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.

One Dimensional Schroedinger Equation Fundamental Solutions Transformation Operator Inte-gral Representation Differential Equation with Discontinues Coefficient Kernel of an Integral Op-erator Integral Equation Method of Successive Approximations
1. Introduction

We consider the differential equation

where is the spectral parameter, is an unknown function, , are real-valued functions, and is the following piecewise-constant function with discontinuity at the

point such that:

Sturm-Liouville equations with potentials depending on the spectral parameter arise in various fields of mathematics and physics (see  - for details). It is well known that in the case the Equation (1) appears for modelling of some problems connected with the scattering of waves and particles in physics  . In this classical case, Jaulent and Jean   have constructed the integral representations of Jost solutions and treated the inverse scattering problem by the Gelfand-Levitan-Marchenko method (see  and  ). Note that this method which is an effective device in the theory of inverse problems  -  , for relativistic scattering problems was first suggested in  and  . Various inverse scattering problems for the Schroedinger equation with an energy dependent potential on the half line and full line were investigated in  -  . Direct and inverse spectral problems in a finite interval for the Equation (1) in the case were first investigated in   . For further discussing of the inverse spectral theory for Equation (1) in a finite interval with we refer to works  -  .

Note that, in the case direct and inverse problems for boundary-value problems generated by an equation of type (1), in various formulations, have been studied in  -  and other works. Inverse scattering problem for Equation (1) with on the half line was investigated and the complete solution of this problem was given in  where the new integral representation, similar to transformation operators  , 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  . The direct and inverse spectral problem for the Equation (1) in the case with some separated boundary conditions on the interval recently has been investigated in    , 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  , 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).

2. Derivation of the Integral Representations for the Solutions

We seek a couple of linearly independent solutions of Equation (1) satisfying the initial conditions

İt is not difficult to show that when the initial value problem (1), (3) has solution

where

and

Consider the integral equation

which is equivalent to the problem (1), (3). Here

By using (4) it is easily obtained that

where

We have

The formula (8) is also written as

Consider the integral Equation (5) and substitute

where will be defined below and is a new unknown function. We have

Taking into our account (8) and the second integral in the right hand side of (10) we require

to be satisfied. Obviously, the last equality will be satisfied if we choose

From (11) we immediately have

Then (10) implies that

where

and

We require that the integral Equation (13) has the solution

where is an unknown function. Substituting the expression (16) of in the Equation (13) we have

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 for which the Equation (16) is written as

Supposing to be zero as and changing orders of integrations at the right hand side of Equation (17) we obtain

According to the uniqueness properties of the Fourier transformation, Equation (18) implies that

Now consider the case. 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, continued as zero for, satisfies some integral equations of type (19) in the corresponding regions. Namely we have the following:

1) if then

2) if then

3) if then

Now we use the method of the successive approximation to show that for every fixed the integral Equation (19), (21)-(23) has a unique solution belonging to. For this reason let us define

We have

where. Similarly we obtain that

that is

for all. Therefore

for all and. Hence the series

absolutely and uniformly converges in the space for each, the sum of this series is a unique solution of the integral Equations (19), (21)-(23) and satisfies the inequality

Therefore we have proved the following theorem:

Theorem 1. For every the solution of Equation (1) satisfying the initial conditions (3) can be represented as

where

and the kernel satisfies (33).

3. Properties of the Kernels

From the integral Equations (19), (21)-(23) we easily compute the following boundary relations for:

1) if then from Equation (19) we have

which implies

Similarly, we find from Equation (19) that

that is

2) Let. Then from integral Equations (21)-(23) we obtain the equation

Now using 1) we easily find that

Hence, combining the formulas (35) and (37) we obtain

From the integral Equations (21)-(23) it is clear that the function has a jump discontinuity at points. Computing the jumps we have

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. Consider the successive approximation (24)-(26). By the differentiation with respect to the variable t we find

Therefore we have

Hence

Further, because of

we can write for all

Note that

where and is a constant. We see that and from (44) we immediately have

for all and. Consequently,

where,

This means that the series

can be differentiated term by term in the space and the sum is also differentiable in this space with

Similarly, from the successive approximation (24)-(26) by differentiation with respect to the variable x we have the series

converges in the space and.

Further,by differentiation integral Equations (19), (21)-(23) we have that

These equations with (47) imply that

where is a constant. Differentiating Equations (48)-(51) once more we have the following partial differential equation for the kernel:

Hence we can formulate the following theorem:

Theorem 2 For all fixed the kernel of the integral representation (34) has the partial derivatives, and satisfy the discontinuous partial differential Equation (53) with the conditions

and the discontinuity conditions

and

where

Cite this paper

Anar AdilogluNabiev,Rauf Kh.Amirov, (2015) Integral Representations for the Solutions of the Generalized Schroedinger Equation in a Finite Interval. Advances in Pure Mathematics,05,777-795. doi: 10.4236/apm.2015.513072