Solution of the spectral problem for the Schrödinger equation with a degenerate polynomial potential of even power
type: Item
instance of: scholarly article
title: Solution of the spectral problem for the Schrödinger equation with a degenerate polynomial potential of even power
zbMATH Open document ID: 0940.34068
DOI: 10.1007/BF02069892
author: A. S. Vshivzev
author: N. V. Norin
author: V. N. Sorokin
published in: Theoretical and Mathematical Physics
publication date: 2000-03-14T00:00:00Z
review text: The authors consider the anharmonic oscillator problem and the related problem of finding the spectrum and wave function for the stationary equation \(\{\partial^2/ \partial x^2-x^{2r}\} \varphi(x)= E\varphi(x)\), with the degenerate potential \(x^{2r}\). This is accomplished in following stages. First, they study the asymptotic behavior of the solutions \(\varphi(x) \sim\exp \{x^{r+ 1}/(r+1)\}\) for \((x\to\infty)\). Then using the time honored technique (Hermite, etc.) they construct orthogonal polynomials \(\Lambda^p_n(x)\) weighted by \[ \exp \{-x^{r+1}\}: \Lambda^p_n(x) =\exp\{x^r\} (\partial/ \partial x)^n\bigl[x^{n+p} \exp\{- x^r\}\bigr]/n! \] allowing them to describe excited states of the spectrum. Finally, with the knowledge of the structure of the solutions they reduce the problem to an algebraic problem in a manner resembling the use of integral transforms in solving ordinary differential equations with constant coefficients. They comment that detailed knowledge of the spectral properties would permit to advance the theory of phase transitions in the ``real'' physical world. They start this project by first examining the eigenvalue problem \(\varphi''(x)- x^{2r}\varphi (x)=\lambda\varphi\) in the \(L_2 (\mathbb{R})\) setting. As \(x\to\infty\) they study a wave function of the form \(\varphi (x)=\exp \{s(x)\}\), obtaining the differential equation: \(\lambda=-[s'' +(s')^2-x^{2r}]\). For large values of \(x\) the approximate solutions are: \(s(x)= \pm|x^{r+1} |/(r+1)\). The spectrum consists of unbounded discrete positive eigenvalues \(0<\lambda_0 <\lambda_1 <\cdots <\lambda_n <\dots\), where the number of eigenvalues smaller than some large number is computed from Titchmarsh-Carleman integral formula. The Taylor expansion of the wave function produces the following infinite series: \(\varphi(x,\lambda)= \sum_0^\infty a_p(\lambda) x^{2p}\), where the recurrence relation is easily established. The polynomials \(a_p(\lambda)\) are invariant with respect to …
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 34L15
zbMATH DE Number: 1382740
zbMATH Keywords: stationary Schrödinger equation
zbMATH Keywords: specral theory
zbMATH Keywords: degenerate potential
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: Mittag-Leffler transform
zbMATH Keywords: Mellin convolution integral
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02069892
OpenAlex ID: W2037380870
description: scientific article; zbMATH DE number 1382740