dorsal/arxiv
View SchemaPseudo-Hermitian Description of PT-Symmetric Systems Defined on a Complex Contour
| Authors | Ali Mostafazadeh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0410012 |
| URL | https://arxiv.org/abs/quant-ph/0410012 |
| Journal | J.Phys. A38 (2005) 3213-3234 |
Abstract
We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT-symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians $H=p^2+x^2(ix)^\nu$ with $\nu\in(-2,\infty)$ that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of $H$ for the cases that $\nu$ is an even integer, so that $H=p^2\pm x^{2+\nu}$, and give a proof of the discreteness of the spectrum of $H$ for all $\nu\in(-2,\infty)$. Furthermore, we study the consequences of defining a square-well Hamiltonian on a wedge-shaped complex contour. This yields a PT-symmetric system with a finite number of real eigenvalues. We present a comprehensive analysis of this system within the framework of pseudo-Hermitian quantum mechanics. We also outline a direct pseudo-Hermitian treatment of PT-symmetric systems defined on a complex contour which clarifies the underlying mathematical structure of the formulation of PT-symmetric quantum mechanics based on the charge-conjugation operator. Our results provide a conclusive evidence that pseudo-Hermitian quantum mechanics provides a complete description of general PT-symmetric systems regardless of whether they are defined along the real line or a complex contour.
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"abstract": "We describe a method that allows for a practical application of the theory of\npseudo-Hermitian operators to PT-symmetric systems defined on a complex\ncontour. We apply this method to study the Hamiltonians $H=p^2+x^2(ix)^\\nu$\nwith $\\nu\\in(-2,\\infty)$ that are defined along the corresponding anti-Stokes\nlines. In particular, we reveal the intrinsic non-Hermiticity of $H$ for the\ncases that $\\nu$ is an even integer, so that $H=p^2\\pm x^{2+\\nu}$, and give a\nproof of the discreteness of the spectrum of $H$ for all $\\nu\\in(-2,\\infty)$.\nFurthermore, we study the consequences of defining a square-well Hamiltonian on\na wedge-shaped complex contour. This yields a PT-symmetric system with a finite\nnumber of real eigenvalues. We present a comprehensive analysis of this system\nwithin the framework of pseudo-Hermitian quantum mechanics. We also outline a\ndirect pseudo-Hermitian treatment of PT-symmetric systems defined on a complex\ncontour which clarifies the underlying mathematical structure of the\nformulation of PT-symmetric quantum mechanics based on the charge-conjugation\noperator. Our results provide a conclusive evidence that pseudo-Hermitian\nquantum mechanics provides a complete description of general PT-symmetric\nsystems regardless of whether they are defined along the real line or a complex\ncontour.",
"arxiv_id": "quant-ph/0410012",
"authors": [
"Ali Mostafazadeh"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.MP"
],
"journal_ref": "J.Phys. A38 (2005) 3213-3234",
"title": "Pseudo-Hermitian Description of PT-Symmetric Systems Defined on a Complex Contour",
"url": "https://arxiv.org/abs/quant-ph/0410012"
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