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Day March 27, 2012

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

Carl M. Bender, Hugh F. Jones

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 \(V=igx^3\), 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 pair of turning points. The extended method gives an extremely accurate approximation to the spectrum of \(V=igx^3\), and more generally it works for potentials of the form \(V=igx^{2N+1}\). 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.
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)