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University of Missouri

УДК 517.956

Spectral dependencies of Evans function for the Schrцdinger equation

A.A. Yurova, V.A. Yurov

USA, MO, Columbia

Introduction

The Evans function E, introduced in the beginning of 1970-s in a series of seminal papers by J. Evans (1-3) has since became one of the sharper tools in the arsenal of a mathematical physicist (cf. for example (4-6)). The interest that Evans function generates is dictated by its very specific properties: defined as a function of spectral parameter for some differential equation problem, the zeroes of Evans function correspond uniquely to the eigenvalues of this problem. Of course, such a property makes one wonder whether the Evans function can in at least some cases be represented as a polynomial over the spectral parameter л. In fact, those special cases for which it was possible to do all the necessary calculations has shown this proposition to be correct. Unfortunately, the general case has proven too difficult to tackle by ordinary means.

In this article we'll try a different approach; instead of trying to simplify the Evans function by itself, we'll try to establish the relationship between the Evans functions of two problems, possessing exactly the same spectrum with one exception: one single point л_s belonging to spectrum of the first problem would be altogether expunged from the spectrum of the second problem. It will be shown that the corresponding Evans functions would indeed differ by the factor of , where л = ?г² and л_s = ?к². The key to success lies in the Darboux transformation (4, 13).

Preliminaries

For this note let us concentrate on the Schrцdinger equation:

(H - г²)Y (г, x) = (??² + V (x)+ г²) Y (г, x)=0, г> 0. (1)

The Evans function for this equation (with a vanishing potential) is known to be equal to:

(2)

where Y (г, x) is the solution to (1), satisfying an additional condition:

Ye^?гx = 1. (3)

Suppose we apply the Darboux transformation to the (1). This would result in a new function Y₁(г, x), s.t.:

Y₁(г, x)= c (?x ? у(к, x)) Y (г, x), (4)

where

у(к, x) = ?_x ln( f (k, x)), (5)

and the function f(к, x) is in turn a solution of

(H ? к²) f (к, x) = ??² + V (x) + к²f (к, x) = 0. (6)

Before we proceed, let us make some further assumptions with regards to f (к, x). Since quantities л = ?г² and л_s = ?к² serves the role of the eigenvalues of corresponding Schrцdinger equations (1), (6), let us assume that л_s is the leftmost point in the discrete spectrum of operator H.

This means that the difference between the discrete spectrums of Schrцdinger problem before and after the Darboux transformation would be in the single point л_s disappearing in the post-Darboux spectrum.

Now, in order to find out how this difference would affect the Evans function (2), we have to find out the exact value of multiple c in (4).

First, since f (к, x) is the eigenfunction of the Schrцdinger equation with corresponding eigenvalue л_s = ?к, it is clear that

and hence,

(7)

Second, according to (3)

?_x (Ye^?гx ) = 0. (8)

Therefore,

?_x (Y(г ,x ))e^?гx - г Y(г ,x ) e^?гx = 0, (9)

and

?_x (Y(г ,x ))e^?гx = г. (10)

Combining these two facts together yields the following formula:

Y₁(г ,x ) e^?гx =c (?_x (Y(г ,x ))e^?гx - у(к, x) Y(г ,x ) e^?гx) =

= c?_x (Y(г ,x ))e^?гx - cу(к, x) Y(г ,x ) e^?гx = x (г - к). (11)

Hence, if we want our new function Y₁ to satisfy the conditions, similar to (3), we have to conclude that

(12)

To finish the preliminaries, let us write down the potential as it would appear after the Darboux transformation:

V₁(к, x) = V (x) ? 2?_x у(к, x), (13)

and the resulting Schrцdinger system:

(??² + V₁(к, x) + г²) Y₁(г, x) = 0, г> 0. (14)

The calculations

Finally, we are ready to implement everything we have gathered so far for the new, post-Darboux, Evans function:

(15)

Using of (12) and (13) in (15) will end up with:

(16)

where primes denote the partial derivatives w.r.t. variable x.

