The method of Hill determinants in PT-symmetric quantum mechanics

TL;DR

This paper extends the Hill-determinant method to PT-symmetric quantum mechanics, demonstrating its effectiveness in ensuring wave function decay and enabling numerical analysis of bound states.

Contribution

It introduces a rigorous application of the Hill-determinant method within PT-symmetric quantum mechanics, including proof of its validity for numerical computations.

Findings

Method guarantees asymptotic decay of wave functions.

Allows numerical truncation of Hill-determinant series.

Applicable to bound state analysis in PT-symmetric systems.

Abstract

Hill-determinant method is described and shown applicable within the so called PT-symmetric quantum mechanics. We demonstrate that in a way paralleling its traditional Hermitian applications and proofs the method guarantees the necessary asymptotic decrease of wave functions as resulting from a fine-tuned mutual cancellation of their asymptotically growing exponential components. Technically, the rigorous proof is needed/offered that in a quasi-variational spirit the method allows us to work, in its numerical implementations, with a sequence of truncated forms of the rigorous Hill-determinant power series for the normalizable bound states.