Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval

Abstract

We reduce the initial value problem for the generalized Schroedinger equation with piecewiseconstant 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.

Half inverse problems for the impulsive quadratic pencil with the discontinouty coefficient

“…Proof The accuracy of the equations (2.15) and (2.16) were obtained in the study of [24] In a similar way, the accuracy of the (2.17) and (2.18) behavior is easily obtained.…”

“…is the Weyl function of the boundary value problem for the equation (1.1) on (0, c 1 ) with boundary conditions U (y) = 0, y ′ (c 1 ) = 0 and without impulsive conditions (1.4) (see [24]). By ( [24]), the Weyl function uniquely species p(x) and q(x) a.e. on (0, c 1 ) and coefficient h.…”

“…There are many studies on the determination of singular operators (coefficients of differential expression and other parameters) according to spectral characteristics (data) of the given problem recently [5], [22].These types of problems create different classes according to the location [1], [28], [9].The inverse spectral problem for the boudary value problem give (1.1)-(1.4) have been previously studied in [24] in general.…”

Inverse Nodal Problems for Pencils of Singular Sturm-Liouville Operators

“…It is clear from the definition given above that the problem (1)-( 3) is equivalent to the problem (4)-( 5), ( 2)-(3), that is, each solution of the problem (1)-( 3) is equivalent to the solution of the problem (4) satisfying the (2),(3) boundary and (5) discontinuity conditions. Let us denote the problem of seeking the solution of (4) equation satisfying ( 2)-( 3) boundary conditions and (5) discontinuity condition with L. By applying the method in the study [17], we obtain the solution of the problem L that satisfies the initial conditions for (6), while |λ| → ∞, according to the x variable,…”

Inverse nodal problems for singular diffusion equation

“…has behavior, where By applying the method in the study [42], we obtain the solution of the problem L that satisfies the initial conditions for (2.2), while |𝜆| → ∞, according to the x variable,…”