$PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum

TL;DR

This paper investigates PT-symmetric non-selfadjoint operators with real discrete spectra, demonstrating conditions for reality and discreteness, and providing examples of non-diagonalizable operators with real spectra.

Contribution

It establishes explicit bounds for the reality of spectra in PT-symmetric operators and presents novel examples of non-diagonalizable operators with real spectra.

Findings

Spectrum of $H(g)$ is real and discrete for $|g|< ho$

Explicitly determined constant $ ho$ for spectral reality

Example of a non-diagonalizable operator with real spectrum

Abstract

Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. $\ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2$ is the quantum harmonic oscillator with rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real coupling constant. We show that if $|g|<\rho$, $\rho$ being an explicitly determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we show that the operator $\ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1$ has real discrete spectrum but is not diagonalizable.