler-Bernoulli Equation

Vassilis Papanicolaou National Technical University of Athens (NTUA)

We develop the theory of the periodic Euler-Bernoulli equation following the spirit of the theory of Hill’s equation. We establish the equality of algebraic and geometric multiplicities at the endpoints of the gaps and pseudogaps. We also propose an Euler-Bernoulli analog of the Hill’s Dirichlet spectrum by considering a multipoint boundary value problem.

Generalized Laguerre Polynomial Solutions for Central Potentials in Quantum Mechanics

Rajendrasinh Himatsinh Parmar Sir P T Science College Modasa, Gujarat-383 315 India

With the help of associated Laguerre polynomial with arbitrary power of its argument, general central potential in the radial Schrödinger equation has been studied. With the appropriate power of its argument the Schrödinger equation with Coulomb, Harmonic and other potential forms and their eigen values have bee obtained. We consider here the general solution of the radial Schrödinger equation as,

u(ϱ) = ϱ^{l+1}e^{−ϱ^{p}/2}L_{n}^{k}(ϱ^{p})

For different choices of p, we obtain the potential forms as well as the eigen value expressions satisfying

− \frac{ℏ^{2}}{2m} \frac{d^{2}u(ϱ)}{dϱ^{2}} + ( \frac{ℏ^{2}}{2m} \frac{l(l + 1)}{ϱ^{2}} + V (ϱ) − E) u(ϱ) = 0.

Joint work with P C Vinodkumar.

Finite time blow-up for a dyadic model of the Euler equations

Natasa Pavlovic University of Illinois at Chicago

This talk will be about joint work with Nets Katz on dyadic models for the equations of fluid motion. We introduce a dyadic model for the Euler and the Navier-Stokes equations. The dyadic model possesses an important feature of the equations, which is conservation (or decay) of energy. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in case when the dissipation degree is sufficiently small.

Spectral Theory of Open Quantum Systems

Claude-Alain Pillet Univ. de Toulon, CPT-CNRS/FRUMAM

The dissipative dynamics of a small quantum system (a N-level atom) interacting with its environment (a radiation field) is usually described by a Markovian semi-group, or equivalently by a quantum master equation. However, this reduced description of the system is only justified in the small coupling regime and on a large time scale: In the so-called van Hove limit. Recently, some progresses have been made in the study of open quantum systems beyond the Markovian approximation. In particular, the problem of return to equilibrium of a system coupled to an environment which is at thermal equilibrium is now fairly well understood (see [JP1], [BFS], [DJ],[M]). The thermal relaxation process is controlled by the complex resonances of a self-adjoint operator: The Liouvillean. In a recent work with V. Jaksic [JP2], we have extended this spectral approach to the study of the long time asymptotics of quantum systems out of equilibrium. Our method allows us to construct natural, non-equilibrium steady states of such systems, and to study their basic properties. Again, the relaxation process is related to the complex resonances of some (non-selfadjoint) operator. I will review some of these results and provide specific examples.

References:

[BFS] V. Bach, J. Froehlich, I.M. Sigal: J. Math. Phys. 41, 3985 (2000). [DJ] J. Derezinski, V. Jaksic: J. Func. Anal. 180, 243 (2001). [JP1] V. Jaksic, C.-A. Pillet: Commun. Math. Phys. 178, 627 (1996). [JP2] V. Jaksic, C.-A. Pillet: To appear in Commun. Math. Phys. [M] M. Merkli: Commun. Math. Phys. 223, 327 (2001).

Minimax approach to solving some inverse problems for Maxwell equations

Yuri Podlipenko Kiev University, Faculty of Cybernetics, Kiev, Ukraine

Problems of optimal reconstruction of solutions and right-hand sides of Maxwell equations under incomplete data are investigated. These problems play an important role in electromagnetics. Depending on a character of an apriori information, stochastic or deterministic approach are possible. The choice is determined by nature of the parameters in the problem, which can be random or not. Moreover the optimality of estimations depends on a criterion with respect to which a given value is evaluated.

We assume that right-hand sides of Maxwell equations are unknown and belong to the given bounded subsets of the space of all square integrable functions in the considered domain and for solving the estimation problems we must have supplementary data (observations) depending on solutions of these equations. We suppose that observation errors are realizations of the stochastic processes, with unknown moment functions of the second order also belonging to certain given subsets.

Our approach is as follows. We are looking for linear with respect to observations optimal estimates of solutions and right-hand sides of Maxwell equations from the condition of minimum of maximal mean square error of estimation taken over the above subsets. We consider constructive methods for obtaining such estimates, which is expressed in terms of solutions of special integro- differential equations.

It should be noted that beyond purely theoretical interest, the mentioned problems can have interesting applications, for automatized measurement data processing systems, for interpretation of electromagnetic observations.