Simplification of this expression would require the following observation:

(17)

In other words, the function у(к, x) satisfies the standard Riccati differential equation:

у² = V + к² - у'. (18)

Let us consider the integral in (16):

using (3) and (7) to evaluate the first term we come to

for whom (3) is indispensible yet again:

which, after application of equations (3), (7) and (10) turns into:

(19)

Recalling the identity (2) we conclude that the absence of eigenvalue к from the spectrum of system (14) results in the following relationship between the Evans functions E1 and E of correspondingly (14) and (1) as follows:

,

which after the simplification gives the following simple identity:

(20)

mathematical physics evans schrodinger spectrum

In other words, the presence of a particular eigenvalue in the spectrum is reflected in the Evans function by a linear factor of in its expansion.

The problem above has been dealing with the eigenvalues of multiplicity 1. It is expected that the higher multiplicities would correspond to the effective increase in power of corresponding factors. Another interesting possibility lies in application of the aforementioned technique to the differential equation of more general nature.

The simple formula (2) becomes obsolete there and should be replaced by the more complex system of Friedholm equations (cf., for example ([7])). The detailed treatise on these ideas would be presented in further papers.

References

1. John W. Evans. Nerve Axon Equations I: Linear approximations // Indiana Univ. Math. J 21 (9) (1972), 877-885

2. John W. Evans. Nerve Axon Equations II: Stability at rest // Indiana Univ. Math. J 22 (1) (1972) 75-90

3. John W. Evans. Nerve axon equations. III. Stability of the nerve impulse // Indiana Univ. Math. J., 22:577-593, 1972/73.

4. Robert L. Pego and Michael I. Weinstein. Eigenvalues, and instabilities of solitary waves. // Philos. Trans. Roy. Soc. London Ser. A, 340(1656):47-94, 1992.

5. Bjorn Sandstede. Stability of travelling waves. // Handbook of dynamical systems, Vol. 2, pages 983-1055. North-Holland, Amsterdam, 2002.

6. Jian Deng and Shunsau Nii. An infinite-dimensional Evans function theory for elliptic boundary value problems. // J. Differential Equations, 244(4):753-765, 2008.

7. F. Gesztesy, Y. Latushkin, K. Makarov. Evans functions, Jost functions and Fredholm determinants // Archive for Rational Mechanics and Analysis vol. 186, N.3, pp.361-421.

Annotation

УДК 517.956

Spectral dependencies of Evans function for the Schrцdinger equation. A.A. Yurova, V.A. Yurov. USA, MO, Columbia, University of Missouri

A.A. Yurova - PhD, docent, Department of Higher Mathematics, Kaliningrad State Technical University, yurov@freemail.ru

V.A. Yurov - PhD, graduate student of Mathematical Department, University of Missouri, Columbia, MO, USA, vayt37@mail.missouri.edu

In this note we are establishing correlation existing between the Evans functions of two Schrцdinger problems differing single point in the discrete spectrum.

Keywords: Evans function, Schrцdinger equation, eigenvalue, Darboux transformation

Аннотация

Форма зависимости функции Эванса уравнения Шрёдингера от спектрального параметра. А.А. Юрова, В.А. Юров

Алла Александровна Юрова - к.ф.-м.н, доцент, кафедра высшей математики, КГТУ, 236039, Калининград, Ленинский пр., д.46, кв.8, тел. 47-38-69, , yurov@freemail.ru

Валериан Артёмович Юров - к.ф.-м.н, аспирант факультета математики, Колумбийский университет, Колледж авеню, д.1, г. Коламбия, штат Миссури, США, vayt37@mail.missouri.edu

В работе устанавливается связь между функциями Эванса двух уравнений Шрёдингера, дискретные спектры которых отличаются на один уровень.

Ключевые слова: функции Эванса, уравнение Шрёдингера, собственное значение, преобразование Дарбу

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