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

Abstract

In this paper, A. Avila's theoremon convergence of the exact quantization scheme of A.~Vo\-rosis related to the reality proofs of eigenvalues of certain $PT$-symmetricboundary value problems.As a result, a special caseof a conjecture of C. Bender, S. Boettcherand 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.}\

On solutions of the Bethe Ansatz for the Quantum KdV model

“…9 See [17] for a recent application of this method our doubts. We are also grateful to Giordano Cotti for having pointed out to us references [21,9] .…”