Asymptotics of bound states, bands and resonances for laterally coupled quantum waveguides and layers

Igor Popov Leningrad Institute of Fine Mechanics and Optics, Leningrad Institute of Fine Mechanics and Optics

Spectral problem for the Dirichlet Laplacian in two-dimensional strips (three-dimensional layers) of widths d_{+}, d_{−}, d_{+} > d_{−} coupled through small openings aω_{i}, i = 1, ...n, (a is small parameter) is considered. Method of matching of asymptotic expansions of solutions of boundary value problem is used. The asymptotics (in a) of a bound state λ_{a} close to the threshold π^{2}/d_{+}^{2} is obtained:

λ_{a} = \{ \begin{matrix}\frac{π^{2}}{d_{+}^{2}} − ( \frac{π^{3}}{d_{+}^{3}} ∑_{i=1}^{n} c_{ω_{i}})^{2}a^{4} + o(a^{4}), & R^{2}, \\ \frac{π^{2}}{d_{+}^{2}} − ( \frac{1}{d_{+}^{2}} exp (− \frac{2d_{+}^{3}}{3π^{2}} (∑_{i=1}^{n} b_{ω_{i}})^{−1}a^{−3}(1 + o(1))), & R^{3}. \end{matrix}

Here c_{ω_{i}} is a capacity of ω_{i} in R^{2}, b_{ω_{i}} is an average virtual mass of ω_{i} in R^{3}. The case of two identical waveguides (layers) is considered too.

The asymptotics of resonances close to N−th threshold is determined for 2D case.. 2D waveguides in a magnetic field and curved waveguides are considered.

Asymptotics of bands for the case of periodic system of coupling windows (period L) for two waveguides is constructed:

[ \frac{π^{2}}{d_{+}^{2}} − \frac{3π^{3}}{2Ld_{+}^{3}} a^{2} + o(a^{2}), \frac{π^{2}}{d_{+}^{2}} − \frac{π^{3}}{2Ld_{+}^{3}} a^{2} + o(a^{2})], R^{2}.

[λ_{1}^{+} − \frac{18π(ψ_{1}^{+0})^{2}b_{ω}}{L} a^{3} + o(a^{3}), λ_{1}^{+} − \frac{6π(ψ_{1}^{+0})^{2}b_{ω}}{L} a^{3} + o(a^{3})], , R^{3}.

Here λ_{1}^{+} is the first transversal eigenvalue, ψ_{1}^{+0} is the value of the normal derivative of the corresponding transversal eigenfunction at the centre of the opening. There is a gap for sufficiently small a. For the case of layers coupled through singly (Λ_{1}) and doubly (Λ_{2}) periodic system of windows there is no gap, and the asymptotics of the lower bound for the continuous spectrum is:

λ_{min} = \frac{π^{2}}{d_{+}^{2}} − \frac{π^{6}b_{ω}^{2}}{Ld_{+}^{6}} a^{6} + o(a^{6}), for Λ_{1},

λ_{min} = \frac{π^{2}}{d_{+}^{2}} − \frac{πb_{ω}|Λ̂|}{d_{+}^{3}} a^{3} + o(a^{3}), for Λ_{2},

where |Λ̂| is a square of the Brillouin zone for Λ_{2}.

Inverse Problems for Elastic Media

Lizabeth Rachele University at Albany, SUNY

We consider dynamic inverse problems for bounded, three- dimensional isotropic elastic and hyperelastic objects with smoothly varying density and elastic properties. Surface data for the inverse problems is modelled by the hyperbolic Dirichlet- to-Neumann map on a finite time interval.

In the case of isotropic elastodynamics we have shown that the density and elasticity parameters are all determined uniquely by the Dirichlet-to-Neumann boundary data.

We also consider uniqueness for hyperelastic media (e.g. elastodynamics with residual stress).

On the localization of the spectral expansions of distributions

Abdumalik Rakhimov National University of Uzbekistan, Tashkent

In the present work it is investigated the problems of the localization of multiple trigonometric Fourier series of the periodic distributions. It is established necessary and sufficient conditions for the localization of the Riesz means of multiple trigonometric Fourier series of distributions from negative periodic Sobolev spaces.

We prove a theorem that characterizes sufficient conditions for summability of the spectral decompositions in a certain class of operators. Then we prove a theorem which showed that conditions found in the previous theorem is also necessary in that class of operators. Though we also prove that for each operator from that class we can weaken obtained condition.

Stability of solving systems Differential Equations of special form in R2

Nasiba Raupova Samarkand State University

Scrutinized common-line system of differential equations with definition At work we are investigating the stability of solving dimensional system right parts that are common-line functions.There is a function of Lyapunova and exposed enough conditions for the given function.

Inverse spectral theory for one-dimensional Schrödinger operators

Christian Remling Universität Osnabrück

We discuss new methods for solving the inverse spectral problem for one-dimensional Schrödinger operators.

Fluctuations of entropy production in anharmonic chains

Luc Rey-Bellet University of Virginia

We discuss the Gallavotti-Cohen fluctuation theorem for a chain of anharmonic oscillators coupled to heat reservoirs.

