WKB Analysis of PT-Symmetric Sturm-Liouville problems

Abstract

Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered the Schroedinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm-Liouville problems, the eigenvalues of the PT-symmetric Sturm-Liouville problem grow like $n^2$ for large $n$. However, the novelty is that a PT eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviors of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies.

11 pages, 8 figures

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Cited by in corpus (4)

• Global stability of the normal state of superconductors in the presence of a strong electric current
• PT-symmetry breaking with divergent potentials: lattice and continuum cases
• Milne quantization for non-Hermitian systems
• Asymptotics of eigenvalue sums when some turning points are complex