Pseudo-Hermiticity, Anti-Pseudo-Hermiticity, and Generalized Parity-Time-Reversal Symmetry at Exceptional Points
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914001388044288 |
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| author | İnce, Nil Mermer, Hasan Mostafazadeh, Ali |
| author_facet | İnce, Nil Mermer, Hasan Mostafazadeh, Ali |
| contents | 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 parity-time-reversal ($PT$) symmetry, where the latter means the existence of an antilinear operator $X:\mathscr{H}\to\mathscr{H}$ satisfying $[X,H]=0$ and $X^2=1$. {The original proof of this result makes use of the anti-pesudo-Hermiticity of every diagonalizable operator $L:\mathscr{H}\to\mathscr{H}$, which means the existence of an antilinear Hermitian bijection $τ:\mathscr{H}\to\mathscr{H}$ satisfying $L^\dagger=τL\,τ^{-1}$. We establish the validity of this result 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 $\mathscr{H}$. {This allows us to generalize the original proof of the equivalence of pseudo-Hermiticity and generalized $PT$-symmetry for diagonalizable operators to block-diagonalizable operators. For a pair of pseudo-Hermitian operators acting respectively in two-dimensional and infinite-dimensional Hilbert spaces, we obtain explicit expressions for the antlinear operators $τ$ and $X$ that realize their anti-pseudo-Hermiticity and generalized $PT$-symmetry at and away from the exceptional points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17687 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudo-Hermiticity, Anti-Pseudo-Hermiticity, and Generalized Parity-Time-Reversal Symmetry at Exceptional Points İnce, Nil Mermer, Hasan Mostafazadeh, Ali Mathematical Physics Quantum Physics 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 parity-time-reversal ($PT$) symmetry, where the latter means the existence of an antilinear operator $X:\mathscr{H}\to\mathscr{H}$ satisfying $[X,H]=0$ and $X^2=1$. {The original proof of this result makes use of the anti-pesudo-Hermiticity of every diagonalizable operator $L:\mathscr{H}\to\mathscr{H}$, which means the existence of an antilinear Hermitian bijection $τ:\mathscr{H}\to\mathscr{H}$ satisfying $L^\dagger=τL\,τ^{-1}$. We establish the validity of this result 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 $\mathscr{H}$. {This allows us to generalize the original proof of the equivalence of pseudo-Hermiticity and generalized $PT$-symmetry for diagonalizable operators to block-diagonalizable operators. For a pair of pseudo-Hermitian operators acting respectively in two-dimensional and infinite-dimensional Hilbert spaces, we obtain explicit expressions for the antlinear operators $τ$ and $X$ that realize their anti-pseudo-Hermiticity and generalized $PT$-symmetry at and away from the exceptional points. |
| title | Pseudo-Hermiticity, Anti-Pseudo-Hermiticity, and Generalized Parity-Time-Reversal Symmetry at Exceptional Points |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2503.17687 |