Consequences of the Connection Formulae for Sturm-Liouville Spectral Functions

Suzanne Riehl University of Northern Iowa

For a special case of the Sturm-Liouville equation −(py′)′+qy = λwy on [0, ∞) with the initial condition y(0) cos α+ p(0)y′(0) sin α = 0, α ∈ [0, π), it is shown that given the spectral derivative ρ′_{α}(μ) for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ_{0}, it is possible to uniquely determine ρ′_{β}(μ), β ∈ [0, π). An explicit formula is derived to accomplish this.

On a new transformation of the one-dimensional Schrödinger equation and higher order WKB-type asymptotics

Alexei Rybkin Univ. of Alaska - Fairbanks

We put forward a new transformation of the one dimensional Schrodinger equation with non-smooth potentials from L_{p} with p > 2. This transformation yields existence of the Weyl solution with higher order WKB-type asymptotic behavior valid in certain domains in the upper half plane asymptotically approaching the real line. We apply our approach to the study of high-energy asymptotics for the Titchmarsh-Weyl m-function improving on many relevant results of others.

Existence Of Solutions Of Differential Equations

Rathinasamy Sakthivel Department of Mathematics, Inha University, South Korea

A large class of scientific and engineering problems is modelled by partial differential equations, integral equations or coupled ordinary and partial differential equations which can be described as differential equations in infinite dimensional spaces using semigroups. In general, nonlinear differential equations, with and without delays, serve as an abstract formulation of many partial nonlinear differential equations which arise in problems connected with heatflow in materials with memory, viscoelasticity and many other physical phenomena. Problems in electrodynamics, reactor kinetics, behaviour of economic systems, weapon fire control from moving platforms are all modelled by nonlinear differential equations. So it becomes important to study the existence of solutions nonlinear differential equations in Banach spaces. All the results are established with the help of fixed point principles.

Nonlinear balance laws in low temperature heat propagation

katarzyna saxton loyola university, new orleans

In the talk we discuss the effects of nonlinearity and memory on the propagation of heat waves at low temperature, which are characterized by processes that are not explained by Fourier’s law. The system describing these thermal properties is of quasilinear, hyperbolic type with nonlinear damping. Large-time behavior, both of smooth solutions and for Riemann problems, has been extensively studied in the p-system with linear damping. In our case the physical situation neither takes the form of a p-system nor allows linear damping.

Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit

Karl Michael Schmidt Cardiff University, Wales

A perturbation decaying to 0 at infinity and not too irregular at 0 introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. This talk reports on a result showing that the distribution of eigenvalues in the gap is given by a quasi-semiclassical asymptotic formula in the slow-decay limit, which for perturbations of power decay is equivalent to the large-coupling limit.

Proof of Bose-Einstein Condensation for Dilute Trapped Gases

Robert Seiringer Princeton University

The ground state of bosonic atoms in a trap has been shown experimentally to display Bose-Einstein condensation (BEC). We prove this fact theoretically for bosons with two-body repulsive potentials in the dilute limit, starting from the basic Schrödinger equation; the condensation is 100that minimizes the Gross-Pitaevskii energy functional. This is the first rigorous proof of BEC in a physically realistic, continuum model.

Homogeneous C*-algebras

Mikhail Shchukin Belarusian State University

Every homogeneous C^{∗}-algebra corresponds to the algebraic fibre bundle. C^{∗}-algebra is called non-trivial if the corresponding algebraic fibre bundle is non-trivial. All C^{∗}-algebras generated by idempotents that studied before corresponded to the trivial algebraic fibre bundles. In the work was showed that every homogeneous separable noncommutative C^{∗}-algebra can be generated by idempotents. It is follows from here that we need to study the topology properties of C^{∗}-algebras generated by idempotents to describe such algebras properly. Also in the work we found the minimal number of idempotent generators for every homogeneous C^{∗}-algebra A with the set of maximal ideals homeomorphic to the sphere S^{2}.

The Energy of Electrons in Heavy Atoms According to Jansen and Hess

Heinz Siedentop Ludwig-Maximilians-Universität München

We show self-adjointness for a relativistic one-electron Hamiltonian used to describe electrons in heavy atoms. The model is a approximate no-pair Hamiltonian in the Furry picture. (Joint work with Edgardo Stockmeyer and Raymond Brummelhuis.)

Fractional Moment Methods for Finite Rank Perturbations

Robert Sims University of California at Irvine

We introduce new methods to prove fractional moment bounds on finite rank perturbations of the discrete Laplacian in ℓ^{2}(ℤ^{d}). Our techniques provide a simple new proof of localization for Anderson-type models in which the single site potential is supported on an arbitrary cube in ℤ^{d}.

Wave Propagation in Cracked Thermoelastic Solid Half-Space

Baljeet Singh Jat College, Rohtak, Haryana, India

Some recent progress on spectral asymptotics for non-self-adjoint operators

Johannes Sjoestrand Centre de Mathematiques, Ecole Polytechnique, F-91128 Palaiseau cedex, France

We start with a short review of recent results by many people about estimates and asymptotics for the distribution of eigenvalues for non-self-adjoint differential operators (with resonances as an important special case), and also about the notion of pseudo-spectrum.

