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arXiv · 1002.0798

Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

Kwang C. Shin
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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 $\arg z=-\fracπ{2} \pm \frac{2π}{m+2}$, where $V(z)=-(iz)^m-P(iz)$ for complex-valued polynomials $P$ of degree at most $m-1\geq 2$. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.

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BibTeXRIS

Kwang C. Shin. 2010-02-03. Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry. https://doi.org/10.3842/sigma.2010.015

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