Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

Details Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

2010-02-03

arxiv.org/abs/1002.0798v1

SIGMA 6 (2010), 015, 9 pages

Physics

Mathematical Physics

Scientific paper

10.3842/SIGMA.2010.015

We study the eigenvalue problem $-u"+V(z)u=\lambda u$ in the complex plane with the boundary condition that $u(z)$ decays to zero as $z$ tends to infinity along the two rays $\arg z=-\frac{\pi}{2} \pm \frac{2\pi}{m+2}$, 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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