Then we discuss the two-dimensional case (starting with joint work with A. Melin) where one often can obtain quite detailed spectral asymptotics by means of Bohr-Sommerfeld quantization conditions, very much as for self-adjoint operators in dimension 1.

Regularization of nonlinear ill-posed problems by dynamical system method

Alexandra Smirnova Georgia State University

Let F be a nonlinear twice Fréchet differentiable map in a Hilbert space H and the equation

(3)  F (z) = 0

be solvable (maybe nonuniquely). Assume that y is a solution to (3), and F′(y) is not boundedly invertible. In this case solving (3) is an ill-posed nonlinear problem. A new approach to solving such a problem is presented. The approach is based on a construction of a dynamical system

(4)  u̇ = ϕ(u, t), u(0) = u_{0},

with a unique global solution u(t) for any u_{0} ∈ B(y, r) that tends to an element u(∞) (in the norm of H), and F (u(∞)) = 0.

If y is a unique solution to (3) in the ball B(y, r), then u(∞) = y. Otherwise u(∞) depends on the choice of u_{0}. The approach is justified, that is, the existence and uniqueness of the global solution to (4) with a suitable choice

of ϕ and u_{0} is proved.

Schrödinger operator on homogeneous metric trees: spectrum in gaps

Michael Solomyak The Weizmann Institute of Science

We study the spectral properties of the Schrödinger operator A_{gV} = A_{0} + gV on a homogeneous rooted metric tree, with a decaying real-valued potential V and a coupling constant g ≥ 0. The spectrum of the free Laplacian A_{0} = −Δ has a band-gap structure with a single eigenvalue of infinite multiplicity in the middle of each finite gap. The perturbation gV gives rise to extra eigenvalues in the gaps. These eigenvalues are monotone functions of g if the potential V has a fixed sign. Assuming that the latter condition is satisfied and that V is symmetric, i.e. depends on the distance to the root of the tree, we carry out a detailed asymptotic analysis of the counting function of the discrete eigenvalues in the limit g → ∞. Depending on the sign and decay of V, this asymptotics is either of the Weyl type or is completely determined by the behaviour of V at infinity.

The results reported were obtained in cooperation with A.V.Sobolev, see A.V. Sobolev and M. Solomyak, Schrödinger operator on homogeneous metric trees: spectrum in gaps, Preprint 2001 (http://xxx.lanl.gov/ps/math/0109016).

Structure and regularity of molecular eigenfunctions and electron densities

Thomas Ostergaard Sorensen Department of Mathematical Sciences, Aalborg University, Denmark

We study the structure of the eigenfunctions of the Hamilton-operator for a molecule near the singularities of the potential. By a product-’Ansatz’, we characterize the singularities of the eigenfunctions up to class C^{1,1}. The obtained information is applied to study the structure and regularity of the corresponding electron density. In particular, we prove that the density is smooth away from the nuclei. This is joint work with S. Fournais, M. and T. Hoffmann-Ostenhof.

A new proof of the Scott correction

Wolfgang Spitzer UC Davis, Mathematics

We introduce new coherent states which we use to prove local semi-classical results for eigenvalues of Schrödinger operators. From there we prove the Scott correction for molecules. This is a joint work with Jan Philip Solovej.

The multi particle Brown-Ravenhall atom

Edgardo Stockmeyer Pontificia Universidad Catolica de Chile

We give a virial theorem for the N particle Brown-Ravenhall atom and we show some of its applications.

Constructing artifact free image families in seismic inversion

Chris Stolk Rice University

In a seismic experiment one generates acoustic waves in the earth by sources at the surface. The wavefield is recorded by an array of receivers, also located at the surface. The purpose is to use the reflections present in the data to construct an image of the subsurface. The data is usually modeled by a linearization. The medium coefficient is written as the sum of a smooth background, in which the geometrical optics approximation can be used, and a singular perturbation that causes the reflections. Given the background medium, from the data a partial, microlocal reconstruction of the medium perturbation can be obtained. To solve the nonlinear inverse problem for the background medium the redundancy in this linear inverse problem needs to be used. In this talk we establish how image families can be constructed for this purpose, when the background medium is strongly refracting, with multivalued travel times.

Discontinuous Solutions of Hamilton-Jacobi Equations: Existence, Uniqueness and Regularity.

Bo Su University of Wisconsin-Madison

In this talk, we address the discontinuous solutions of Hamilton-Jacobi equations. The existence of L^{∞} solutions is proven for general Hamiltonians. Then we clarify the connections in between the existing notions including the classical semicontinuous viscosity solution by Ishii. We look at the important special class of Hamiltonians and show the uniqueness of discontinuous solutions including the classical semicontinuous viscosity solutions, L^{∞} solutions. We prove the new interesting regularity property for locally strictly convex Hamiltonians H such as H(p) = (1 + |p|^{2})^{\frac{1}{2}} . In other words, with discontinuous initial data, the discontinuous solutions of u_{t} + H(Du) = 0 becomes Lipchitz continuous in finite time. The difference between the regularization effect of u_{t} + |Du|^{2} = 0 and u_{t} + (1 + |Du|^{2})^{\frac{1}{2}} = 0 is similar to the difference between the difference of regularity of u_{t} − Δu = 0 and u_{t} − Δu^{2} = 0.

