$\mathcal{PT}$-symmetric strings

TL;DR

This paper investigates the spectral properties of inhomogeneous $ ext{PT}$-symmetric strings, providing exact solutions for specific models, deriving sum rules for eigenvalues, and numerically analyzing eigenvalue transitions from real to complex.

Contribution

It introduces an exactly solvable $ ext{PT}$-symmetric string model and derives explicit sum rules for eigenvalues in more general cases, advancing understanding of spectral behavior.

Findings

Exact isospectral $ ext{PT}$-symmetric string model identified.

Explicit sum rules for eigenvalues derived for general $ ext{PT}$-symmetric strings.

Numerical estimates of critical parameters where eigenvalues become complex obtained.

Abstract

We study both analytically and numerically the spectrum of inhomogeneous strings with $\mathcal{PT}$-symmetric density. We discuss an exactly solvable model of $\mathcal{PT}$-symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules $Z(p) \equiv \sum_{n=1}^\infty 1/E_n^p$, with $p=1,2,\dots$ and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex.