Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

TL;DR

This paper investigates the spectral properties of certain complex anharmonic oscillators, providing conditions for infinitely many real eigenvalues and deriving asymptotic formulas for these eigenvalues.

Contribution

It offers a new criterion for the existence of infinitely many real eigenvalues in complex anharmonic oscillators with polynomial potentials.

Findings

Derived asymptotic formulas for eigenvalues.

Established necessary and sufficient conditions for infinitely many real eigenvalues.

Analyzed eigenvalue distribution in the complex plane.

Abstract

We study the eigenvalue problem $-u"+V(z)u=\lambda u$ in the complex plane with the boundary condition that $u(z)$ decays to zero as $z$ tends to infinity along the two rays $\arg z=-\frac{\pi}{2} \pm \frac{2\pi}{m+2}$, where $V(z)=-(iz)^m-P(iz)$ for complex-valued polynomials $P$ of degree at most $m-1\geq 2$. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.