Quantal problems with partial algebraization of the spectrum

M. A. Shifman, A. V. Turbiner

Research output: Contribution to journalArticle

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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 languageEnglish (US)
Pages (from-to)347-365
Number of pages19
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - Dec 1 1989

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