Cauchy Problem for Schroedinger Equation

Ikrom Yarmukhamedov Samarkand State University, Uzbekistan

Let E^{3} be a three-dimensional real Euclidean space, y = (y_{1}, y_{2}, y_{3}), x = (x_{1}, x_{2}, x_{3}) ∈ E^{3}. D bounded single-connected domain in E^{3} with boundary ∂D, consisting of a compact connected part T of the plane y_{3} = 0 and smooth Lyapunov surface S, laying in the semispace y_{3} ≥ 0, ∂D = S ⋃ T, D̄ = D ⋃ ∂D; \frac{d}{dn} - operator of differentiation on external normal, λ(y) ∈ C^{2}(D̄). Δ is a three-dimensional Laplace operator

Problem (Cauchy). Let U(y) ∈ C^{2}(D) and

ΔU(y) − λ^{2}(y)U(y) = 0,  y ∈ D  (1)
U(y) = f(y),  \frac{∂U}{∂n} (y) = g(y), y ∈ S  (2)

where f(y) and g(y) are given functions of the classes C^{1}(S) and C(S), respectively. It is required to compute U(y) in interior points of D.

Since specified problem is ill-posed we will assume that solution exists, belongs to the class C^{2}(D) ⋂ C^{1}(D̄) and satisfies the condition

|U(y)| + |\frac{d}{dn}U(y)| ≤ M, y ∈ T,

where M given positive number.

In this assumption we shall build approximate solution for the problem (1)-(2). Denote

−\frac{2}{π^{2}} e^{σx_{3}^{2}}Φ_{σ}(x, y) = ∫_{0}^{∞}Im [\frac{e^{σw^{2}}}{w − x_{3}} ] \frac{cos λudu}{\sqrt{u^{2} + s}},

G_{σ}(x, y) = Φ_{σ}(x, y)+e^{−σx_{3}^{2}} ∫_{D} Γ(x, t)Φ_{σ}(t, y)e^{σt_{3}^{2}}dt, U_{σδ}(x) = ∫_{S} [g_{δ}G_{σ} − f_{δ} \frac{dG_{σ}}{dn} ] ds_{y},

here t = (t_{1}, t_{2}, t_{3}) ∈ R^{3} s = (y_{1} − x_{1})^{2} + (y_{2} − x_{3})^{2}, w = i\sqrt{u^{2} + s} + y_{3}, Γ(x, t) is a resolvent of the kernel of the integral equation of special kind [1]. f_{δ}(y), g_{δ}(y) ∈

C(S) and sup_{S} |f(y) − f_{δ}(y)| + sup_{S} |g(y) − g_{δ}(y)| < δ, σ = \frac{1}{x_{0}} ln \frac{M}{δ} , x_{0} = max_{D̄} x_{3},

δ < Mexp(−c|D|), |D|- volume of D and c = const > 0. Under this conditions the following inequality holds

|U(x) − U_{σδ}| ≤ Ψ(σ)M^{1− \frac{x_{3}}{x_{0}}} δ^{\frac{x_{3}}{x_{0}}} , Ψ(σ) = O(\sqrt{σ}), σ → ∞,

and

lim_{δ→0} U_{σδ} = U(x).