Spectral dependencies of Evans function for the Schrodinger equation
The specific properties of the Evans function, its use as a tool in mathematical physics. Application of the Darboux transformation to the Schrodinger equation. Solution of the standard Riccati differential equation. Investigation of a discrete spectra.
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University of Missouri
Spectral dependencies of Evans function for the Schrцdinger equation
A.A. Yurova, V.A. Yurov
USA, MO, Columbia
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 лs 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 лs = ?к2. The key to success lies in the Darboux transformation (4, 13).
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:
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)
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 лs = ?к2 serves the role of the eigenvalues of corresponding Schrцdinger equations (1), (6), let us assume that лs 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 лs 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 лs = ?к, it is clear that
Second, according to (3)
?x (Ye?гx ) = 0. (8)
?x (Y(г ,x ))e?гx - г Y(г ,x ) e?гx = 0, (9)
?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
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)
Finally, we are ready to implement everything we have gathered so far for the new, post-Darboux, Evans function:
Using of (12) and (13) in (15) will end up with:
where primes denote the partial derivatives w.r.t. variable x.
Simplification of this expression would require the following observation:
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:
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:
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 ()). The detailed treatise on these ideas would be presented in further papers.
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.
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, firstname.lastname@example.org
V.A. Yurov - PhD, graduate student of Mathematical Department, University of Missouri, Columbia, MO, USA, email@example.com
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
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, , firstname.lastname@example.org
Валериан Артёмович Юров - к.ф.-м.н, аспирант факультета математики, Колумбийский университет, Колледж авеню, д.1, г. Коламбия, штат Миссури, США, email@example.com
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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