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On the Reality of the Eigenvalues for a Class of -Symmetric Oscillators
Authors:K. C. Shin
Affiliation:(1) Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. E-mail: kcshin@math.uiuc.edu, US
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. Received: 16 January 2002 / Accepted: 1 May 2002 Published online: 6 August 2002
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