Abstract. A new two-parameter family of quasi-exactly solvable quartic polynomial potentials is introduced. Heretofore, 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, -symmetric Hamiltonians whose spectra are real, discrete, and bounded below. Replacing hermiticity by the weaker condition of 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 Jth-degree polynomial.

Print publication: Issue 14 (10 April 1998)
Received 19 January 1998