Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 015, 9 pages      arXiv:1002.0798      https://doi.org/10.3842/SIGMA.2010.015
Contribution to the Proceedings of the 5-th Microconference Analytic and Algebraic Methods V

Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

Kwang C. Shin
Department of Mathematics, University of West Georgia, Carrollton, GA, 30118, USA

Received October 11, 2009, in final form January 28, 2010; Published online February 03, 2010

Abstract
We study the eigenvalue problem −u''+V(z)u=λu in the complex plane with the boundary condition that u(z) decays to zero as z tends to infinity along the two rays arg z=−π/2± 2π(m+2), where V(z)=−(iz)m−P(iz) for complex-valued polynomials P of degree at most m−1≥2. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.

Key words: anharmonic oscillators; asymptotic formula; infinitely many real eigenvalues; PT-symmetry.

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