Detecting Broken PT-Symmetry

A fundamental problem in the theory of PT-invariant quantum systems is to determine whether a given system “respects” this symmetry or not. If not, the system usually develops non-real eigenvalues. It is shown in this contribution how to algorithmically detect the existence of complex eigenvalues for a given PT-symmetric matrix. The procedure uses classical results from stability theory which qualitatively locate the zeros of real polynomials in the complex plane. The interest and value of the present approach lies in the fact that it avoids diagonalization of the Hamiltonian at hand. 1 Motivation When dealing with a non-hermitean operator such as H = p + ix, (1) one needs to address two questions which do not arise for a Hermitean operator: Q1: Is the operator H is diagonalizable? Q2: Does the operator H have real eigenvalues only? For a randomly picked non-hermitean operator the answers to both questions are unlikely to be positive: it will neither have a complete set of eigenfunctions nor a real spectrum. However, operators with PT-symmetry [1], [H,PT] = 0 , (2) invariant under simultaneous application of parity P and time-reversal T, behave somewhat ‘better.’ PT-invariant operators tend to be diagonalizable but for the rare occurrence of exceptional points, and each of their eigenvalues must be either real or have a complex conjugate counterpart. Positive answers to Q1 and Q2 are necessary in order to attempt a consistent quantum mechanical interpretation of the operator H since it can be similar to a hermitean operator only then [2]. To answer these question for a given PT-invariant Hamiltonian is by no means straightforward. It is known, for example, that the operator H in Eq. (1) does have only real eigenvalues [3] while the (likely) completeness of its eigenfunctions has, apparently, not yet been established rigorously. Perturbative results allow one to confirm that the spectrum of an initially hermitean operator such as the Hamiltonian of a particle in an oscillator-type potential remains real if a sufficiently weak PT-symmetric term is added [4]. As long as no degeneracies develop, this approach also makes plausible the existence of a complete set of eigenfunctions; they are, however, not necessarily