Generation of families of spectra in PT-symmetric quantum mechanics and scalar bosonic field theory

TL;DR

This paper systematically explores the generation of spectral families in PT-symmetric quantum mechanics and scalar field theory, highlighting similarities, differences, and the role of Stokes' wedges using WKB approximation.

Contribution

It introduces a method to determine the number of spectral families based on Stokes' wedges and compares quantum-mechanical and field-theoretical cases.

Findings

Number of spectral families relates to noncontiguous PT-symmetric Stokes' wedges.

Eigenvalues are real in certain regions as shown by WKB approximation.

Differences between quantum mechanics and field theory lie in accessible decay regions.

Abstract

This paper explains the systematics of the generation of families of spectra for the PT-symmetric quantum-mechanical Hamiltonians $H=p^2+x^2(ix)^\epsilon$, $H=p^2+(x^2)^\delta$, and $H=p^2-(x^2)^\mu$. In addition, it contrasts the results obtained with those found for a bosonic scalar field theory, in particular in one dimension, highlighting the similarities and differences to the quantum-mechanical case. It is shown that the number of families of spectra can be deduced from the number of noncontiguous pairs of Stokes' wedges that display PT-symmetry. To do so, simple arguments that use the WKB approximation are employed, and these imply that the eigenvalues are real. However, definitive results are in most cases presently only obtainable numerically, and not all eigenvalues in each family may be real. Within the approximations used, it is illustrated that the difference between the quantum-mechanical and the field-theoretical cases lies in the number of accessible regions in which the eigenfunctions decay exponentially. This paper reviews and implements well-known techniques in complex analysis and PT-symmetric quantum theory.