Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

Abstract

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 $m-1\ge 2$. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues.

References in corpus (4)

• $PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum
• Scattering in the PT-symmetric Coulomb potential
• Asymptotically vanishing PT-symmetric potentials and negative-mass Schroedinger equations
• Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra

Cited by in corpus (1)

• Anharmonic oscillators in the complex plane, $\mathcal{P}\mathcal{T}$-symmetry, and real eigenvalues