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Communications in Mathematical Physics

, Volume 126, Issue 2, pp 347–365 | Cite as

Quantal problems with partial algebraization of the spectrum

  • M. A. Shifman
  • A. V. Turbiner
Article

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 isSL(2,R). It is shown that other groups lead to a partial algebraization in multidimensional quantal problems. In particular,SL(2,RSL(2,R),SO(3) andSL(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 ofSL(2,R) we get a hamiltonian describing the motion of a spinor particle.

Keywords

Neural Network Statistical Physic Wave Function Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • M. A. Shifman
    • 1
  • A. V. Turbiner
    • 1
  1. 1.Institute of Theoretical and Experimental PhysicsMoscowUSSR

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