Schr\"odinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues

TL;DR

This paper analyzes eigenvalue problems with polynomial potentials in complex Schrödinger equations, deriving asymptotics of eigenvalues and applying results to inverse spectral problems, while establishing reality and positivity of eigenvalues for symmetric cases.

Contribution

It provides asymptotic expansions for eigenvalues of complex polynomial Schrödinger operators and applies these to reconstruct potential coefficients and analyze eigenvalue reality.

Findings

Eigenvalues have explicit asymptotic expansions.

Reconstruction of polynomial potential coefficients from eigenvalue asymptotics.

Eigenvalues are real and positive for symmetric PT-symmetric potentials, with finitely many exceptions.

Abstract

For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problem $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2}$ in the complex plane, where $P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $\lambda_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues. Also, we show for arbitrary $\mathcal{PT}$-symmetric polynomial potentials of degree $m\geq 3$ and all symmetric decaying boundary conditions that the eigenvalues are all real and positive, with only finitely many exceptions.