PT symmetry breaking and exceptional points for a class of inhomogeneous complex potentials

TL;DR

This paper investigates PT-symmetric Hamiltonians with inhomogeneous complex potentials, analyzing eigenvalue phase transitions, exceptional points, and Jordan structures, revealing novel properties and a new exactly-solvable model.

Contribution

It introduces a new class of inhomogeneous complex potentials, explores their phase diagrams, and uncovers novel properties of associated polynomials and solvable limits.

Findings

Real eigenvalues merge at exceptional points

Identification of a new exactly-solvable inhomogeneous complex square well

Insights into phase transition to infinitely-many complex eigenvalues

Abstract

We study a three-parameter family of PT-symmetric Hamiltonians, related via the ODE/IM correspondence to the Perk-Schultz models. We show that real eigenvalues merge and become complex at quadratic and cubic exceptional points, and explore the corresponding Jordon block structures by exploiting the quasi-exact solvability of a subset of the models. The mapping of the phase diagram is completed using a combination of numerical, analytical and perturbative approaches. Among other things this reveals some novel properties of the Bender-Dunne polynomials, and gives a new insight into a phase transition to infinitely-many complex eigenvalues that was first observed by Bender and Boettcher. A new exactly-solvable limit, the inhomogeneous complex square well, is also identified.