
SIGMA 6 (2010), 015, 9 pages arXiv:1002.0798
http://dx.doi.org/10.3842/SIGMA.2010.015
Contribution to the Proceedings of the 5th Microconference Analytic and Algebraic Methods V
Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PTSymmetry
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 complexvalued 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; PTsymmetry.
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