Time-dependent solvable models in quantum mechanics

Alina Suzko Joint Institute for Nuclear Research

A method is elaborated of constructing time-dependent periodic potentials for wich a system of Schrödinger equations admits exact solutions. Time-independent soluble problems are transformed into time-dependent ones by a set of unitary time-dependent transformations and a proper choice of initial states. A class of potentials with corresponding cyclic solutions are constructed in a closed form for a system of two coupled Schrödinger equations. It is shown how to generate a whole family of time-dependent models from one time-independent model by employing a family of special time-dependent transformations.

Local Limit Theorems and Recurrence for the Planar Lorentz Process

Domokos Szasz Budapest University of Technology and Economics

Assuming the CLT, its local version is established for the periodic Lorentz process in ℝ^{d}. This result implies a.) the local CLT and b.) the recurrence for the planar Lorentz process with a finite horizon, thus improving earlier, so-called quasi-versions of the statements obtained by Krámli-Szász in 1985. For the proof of the local CLT, a Markov approximation technique is worked out to be applicable to L. S. Young’s 1998 tower construction. For the Fourier transform of the Perron-Frobenius operator, the necessary analysis of the spectrum was carried out via Doeblin-Fortet inequalities generalizing the 2000 method of Aaaronson-Denker. The challenging arithmeticity questions associated with the local CLT require particular treatment. It is worth noting that for the same case (d = 2, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt, 1998 and J.-P. Conze, 1999, could already establish recurrence. The results are joint with Tamás Varjú.

Analysis of coupled systems of kinetic equations and conservation laws

Moulay Tidriri Iowa State University

We give a survey of recent progress on the rigorous derivation and analysis of the coupling of kinetic equations and their hydrodynamic conservation laws limits for some BGK-type models.

Concentration compactness and nonlinear subelliptic problems without compactness

Kyril Tintarev Uppsala University, Sweden

An abstract version of concentration compactness on Hilbert spaces applies to to actions of Lie groups. Using the concentration-compactness argument we prove existence of solutions for semilinear problems involving sub-Laplacians on whole Lie groups and on their certain non-compact subsets. The main assumption on the group and the sub-Laplacian is compactness of Folland-Stein embedding in the local case. This is a joint work with Ian Schindler

1d periodic differential operator of order 4

Vadim Tkachenko Ben-Gurion University of the Negev, Beer-Sheva, Israel

We consider a differential operator

L = \frac{d^{4}}{dx^{4}} + \frac{d}{dx} p(x) \frac{d}{dx} + q(x), x ∈ R^{1}

with 1-periodic functions p(x) and q(x). We prove that the characteristic equation for its Floquet multipliers is inverse and hence its spectrum in L^{2}(R^{1}) may be described using some hyper-elliptic Riemann surface.

We prove the following uniqueness theorem:

Let U(λ) be the monodromy matrix of operator L and let its characteristic determinant det(U(λ) − ρI) be the same as of operator L_{0} = \frac{d^{4}}{dx^{4}} . Then p(x) ≡ q(x) ≡ 0.

Exponentially Small Error Estimates of Quasiclassical Eigenvalues

Julio Toloza Virginia Tech

We study the behavior of truncated Rayleigh-Schrödinger series for the low-lying eigenvalues of the time-independent Schrödinger equation, in the semiclassical limit ℏ ↘ 0. We prove that, under certain hypotheses on the potential V (x) and for small ℏ > 0, the perturbation series for the eigenvalues admits an exponentially accurate truncation. That is, the difference between the truncated series and the actual eigenvalue can be made smaller than exp(−C/ℏ) for a positive constant C. An analogous statement is shown for the corresponding eigenfuctions.

Exact Soluiton of the Dirichlet Problem for Nonlinear Poisson Equation on the Plane

Igor Tralle Center for Quantum Devices, Department of Electrical and Computer Engineering, Nortwestern University, Evanston, Ill, USA (on leave from Institute of Physics, University of Rzeszow, Poland)

The regular method of searching for the solution of Dirichlet problem for nonlinear Poisson equation (NPE) on the plane is proposed. It is shown the Dirichlet problem on the plane can be solved for almost arbitrary simply connected domain which can be mapped conformally into the unit circle. If the function which maps this domain conformally into the cirlce is known, the solution can be expressed in closed form which is derived in the work. The approach is based on the observation that there exist Baeklund transformation which interrelates the solutions of NPE with the solutions of Laplace equation. Hence, in order to solve the Dirichlet problem for NPE, one should solve an appropriate auxiliary Dirichlet problem for Laplace equation. Such an auxiliary Dirichlet problem can be easily posed and solved for the Laplace equation in the rectangle and cirlce. The last one gives the possibility to solve Dirichlet problem for NPE in alsmost arbitrary simply connected domain on the plane. Indeed, is is known (Riemann Theorem), that simply conncted domain with the boundary comprising more than one point can be mapped conformally onto the unit circle. Thus, if one knows the function wich maps conformally the initial domain onto the unit circle, one can obtain the solution of Dirichlet problem for NPE in this domain in closed form by means of the formulae derived in the work.

