On the Reality of the Eigenvalues for a Class of -Symmetric Oscillators
By Öö Þ ¾ ¦ ¾ Ñ·¾¸ûûûöö È ´þµ ½ Þ Ñ ½ · ¾ Þ Ñ ¾ · ¡ ¡ ¡ Öööð, Ôóðýòóñññð Òò, Ñ º, Ï Ôöóú, Øøøø Óö ×óññ ½ Ñ ¾ ¸ Û Ú ´ Μ ¼ Óö Ðð, È þµ ½ Þ Ñ ½ · ¾ Þ Ñ ¾ · ¡ ¡ ¡ · Ñ ½ Þ Ûûøø Ðð. 2002
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Abstract We study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays , where is a real polynomial and . We prove that if for some we have for all , then the eigenvalues are all positive real. We then sharpen this to a larger class of polynomial potentials. In particular, this implies that the eigenvalues are all positive real for the potentials when with , and with the boundary conditions that decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin.
Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/2c6af56df9e1149e8e2521bec7d5d690b29ca62b
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In Atomic and Molecular Physics, and Optics
In Statistical and Nonlinear Physics
Citations in the corpus
- The ODE/IM correspondence (2001, background): “Finally, we remark that Shin has extended the Bethe-ansatz-inspired ideas of the main proof given above to establish spectral reality for a somewhat wider class of nonHermitian potentials [89,92].”
- Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries (2002, background): “The main reason for the recent interest in PT -symmetry [1] is that the eigenvalues of every PT -symmetric Hamiltonian are real or come in complex conjugate pairs.”