WKB Analysis of PT-Symmetric Sturm-Liouville problems. II

Abstract

In a previous paper it was shown that a one-turning-point WKB approximation gives an accurate picture of the spectrum of certain non-Hermitian PT-symmetric Hamiltonians on a finite interval with Dirichlet boundary conditions. Potentials to which this analysis applies include the linear potential $V=igx$ and the sinusoidal potential $V=ig\sin (\alpha x)$. However, the one-turning-point analysis fails to give the full structure of the spectrum for the cubic potential $V=ig{x}^3$, and in particular it fails to reproduce the critical points at which two real eigenvalues merge and become a complex-conjugate pair. The present paper extends the method to cases where the WKB path goes through a {\it pair} of turning points. The extended method gives an extremely accurate approximation to the spectrum of $V=ig{x}^3$, and more generally it works for potentials of the form $V=ig{x}^{2N+1}$. When applied to potentials with half-integral powers of $x$, the method again works well for one sign of the coupling, namely that for which the turning points lie on the first sheet in the lower-half plane.

11 pages, 10 figures

References in corpus (4)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• Bifurcation diagram and pattern formation in superconducting wires with electric currents
• MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

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• Metric Operator For The Non-Hermitian Hamiltonian Model and Pseudo-Supersymmetry
• Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator
• Explicit energy expansion for general odd degree polynomial potentials