NEW FINDINGS IN QUANTUM MECHANICS (PARTIAL ALGEBRAIZATION OF THE SPECTRAL PROBLEM)

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Abstract
We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (like the famous harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that an (arbitrary) 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 SU(2). The simplest one-dimensional system admitting algebraization for a part of the spectrum is the anharmonic oscillator with the x6 anharmonicity and a relation between the coefficients in front of x2 and x6. We review also more complicated cases with the emphasis on pedagogical aspects. The groups SU (2)× SU (2), SO(3) and SU(3) generate two-dimensional problems with the partial algebraization of the spectrum. Typically we get Schrödinger-type equations in curved space. An intriguing relation between the algebraic structure of the Hamiltonian and the geometry of the space emerges. Another interesting development is the use of the graded algebras which allow one to construct multi-component quasi-exactly-solvable Hamiltonians.

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