Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra

Abstract

The spectrum of the Hermitian Hamiltonian $H=p^2+V(x)$ is real and discrete if the potential $V(x)\to \infty$ as $x\to \pm \infty$. However, if $V(x)$ is complex and PT-symmetric, it is conjectured that, except in rare special cases, $V(x)$ must be analytic in order to have a real spectrum. This conjecture is demonstrated by using the potential $V(x)=(ix{)}^a|x{|}^b$, where $a,b$ are real.

8 Pages, 2 figures

References in corpus (5)

• Making Sense of Non-Hermitian Hamiltonians
• The ODE/IM Correspondence
• Interactions of Hermitian and non-Hermitian Hamiltonians
• Fragile PT-symmetry in a solvable model
• PT-symmetric deformations of Calogero models

Cited by in corpus (4)

• Real-Time Transport in Open Quantum Systems From $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• A new solvable complex PT-symmetric potential
• PT-symmetric dynamical confinement: Fermi acceleration, quantum force and Berry phase
• Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry