Anharmonic oscillators in the complex plane, $\mathcal{PT}$-symmetry, and real eigenvalues

TL;DR

This paper investigates eigenvalue problems for anharmonic oscillators in the complex plane, establishing conditions for real eigenvalues under $ ext{PT}$-symmetry and providing asymptotic eigenvalue expansions.

Contribution

It introduces new spectral results for complex anharmonic oscillators, including criteria for real eigenvalues and asymptotic eigenvalue formulas, extending understanding of $ ext{PT}$-symmetric quantum systems.

Findings

Eigenvalues are real and positive under $ ext{PT}$-symmetry.

Infinitely many real eigenvalues occur if and only if the problem is $ ext{PT}$-symmetric or its translation.

Asymptotic expansions of eigenvalues are derived.

Abstract

For integers $m\geq 3$ and $1\leq\ell\leq m-1$, we study the eigenvalue problems $-u^{\prime\prime}(z)+[(-1)^{\ell}(iz)^m-P(iz)]u(z)=\lambda u(z)$ 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 $P$ is a polynomial of degree at most $m-1$. We provide asymptotic expansions of the eigenvalues $\lambda_{n}$. Then we show that if the eigenvalue problem is $\mathcal{PT}$-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when $\gcd(m,\ell)=1$, the eigenvalue problem has infinitely many real eigenvalues if and only if its translation or itself is $\mathcal{PT}$-symmetric. Also, we will prove some other interesting direct and inverse spectral results.