Abstract
We reduce the initial value problem for the generalized Schroedinger equation with piecewise-constant leading coefficient to the system of Volterra type integral equations and construct new useful integral representations for the fundamental solutions of the Schroedinger equation. We also investigate some significant properties of the kernels of these integral representations. The integral representations of fundamental solutions enable to obtain the basic integral equations, which are a powerful tool for solving inverse spectral problems.
Highlights
We consider the differential equation− y′′ + (q ( x) + 2λ= p ( x)) y λ2ρ ( x) y, 0 ≤ x ≤ π, (1)where λ is the spectral parameter, y = y ( x, λ ) is an unknown function, q ( x) ∈ L2 (0, π), p ( x) ∈W21 (0, π) are real-valued functions, and ρ ( x) is the following piecewise-constant function with discontinuity at the point a ∈ (0, π) such that a > απ : α +1How to cite this paper: Nabiev, A.A. and Amirov, R.Kh. (2015) Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval
Jaulent and Jean [7] [8] have constructed the integral representations of Jost solutions and treated the inverse scattering problem by the Gelfand-Levitan-Marchenko method
We reduce the differential Equation (1) with initial conditions (3) to the system of Volterra type integral equations and we construct new useful integral representations for the fundamental solutions of the Equation (1)
Summary
It is well known that in the case ρ ( x) = 1 the Equation (1) appears for modelling of some problems connected with the scattering of waves and particles in physics [4] In this classical case, Jaulent and Jean [7] [8] have constructed the integral representations of Jost solutions and treated the inverse scattering problem by the Gelfand-Levitan-Marchenko method (see [9] and [10]). The direct and inverse spectral problem for the Equation (1) in the case p ( x) = 0 with some separated boundary conditions on the interval (0, π) recently has been investigated in [39] [40] [42], where the new integral representations for solutions have been constructed. The constructed integral representations of fundamental solutions play an important role in the derivation of main integral equations which are a powerful tool for solving inverse spectral problems for the Equation (1)
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