Systems of Conservation Laws in Continuum Physics

Konstantina Trivisa University of Maryland

Most of the partial differential equations arising in elasticity, plasticity, fluid mechanics, particle physics, semiconductors, combustion and other applied areas can be formulated as conservation laws. Solutions to hyperbolic conservation laws may be visualized as propagating waves. When the system is nonlinear, the profiles of compression waves get progressively steeper and eventually break, generating jump discontinuities which propagate as shocks. Hence, inevitably the theory must deal with weak solutions. In this talk we consider various models of hyperbolic and viscous conservation laws and we present results on the global existence, uniqueness and asymptotic behavior of solutions even in the case of large discontinuous initial data.

Chaos and the coexistence of attractors in a model system of coupled difference equations

Zdzislaw Trzaska Warsaw University of Technology, Warsaw, Poland

The dynamical behavior of a semi-discrete model of conservation laws featuring interactions in chain is studied in two different parameter regions. The model corresponds to a partial differential equation of conservation laws. For sinusoidal forcing term it is represented by two coupled nonlinear difference equations of the first order, namely

(5)  x(k + 1) = f_{1}(x(k), y(k), w)

(6)  y(k + 1) = f_{2}(x(k), y(k), w)

where nonlinear functions f_{1}(.) and f_{2}(.) satisfy a set of appropriate conditions and w is taken as an external control parameter. The paper aim is focussed on the complexity in system behavior when the external control parameter is varied. The coexistence of attractors between various complex oscillations and between periodic oscillations and chaos has been found. The considered problem is characterized by the following classes of trajectories: i) quasiperiodic trajectories characterized by an irrational number of rotations W which are related to non-commensurable structures, appearing in result of external excitation with period T_{0} on the lattice with non-commensurable period W . When a critical values T_{c} is attained then the continuity of the trajectories is abrupt; ii) periodic trajectories with n ≥ 3 for which the conditions x(k + n) = x(k) and y(k + n) = y(k) are fulfilled; iii) homo- and hetero- cyclic trajectories describing structures, which asymptotically go to periodic at k→∞ but containing for limited k some deformed distributions; iv) choatic trajectories with corresponding structures having some disorder. The chaos in such models as that studied in this paper appears very specific and estimated changes in the variable distributions as a rule are due to the tripling and/or fourthing of the elementary structures.

On von Neumann’s problem

Eduard Tsekanovskii Niagara University, USA

The solution of von Neumann’s problem on parametrization of all nonnegative selfadjoint extensions of a nonnegative symmetric operator in terms of his classical formulas is obtained as well as applications to point-intractions in ℝ^{3}, rank one perturbations and description of all canonical resolvents. New integral representations of the M.Kreĭn-I.Ovcharenko Q_{M} and Q_{μ} operator-valued functions in terms of H_{−2} construction are established.

This is joint work with Yu.Arlinskii.

On the sum of eigenvalues of the discrete Laplace operator

Daniel Ueltschi University of California, Davis

The ground state energy of N non-interacting, spinless electrons in a finite domain D is given by the sum of the N lowest eigenvalues of the Laplace operator with Dirichlet boundary conditions. Li and Yau proved a lower bound in 1983, namely that this sum is larger than a ‘bulk term’ that is proportional to |D|.

Motivated by a study of the Falicov-Kimball model, we considered the analogous problem on a lattice. One can show that the ground state energy of N electrons in an arbitrary domain D ⊂ ℤ^{d} goes like a bulk term proportional to |D|, plus a positive term proportional to the boundary of D.

This work was done in collaboration with Jim Freericks and Elliott Lieb and is archived in math-ph/0107003.

Asymptotic properties of the generalized eigenfunctions of relativistic Schrödinger operators

Tomio Umeda Himeji Institute of Technology

This talk is devoted to a discussion about asymptotic properties of the generalized (or “continuum”) eigenfuctions φ^{±}(x, k) of relativistic Schrödinger operators H = \sqrt{−Δ}+V (x) in R^{3} with V (x) decreasing rapidly at infinity. Based on the construction of φ^{±}(x, k) via the limiting absorption principle for H, we show that: (i) φ^{±}(x, k) are distorted plane waves; (ii) φ^{±}(x, k) are characterized as solutions to the Lippmann-Schwinger type integral equations, of which kernels have severe singularity. We also show that φ^{±}(x, k) are the sums of plane waves and solutions to the equations (\sqrt{−Δ} − |k|)u = f with radiation conditions, and discuss regularity of φ^{±}(x, k) with respect to the variable x. Under a more restrictive condition on V, we show that φ^{±}(x, k) are asymptotically the sums of plane waves and spherical waves.

