### Abstract

We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (e.g., harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that a part of the eigenvalues, and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the hamiltonian. For one-dimensional motion this hidden symmetry is SL(2, R). It is shown that other groups lead to a partial algebraization in multidimensional quantal problems. In particular, SL(2, R)×SL(2, R), SO(3) and SL(3, R) are relevant to two-dimensional motion inducing a class of quasi-exactly-solvable two-dimensional hamiltonians. Typically they correspond to systems in a curved space, but sometimes the curvature turns out to be zero. Graded algebras open the possibility of constructing quasi-exactlysolvable hamiltonians acting on multicomponent wave functions. For example, with a (non-minimal) superextension of SL(2, R) we get a hamiltonian describing the motion of a spinor particle.

Original language | English (US) |
---|---|

Pages (from-to) | 347-365 |

Number of pages | 19 |

Journal | Communications in Mathematical Physics |

Volume | 126 |

Issue number | 2 |

DOIs | |

State | Published - Dec 1 1989 |

## Fingerprint Dive into the research topics of 'Quantal problems with partial algebraization of the spectrum'. Together they form a unique fingerprint.

## Cite this

*Communications in Mathematical Physics*,

*126*(2), 347-365. https://doi.org/10.1007/BF02125129