$\mathcal{P}\mathcal{T}$-symmetric strings

Abstract

We study both analytically and numerically the spectrum of inhomogeneous strings with $\mathcal{P}\mathcal{T}$-symmetric density. We discuss an exactly solvable model of $\mathcal{P}\mathcal{T}$-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.

13 pages, 7 figures

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Cited by in corpus (2)

• Exact sum rules for inhomogeneous systems containing a zero mode
• "Striped" Rectangular Rigid Box with Hermitian and non-Hermitian $\mathcal{P}\mathcal{T}$ Symmetric Potentials