Wegner estimate with local continuity requirements on the coupling constants

Ivan Veselic Ruhr-Universität Bochum, Germany

(joint work with W. Kirsch)

The talk concerns spectral properties of random Schrödinger operators. We prove a Wegner estimate for alloy type models near the infimum of the spectrum. The single site potential is assumed to be non-positive and the coupling constants non-negative. The distribution of a single coupling constant needs to posses a bounded density only near the upper edge of its support.

The regularization of the Kepler problem and the Bargmann Transform

Carlos Villegas-Blas Universidad Nacional Autonoma de Mexico

There are two known ways to regularize the Kepler problem: via the Kustaanheimo-Steifel transformation or the Moser transformation. M. Kummer showed how to relate these two regularizations by means of a transformation mapping a reduced subspace of C^{4} onto the cotangent bundle of the 3-sphere (with the zero section removed). The goal of this talk is to show how a canonical transformation obtained from a Bargmann transform for the 3-sphere (introduced by the author) links the two mentioned regularizations in the analog way as the transformation studied by M. Kummer does it.

Harmonic analysis methods for incompressible flows of an ideal fluid

Mikhail Vishik The University of Texas at Austin

We formulate uniqueness results for the Euler equations of an ideal incompressible fluid and give some examples of (classes of) flows in terms of their Littlewood-Paley decomposition.

Pauli operator and Aharonov-Casher theorem for measure valued magnetic fields

Vitali Vougalter University of British Columbia

We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual A ∈ L_{loc}^{2} condition on the vector potential which does not allow to consider such singular fields. We extend the Aharonov-Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted L^{2} estimate on a singular integral operator.

Paley-Wiener Theorem and Sturm-Liouville Transforms

Kim Tuan Vu Kuwait University

This talk is devoted to a class of integral transforms arising from a singular Sturm-Liouville problem

y′′ − q(x)y = −λy, x ∈ (a, b)

in the limit-point case at one end or both ends of the interval (a, b). We present a complete solution of the problem of characterization of the image of a function that has compact support and also of a function that vanishes on some interval under this class of transforms. The characterizations are obtained with no restriction on q(x) other than being locally integrable and are much simpler than the previous characterizations given by the author and his associates.

Enchanced Binding in non-relativistic QED

Semjon Vugalter University of Munich

For a spinless particle coupled to a photon field we prove that even if the Schroedinger operator without photon field does not have eigenvalues the system can have a ground state. We describe the coupling by means of the Pauli-Fierz Hamiltonian and our result holds in the case where the coupling constant is small.

Exotherimic Reacting Euler Equations

David Wagner University of Houston

We present technical improvements in the global existence of entropy solutions for exothermically reacting Euler equations. In particular we recognize that the lower order reation term causes the solution to drift in U-space, and this drift imposes conditions on the initial data. In addition we derive a relationship between ϵ or γ − 1 and the total variation of the initial data; this relationship is more explicit than the one presented in Temple’s thesis.

The Time-Dependent Approach to Inverse Scattering

Ricardo Weder Universidad Nacional Autónoma de México

In this talk I present the time-dependent approach to inverse scattering, that has been applied to many linear and non-linear scattering problems. The cases of potential scattering and of the non-linear Schroedinger equation with a potential are discussed in detail. We prove that the scattering operator uniquely determines the potential and the nonlinearity, and we give formulae for the reconstruction of both. The results for other two-body and N-body problems are briefly reviewed. The proofs follow closely physical intuition. Key observations are that in the linear case at high energies translation of wave packets dominates over spreading during the interaction time and that in the non-linear case at low energies the dispersive effect of the linear part dominates the nonlinearity.

On the Stabilizability of Systems with Control Delay Via Output Feedback

Jiang Wei University of Anhui, Hefei, China

In this paper, we will discusses the stabilizability of the systems with control delay. Firstly we give a transformation, by which we transform the discussed systems with control delay to be the systems without control delay. We give the relationship of stabilizability of these two kind systems. Then we discuss the stabilizability via output feedback of the the systems without control delay, we give the conditions for the stabilizability of systems with control delay.

Fractional integro-differentiation by Chen-Hadamard

Mahmadiyor Yahshiboev

Department of Mathematics, Samarkand State University, University Boulevard 15, 703004,Samarkand, Uzbekistan

The fractional differentiation by Chen-Hadamard

D_{c}^{α}f = \frac{1}{μ(α, l)} ∫_{0}^{∞} \frac{(Δ̃_{t}^{l}f_{c+})(x) + (Δ̃_{t^{−1}}^{l}f_{c−})(x)}{|ln t|^{1+α}} \frac{dt}{t} ,

is discussed, where Δ̃_{t}^{l} = ∑_{k=1}^{l}(−1)^{k}\binom{k}{l}f(x · t^{k}), l > α > 0, c > 0, f ∈ L_{p} (R_{+}^{1}, \frac{dx}{x} ) ( or f ∈ L_{p}^{loc} (R_{+}^{1}, \frac{dx}{x} )),

f_{c+}(x) = \{ \begin{matrix}f(x), & x > c \\ 0, & x < c, \end{matrix} f_{c−}(x) = \{ \begin{matrix}0, & x > c \\ f(x), & x < c \end{matrix} .

