Entire functions, PT-symmetry and Voros’s quantization scheme

A.E. Eremenko

doi:10.30970/ms.54.2.203-210

A.E. Eremenko Department of Mathematics, Purdue University West Lafayette, IN

DOI: https://doi.org/10.30970/ms.54.2.203-210

Keywords: entire functions; P T -symmetry; exact quantization scheme; Stokes multipliers

Abstract

In this paper, A. Avila's theorem
on convergence of the exact quantization scheme of A.~Vo\-ros
is related to the reality proofs of eigenvalues of certain $PT$-symmetric
boundary value problems.
As a result, a special case
of a conjecture of C. Bender, S. Boettcher
and P. Meisinger on reality of eigenvalues is proved.
In particular the following Theorem~2 is proved:
{\sl Consider the eigenvalue problem
$$-w''+(-1)^\ell(iz)^mw=\lambda w,$$
where $m\geq 2$ is real, and $(iz)^m$ is the principal branch,
$(iz)^m>0$ when $z$ is on the negative imaginary ray,
with boundary conditions $w(te^{i\beta})\to 0,\ t\to\infty,$
where
$ \beta=\pi/2\pm\frac{\ell+1}{m+2}\pi.$
If $\ell=2$, and $m\geq 4$, then all eigenvalues are positive.
}\

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