On the reality of the eigenvalues for a class of PT-symmetric oscillators

Abstract

We study the eigenvalue problem -u"(z)-[(iz)^m+P(iz)]u(z)=λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays \arg z=-\fracπ{2}\pm \frac{2π}{m+2}, where P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z is a real polynomial and m\geq 2. We prove that if for some 1\leq j\leq\frac{m}{2}, we have (j-k)a_k\geq 0 for all 1\leq k\leq m-1, then the eigenvalues are all positive real. We then sharpen this to a slightly larger class of polynomial potentials. In particular, this implies that the eigenvalues are all positive real for the potentials αiz^3+βz^2+γiz when α,βand γare all real with α\not=0 and αγ\geq 0, and with the boundary conditions that u(z) decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin.

22 pages, one figure. Reason for update--in order to more clearly explain which parts of the proof follow the earlier work of Dorey, Dunning and Tateo. Main change on page 3, minor changes on page 12 and 18

References in corpus (1)

• A Class of Exactly-Solvable Eigenvalue Problems

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