dorsal/arxiv
View SchemaAn Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
| Authors | H. F. Jones, J. Mateo |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0601188 |
| URL | https://arxiv.org/abs/quant-ph/0601188 |
| DOI | 10.1103/PhysRevD.73.085002 |
| Journal | Phys.Rev.D73:085002,2006 |
Abstract
The potential -x^4, which is unbounded below on the real line, can give rise to a well-posed bound state problem when x is taken on a contour in the lower-half complex plane. It is then PT-symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and integral systems, was subsequently constructed for the general class of potentials -(ix)^N. For PT-symmetric but non-Hermitian Hamiltonians the natural PT metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator Q. Further, with the help of this operator an equivalent Hermitian Hamiltonian h can be constructed. This programme has been carried out exactly for a few soluble models, and the first few terms of a perturbative expansion have been found for the potential m^2x^2+igx^3. However, until now, the -x^4 potential has proved intractable. In the present paper we give explicit, closed-form expressions for Q and h, which are made possible by a particular parametrization of the contour in the complex plane on which the problem is defined. This constitutes an explicit proof of the reality of the spectrum. The resulting equivalent Hamiltonian has a potential with a positive quartic term together with a linear term.
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"abstract": "The potential -x^4, which is unbounded below on the real line, can give rise\nto a well-posed bound state problem when x is taken on a contour in the\nlower-half complex plane. It is then PT-symmetric rather than Hermitian.\nNonetheless it has been shown numerically to have a real spectrum, and a proof\nof reality, involving the correspondence between ordinary differential\nequations and integral systems, was subsequently constructed for the general\nclass of potentials -(ix)^N. For PT-symmetric but non-Hermitian Hamiltonians\nthe natural PT metric is not positive definite, but a dynamically-defined\npositive-definite metric can be defined, depending on an operator Q. Further,\nwith the help of this operator an equivalent Hermitian Hamiltonian h can be\nconstructed. This programme has been carried out exactly for a few soluble\nmodels, and the first few terms of a perturbative expansion have been found for\nthe potential m^2x^2+igx^3. However, until now, the -x^4 potential has proved\nintractable. In the present paper we give explicit, closed-form expressions for\nQ and h, which are made possible by a particular parametrization of the contour\nin the complex plane on which the problem is defined. This constitutes an\nexplicit proof of the reality of the spectrum. The resulting equivalent\nHamiltonian has a potential with a positive quartic term together with a linear\nterm.",
"arxiv_id": "quant-ph/0601188",
"authors": [
"H. F. Jones",
"J. Mateo"
],
"categories": [
"quant-ph",
"hep-th"
],
"doi": "10.1103/PhysRevD.73.085002",
"journal_ref": "Phys.Rev.D73:085002,2006",
"title": "An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential",
"url": "https://arxiv.org/abs/quant-ph/0601188"
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