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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.

We consider the differential equation

where

point

Sturm-Liouville equations with potentials depending on the spectral parameter arise in various fields of mathematics and physics (see [

Note that, in the case

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).

We seek a couple of linearly independent solutions

İt is not difficult to show that when

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

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

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

Supposing

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

Now consider the case

Now, similar to previous case we obtain from the Equation (20) that the function

1) if

2) if

3) if

Now we use the method of the successive approximation to show that for every fixed

We have

where

that is

for all

for all

absolutely and uniformly converges in the space

Therefore we have proved the following theorem:

Theorem 1. For every

where

and the kernel

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

1) if

which implies

Similarly, we find from Equation (19) that

that is

2) Let

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

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

Therefore we have

Hence

Further, because of

we can write for all

Note that

where

for all

where

This means that the series

can be differentiated term by term in the space

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

converges in the space

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

These equations with (47) imply that

where

Hence we can formulate the following theorem:

Theorem 2 For all fixed

and the discontinuity conditions

and

where

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