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


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.


Neural Network Statistical Physic Wave Function Complex System Nonlinear Dynamics 


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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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