WKB analysis of PT-symmetric Sturm-Liouville problems

Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered the Schrodinger 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 eige...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2012-11, Vol.45 (44), p.1-11
Main Authors: Bender, Carl M, Jones, Hugh F
Format: Article
Language:English
Subjects:
Asymptotic properties
Boundary conditions
Conjugates
Critical point
Eigenvalues
Mathematical analysis
Planes
Platinum
Schroedinger equation
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Summary:Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered the Schrodinger 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 super(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 behaviours of the real eigenvalues and the locations of the critical points. Hie method turns out to be surprisingly accurate even at low energies.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/45/44/444004