Are Pseudo-Hermiticity and Generalized PT-Symmetry Equivalent at Exceptional Points?

• URL: http://arxiv.org/abs/2503.17687v2
• Date: Tue, 25 Mar 2025 09:21:07 GMT
• Title: Are Pseudo-Hermiticity and Generalized PT-Symmetry Equivalent at Exceptional Points?
• Authors: Nil İnce, Hasan Mermer, Ali Mostafazadeh,
• Abstract summary: We show that for a diagonalizable linear operator $H:mathscrHtomathscrH$, the pseudo-Hermiticity of $H$ is equivalent to its generalized PT-symmetry.<n>We also establish the equivalence of pseudo-Hermiticity and generalized $PT$-symmetry for arbitrary linear operators acting in a finite-dimensional Hilbert space.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: For a diagonalizable linear operator $H:\mathscr{H}\to\mathscr{H}$ acting in a separable Hilbert space $\mathscr{H}$, i.e., an operator with a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of eigenvectors that form a Reisz basis of $\mathscr{H}$, the pseudo-Hermiticity of $H$ is equivalent to its generalized PT-symmetry, where the latter means the existence of an antilinear operator $X:\mathscr{H}\to\mathscr{H}$ satisfying $[X,H]=0$ and $X^2=1$. We show that this equivalence is generally valid for block-diagonalizable operators, i.e., those which have a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of generalized eigenvectors that form a Jordan Reisz basis of $\mathscr{H}$. In particular, we establish the equivalence of pseudo-Hermiticity and generalized $PT$-symmetry for arbitrary linear operators acting in a finite-dimensional Hilbert space.

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