Quadratic PT-symmetric operators with real spectrum and similarity to self-adjoint operators

TL;DR

This paper characterizes when PT-symmetric elliptic quadratic differential operators with real spectra are similar to self-adjoint operators, linking this property to the structure of the associated fundamental matrix.

Contribution

It provides a precise criterion connecting the similarity to self-adjointness with the absence of Jordan blocks in the fundamental matrix.

Findings

Operators are similar to self-adjoint operators if and only if the fundamental matrix has no Jordan blocks.

The paper establishes a clear condition for the spectral reality of PT-symmetric quadratic operators.

It advances understanding of the spectral properties of PT-symmetric operators in mathematical physics.

Abstract

It is established that a PT-symmetric elliptic quadratic differential operator with real spectrum is similar to a self-adjoint operator precisely when the associated fundamental matrix has no Jordan blocks.