Solution of the spectral problem for the Schrödinger equation with a degenerate polynomial potential of even power
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 the action of group \(Z_{r+1}\). They construct the Jacobi matrix that is almost symmetric. A generating function \(\Phi(\lambda,w)\) is introduced. There is some manipulation of entries in the Jacobi matrix, a transformation that is a rather clever generalization of the Laplace transform \(\varphi(x,\lambda) \to\Phi (\lambda,w)\), which is followed by the Mittag-Leffler transform. Just these three pages warrant in the opinion of the reviewer at least one, maybe two separate short research papers. There has been much recent interest in material with constitutive equations described by fractional order derivatives where such Mittag-Leffler transforms have been used and in the opinion of the reviewer ideas of this paper are applicable. After several substitutions the authors derive an equation with a hypergeometric-type differential equation plus a term which is a Mellin convolution integral. This is in turn reduced to an infinite system of algebraic equations. Part 2 of the paper deals with an anharmonic oscillator with the potential \((x^2-\varepsilon)^2\). The authors look for odd and even solutions to the eigenvalue equation: \(\psi''-(x^2-\varepsilon)^2\psi= \lambda\psi\). As \(\varepsilon\to 0\), the degeneracy of order \(x^4\) occurs, corresponding to breaking of symmetry. Calculations of the spectrum revealed a complicated dependence of the eigenvalues on the parameter \(\varepsilon\). The last part of the paper contains a summary of numerical studies for the potential \(x^{2r}\), with \(r=2,3, \dots, 18\). This is an impressive work containing a mix of some of the most difficult aspects of 19th century mathematis, which are no longer offered in our classrooms, with modern approaches of functional analysis and group theory.
scientific article; zbMATH DE number 1382740
scientific article