Since the Chen construction can be applied to functions with an arbitrary growth when x → ∞ or x → 0, this construction is more convenient when applied to such functions than the integro-differentiation by Hadamard [1] itself. As usual, the fractional derivative is to be treated as a certain limit. To this end, several types of different ”truncation” of the Chen-Hadamard fractional derivative are introduced, denoted by

D_{c,ρ}^{α}f,  D^{∗}_{c,ρ}^{α} f,  D_{c,ρ̃}^{α}f,

where \frac{1}{\sqrt[ℓ]{e}} < ρ < 1, ρ̃ = |ln \frac{x}{c} | ln \frac{1}{ρ}, ℓ > α > 0. The following inversion theorem is valid, where J_{c}^{α}φ stands for the corresponding Chen-Hadamard fractional integral.

Theorem 1. Let f = J_{c}^{α}φ, φ ∈ L_{p} (R_{+}^{1}, \frac{dx}{x} ) (or φ ∈ L_{p}^{loc} (R_{+}^{1}, \frac{dx}{x} )), 1 ≤ p < ∞, α > 0, c > 0. Then

lim_{ρ→1} D_{c,ρ}^{α}f = φ,  (1)
lim_{ρ→1} D^{∗}_{c,ρ}^{α} f = lim_{ρ→1} D_{c,ρ̃}^{α}f = φ,  (2)
The  limits  in  (1)-(2)  can  be  understood  both  in  L_{p} (R_{+}^{1}, \frac{dx}{x} )

or L_{p}^{loc} (R_{+}^{1}, \frac{dx}{x} ), correspondingly, 1 ≤ p < ∞, except for the case p = 1 in (2), or almost everywhere.

Theorem 2 For the function f(x) to be presented as f(x) = (J_{c}^{α}φ)(x), where φ ∈ L_{p}^{loc}(R_{+}^{1}, \frac{dx}{x}); α > 0, c > 0, 1 < p < ∞, it is necessary and sufficient that | ln \frac{x}{c} |^{−α}f(x) ∈ L_{p}^{loc}(R_{+}^{1}, \frac{dx}{x}), and

lim_{ρ→1} D_{c,ρ̃}^{α}f

exists in L_{p}^{loc}(R_{+}^{1}, \frac{dx}{x}).

References

[1] S.G. Samko, A.A. Kilbas, O.I. Marichev. Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach. Sci. Publ., London-N.York (1993) (Russian Ed.:Fractional Integrals and Derivatives and Some of Their Applications. Nauka i Tekhnika, Minsk (1987).

Local decay for time period Schrödinger equations

Kenji Yajima Department of Mathematical Sciences, University of Tokyo

We consider Schroedinger equations with potentials which are 2π periodic in time and decay suitably at spatial inifinity. Assume that the Floquet operator U(2π, 0) does not have eigenvalues except 1. We show that, for suitably decaying initial states, the solution has an asymptotic expansion as t → ∞, extending the results of Jensen-Kato and Murata. This is a joint work with A. Galtbayar and A. Jensen.

On a nonlinear PDE arising in magnetic resonance electrical impedance tomography (MREIT)

Jeong-Rock Yoon

We consider the fundamental questions, such as existence and uniqueness, of a mathematical model arising in MREIT system, which is electrical impedance tomography technique integrated with magnetic resonance imaging. The mathematical model for MREIT is the Neumann problem of a nonlinear ellptic partial differential equation ∇ · ( \frac{a(x)}{|∇u(x)|} ∇u(x)) = 0. We show that this Neumann problem belongs to one of two cases: either infinitely many solutions or no solution exist. This explains rigorously the reason why we should utilize two distinct measurements. With two measurements, we prove a uniqueness result on the edge detection of a piecewise continuous conductivity distribution.

The p-system with Large Data

Robin Young Univ of Massachusetts

We describe recent progress towards proving global existence of solutions to the p-system of isentropic gas dynamics. The Cauchy data may be large in amplitude and in total variation. We give precise conditions for the formation and annihilation of a vacuum. We then describe a modified front tracking scheme for constructing large amplitude solutions.

A Singular Integral Equation with Sum-Difference Kernel Arising in Contact Problems with Friction

Olesya Zhupanska University of Florida

We consider an integral equation derived from the contact problem for elastic half-space indented with friction by a rigid cylinder. The mixed boundary-value problem of planar elasticity was formulated in bipolar coordinates. Elastic displacement and stresses in the half-space were represented by two harmonic Papkovich-Neuber functions in the form of the Fourier expansions. The problem was reduced to the singular integral equation with sum-difference kernel with respect to unknown normal pressure in the slip contact zones. An exact analytical solution of this equation was constructed by the Wiener-Hopf technique.