Advertisement

Communications in Mathematical Physics

, Volume 229, Issue 3, pp 543–564 | Cite as

On the Reality of the Eigenvalues for a Class of -Symmetric Oscillators

  • K. C. Shin

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.

Keywords

Boundary Condition Real Axis Positive Real Negative Real Axis Symmetric Oscillator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • K. C. Shin
    • 1
  1. 1.Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. E-mail: kcshin@math.uiuc.eduUS

Personalised recommendations