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

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.


Introduction
We consider the differential equation , 0 π, y q x p x y x y x λ λ ρ where λ is the spectral parameter, ( ) Sturm-Liouville equations with potentials depending on the spectral parameter arise in various fields of mathematics and physics (see [1]- [3] for details).It is well known that in the case ( ) 1 x ρ = the Equation (1) ap- pears 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]).Note that this method which is an effective device in the theory of inverse problems [11]- [16], for relativistic scattering problems was first suggested in [5] and [6].Various inverse scattering problems for the Schroedinger equation with an energy dependent potential on the half line and full line were investigated in [24]- [30].Direct and inverse spectral problems in a finite interval for the Equation (1) in the case ( ) x ρ = were first investigated in [17] [18].
For further discussing of the inverse spectral theory for Equation (1) in a finite interval with ( ) x ρ = we refer to works [19]- [23].
Note that, in the case ( ) 0 p x = direct and inverse problems for boundary-value problems generated by an equation of type (1), in various formulations, have been studied in [32]- [37] and other works.Inverse scattering problem for Equation (1) with ( ) 0 p x = on the half line [ ) 0, +∞ was investigated and the complete solution of this problem was given in [38] where the new integral representation, similar to transformation operators [9], was obtained for the Jost solution of the discontinuous Sturm-Liouville equation.Direct and inverse scattering problems on the half-line for the discontinuous Sturm-Liouville equation with eigenparameter dependent boundary conditions have been investigated in [41].The direct and inverse spectral problem for the Equation (1) in the case ( ) 0 p x = with some separated boundary conditions on the interval ( ) 0, π recently has been investi- gated in [39] [40] [42], where the new integral representations for solutions have been also constructed.The inverse spectral problem of recovering pencils of second-order differential operators on the half-axis with turning points was studied in [31], where the properties of spectral characteristics were established, formulation of the inverse problem was given and a uniqueness theorem for solution of the inverse problem is proven.But the spectral problems for Equation (1) in a finite interval, especially, inverse spectral problems and full-line inverse scattering problems requiring the recovery of the potential functions by the Gelfand-Levitan-Marchenko methods have not been studied yet and there isn't any serious work published in this direction.
In this work, 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).In Section 2, we consider a pair of linearly independent solutions of the Equation (1) with initial conditions at zero.We seek special Fourier-type integral forms for these solutions.To prove the existence such forms, we derive the system of Volterra type integral equations for the kernel functions.Then we solve these systems by the successive approximation method.In Section 3, we investigate some significant properties of the kernels of these integral representations.Namely, we find an important relationship between the kernels of the integrals and the coefficient of the Equation (1).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).

Derivation of the Integral Representations for the Solutions
We seek a couple of linearly independent solutions ( ) İt is not difficult to show that when ( ) ( ) 0 q x p x ≡ ≡ the initial value problem (1), (3) has solution , e e , where .
x j j j y x e x x t q t y t t j which is equivalent to the problem (1), (3).Here where , , e The formula ( 8) is also written as Consider the integral Equation ( 5) and substitute , e e , , where ( ) j R x ± will be defined below and ( ) Taking into our account (8) and the second integral in the right hand side of (10) we require From (11) we immediately have Then (10) implies that , , e d , where ( ) , , 2 2 2 , We require that the integral Equation ( 13) has the solution , , e d , where ( ) , j K x t is an unknown function.Substituting the expression ( 16) of ( ) Now using the formulas (8), (8') we transform the right hand side of Equation ( 16) to the form of the Fourier integral.
First consider the case 0 x a ≤ ≤ for which the Equation ( 16) is written as Supposing ( ) According to the uniqueness properties of the Fourier transformation, Equation ( 18) implies that x t x t x s j j j j j t x s Now consider the case x a > .In this case, according to formulas ( 8) and (8'), the Equation ( 16) yields Now, similar to previous case we obtain from the Equation ( 20) that the function ( ) ( ) , satisfies some integral equations of type (19) in the corresponding regions.Namely we have the following: 1) if ( ) ( ) .
3) if ( ) ( ) x q s s K s q s s K s q s s K s Now we use the method of the successive approximation to show that for every fixed [ ] 0, π x ∈ the integral Equation ( 19), ( 21)-( 23) has a unique solution ( ) , x t x t j j j j j j j K x t q s R s s q s R s s x t a j j j a j j j j K x t q s R s s q s R s s q s R s s We have where Similarly we obtain that for all [ ] 0, π x ∈ and 0,1, 2, n =  .Hence the series absolutely and uniformly converges in the space , the sum ( ) ,.
j K x of this series is a unique solution of the integral Equations ( 19), ( 21)-( 23) and ( ) , d e 1.
Therefore we have proved the following theorem: Theorem 1.For every λ the solution ( ) , j y x λ of Equation (1) satisfying the initial conditions (3) can be represented as , e e , e d ,

Properties of the Kernels
From the integral Equations ( 19), ( 21)-( 23) we easily compute the following boundary relations for ( ) Similarly, we find from Equation ( 19) that 2) Let x a > .Then from integral Equations ( 21)-( 23) we obtain the equation Hence, combining the formulas ( 35) and ( 37) we obtain From the integral Equations ( 21)-( 23) it is clear that the function ( ) x a > has a jump discontinuity at points ( ) and Finally, from ( 23) we find that Hence, combining the formulas ( 36) and ( 41) we obtain Now we investigate the additional properties of the function ( ) ,. j K x .Consider the successive approximation ( 24)- (26).By the differentiation with respect to the variable t we find ( ) ( ) Therefore we have Further, because of Note that where This means that the series Similarly, from the successive approximation ( 24)-( 26) by differentiation with respect to the variable x we have the series ∈ − .Further,by differentiation integral Equations ( 19), ( 21)- (23) we have that , d , ,0 , , .
x s s j j j j x These equations with (47) imply that where 1 0 C > is a constant.Differentiating Equations ( 48)-( 51) once more we have the following partial diffe- rential equation for the kernel ( ) , j K x t : are real-valued functions, and ( )x ρis the following piecewise-constant function with discontinuity at the point . Obviously, the last equality will be satisfied if we choose ) ) , , , , , , d . ,