New types of solvability in PT symmetric quantum theory

TL;DR

This paper explores new solvability features in PT symmetric quantum theory, introducing semi-exact solvability and variational methods for non-Hermitian Hamiltonians with real spectra, with implications for superintegrability.

Contribution

It introduces two novel forms of solvability—semi-exact solvability and variational tractability—in PT symmetric quantum systems, expanding understanding of non-Hermitian Hamiltonians.

Findings

Description of semi-exact solvability (SES) in PT symmetric systems

Development of variational methods for non-Hermitian Hamiltonians

Discussion of potential links between PT symmetry, solvability, and superintegrability

Abstract

The characteristic anti-linear (parity/time reversal, PT) symmetry of non-Hermitian Hamiltonians with real energies is presented as a source of two new forms of solvability of Schr\"{o}dinger's bound-state problems. In detail we describe (1) their very specific semi-exact solvability (SES) and (2) their innovated variational tractability. SES technicalities are discussed via charged oscillator example. In a broader context, speculations are added concerning possible relationship between PT symmetry, solvability and superintegrability.