Complex WKB Analysis of a PT Symmetric Eigenvalue Problem

Abstract

The spectra of a particular class of PT symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper we present an explanation for these spectral properties in terms of quantisation conditions obtained from the complex WKB method. In particular, we consider the relation of the quantisation conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al. in [1,2].

22 pages, 13 figures. Added references, minor revisions

References in corpus (10)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• PT-Symmetric Quantum Mechanics
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Supersymmetry and the spontaneous breakdown of PT symmetry
• The ODE/IM Correspondence
• Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations
• Spectral determinants for Schroedinger equation and Q-operators of Conformal Field Theory
• On the relation between Stokes multipliers and the T-Q systems of conformal field theory
• Functional relations in Stokes multipliers - Fun with $x^6+\alpha x^2$ potential-
• Aspects of the ODE/IM correspondence

Cited by in corpus (2)

• PT symmetry breaking and exceptional points for a class of inhomogeneous complex potentials
• Eigenvalue and Eigenfunction for the $PT$-symmetric Potential $V=-(ix{)}^N$