Direct And Inverse Spectral Problems For Differential Operators With Singular Boundary Conditions

Gerhard Freiling Universität Duisburg-Essen, Campus Duisburg, Germany

We consider direct and inverse spectral problems for a class of singular differential equations of the form

−\frac{d}{dt}(p_{2}(t)\frac{dz}{dt} ) + p_{1}(t)z(t) = λp_{0}(t)z(t), t ∈ (t_{0}, t_{1}).

Here λ is the spectral parameter, and the real-valued functions p_{k}(t) have zeros or/and singularities at the end points of the interval (t_{0}, t_{1}). More precisely,

p_{k}(t) = (t − t_{0})^{s_{k0}}(t_{1} − t)^{s_{k1}}p_{k0}(t),

where s_{km} are real numbers, p_{10}(t) ∈ C[t_{0}, t_{1}], p_{k0}(t) ∈ C^{2}[t_{0}, t_{1}], p_{k0}(t) > 0 for k = 0, 2, t ∈ [t_{0}, t_{1}]. We assume that s_{2m} < s_{0m} + 2, s_{2m} ≤ s_{1m} + 2, m = 0, 1, i.e. we consider the case of the so-called regular singularities.

In the first part of our lecture we provide a general method for defining twopoint singular boundary conditions in the above-mentioned general case. Using this definition we derive asymptotic estimates for the eigenvalues of a corresponding class of boundary value problems and various properties of the set of eigenfunctions.

Subsequently we study the corresponding inverse spectral problems for the case of separated singular boundary conditions and also for a class of non-separated singular boundary conditions. For each class we prove a uniqueness theorem and give a procedure for constructing the solution of the inverse problem.

This is a report on joint work with V. Yurko, Saratov, Russia.