dorsal/arxiv
View SchemaQuasi-exactly solvable quartic potential
| Authors | Carl M. Bender, Stefan Boettcher |
|---|---|
| Categories | |
| ArXiv ID | physics/9801007 |
| URL | https://arxiv.org/abs/physics/9801007 |
| DOI | 10.1088/0305-4470/31/14/001 |
| Journal | J.Phys. A31 (1998) L273-L277 |
Abstract
A new two-parameter family of quasi-exactly solvable quartic polynomial potentials $V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x$ is introduced. Until now, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial potential is sextic. This belief is based on the assumption that the Hamiltonian must be Hermitian. However, it has recently been discovered that there are huge classes of non-Hermitian, ${\cal PT}$-symmetric Hamiltonians whose spectra are real, discrete, and bounded below [physics/9712001]. Replacing Hermiticity by the weaker condition of ${\cal PT}$ symmetry allows for new kinds of quasi-exactly solvable theories. The spectra of this family of quartic potentials discussed here are also real, discrete, and bounded below, and the quasi-exact portion of the spectra consists of the lowest $J$ eigenvalues. These eigenvalues are the roots of a $J$th-degree polynomial.
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"abstract": "A new two-parameter family of quasi-exactly solvable quartic polynomial\npotentials $V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x$ is introduced. Until now,\nit was believed that the lowest-degree one-dimensional quasi-exactly solvable\npolynomial potential is sextic. This belief is based on the assumption that the\nHamiltonian must be Hermitian. However, it has recently been discovered that\nthere are huge classes of non-Hermitian, ${\\cal PT}$-symmetric Hamiltonians\nwhose spectra are real, discrete, and bounded below [physics/9712001].\nReplacing Hermiticity by the weaker condition of ${\\cal PT}$ symmetry allows\nfor new kinds of quasi-exactly solvable theories. The spectra of this family of\nquartic potentials discussed here are also real, discrete, and bounded below,\nand the quasi-exact portion of the spectra consists of the lowest $J$\neigenvalues. These eigenvalues are the roots of a $J$th-degree polynomial.",
"arxiv_id": "physics/9801007",
"authors": [
"Carl M. Bender",
"Stefan Boettcher"
],
"categories": [
"math-ph",
"cond-mat",
"hep-th",
"math.MP",
"nlin.SI",
"quant-ph"
],
"doi": "10.1088/0305-4470/31/14/001",
"journal_ref": "J.Phys. A31 (1998) L273-L277",
"title": "Quasi-exactly solvable quartic potential",
"url": "https://arxiv.org/abs/physics/9801007"
},
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