On the eigenvalues of some nonhermitian oscillators

TL;DR

This paper investigates the eigenvalues of nonhermitian oscillators, exploring PT-symmetry, resonances, and computational methods like complex rotation and Riccati-Padé, revealing how optimal rotation angles relate to oscillator symmetry and eigenvalue properties.

Contribution

It provides a detailed analysis of eigenvalues in nonhermitian oscillators, linking PT-symmetry, resonances, and computational techniques, and clarifies the effects of rotation angles on eigenvalue spectra.

Findings

Optimal rotation angles convert oscillators into PT-symmetric or Hermitian forms.

PT-symmetric oscillators have real positive eigenvalues and resonances under various boundary conditions.

Riccati-Padé method effectively computes eigenvalues and resonances of nonhermitian oscillators.

Abstract

We consider a class of one-dimensional nonhermitian oscillators and discuss the relationship between the real eigenvalues of PT-symmetric oscillators and the resonances obtained by different authors. We also show the relationship between the strong-coupling expansions for the eigenvalues of those oscillators. Comparison of the results of the complex rotation and the Riccati-Pad\'{e} methods reveals that the optimal rotation angle converts the oscillator into either a PT-symmetric or an Hermitian one. In addition to the real positive eigenvalues the PT-symmetric oscillators exhibit real positive resonances under different boundary conditions. They can be calculated by means of the straightforward diagonalization method. The Riccati-Pad\'e method yields not only the resonances of the nonhermitian oscillators but also the eigenvalues of the PT-symmetric ones.