Quasi-exactly solvable quartic potential
Bender, Carl M. ; Boettcher, Stefan
arXiv, 9801007 / Harvested from arXiv
Text on arXiv
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.
Publié le : 1998-01-07
Classification:  Mathematical Physics,  Condensed Matter,  High Energy Physics - Theory,  Nonlinear Sciences - Exactly Solvable and Integrable Systems,  Quantum Physics 
@article{9801007,
     author = {Bender, Carl M. and Boettcher, Stefan},
     title = {Quasi-exactly solvable quartic potential},
     journal = {arXiv},
     volume = {1998},
     number = {0},
     year = {1998},
     language = {en},
     url = {http://dml.mathdoc.fr/item/9801007}
}

Bender, Carl M.; Boettcher, Stefan. Quasi-exactly solvable quartic potential. arXiv, Tome 1998 (1998) no. 0, . http://gdmltest.u-ga.fr/item/9801007/