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# Title: Quasi-exactly solvable quartic potential

Authors: Carl M. Bender (Dept. of Physics, Washington University), Stefan Boettcher (CNLS, Los Alamos and CTSPS, Clark Atlanta University)
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.
 Comments: 3 Pages, RevTex, 1 Figure, encapsulated postscript Subjects: Mathematical Physics (math-ph); Condensed Matter (cond-mat); High Energy Physics - Theory (hep-th); Exactly Solvable and Integrable Systems (nlin.SI); Quantum Physics (quant-ph) Journal reference: J.Phys. A31 (1998) L273-L277 DOI: 10.1088/0305-4470/31/14/001 Cite as: arXiv:physics/9801007 [math-ph] (or arXiv:physics/9801007v1 [math-ph] for this version)

## Submission history

From: Stefan Boettcher [view email]
[v1] Thu, 8 Jan 1998 04:35:16 GMT (10kb)

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