On the Reality of the Eigenvalues for a Class of -Symmetric Oscillators
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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.
KeywordsBoundary Condition Real Axis Positive Real Negative Real Axis Symmetric Oscillator
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