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Quasi-exactly-solvable problems and sl(2) algebra. (English) Zbl 0683.35063
Summary: Recently discovered quasi-exactly-solvable problems of quantum mechanics are shown to be related to the existence of the finite-dimensional representation of the group SL(2,Q), where $$Q=R,C$$. It is proven that the bilinear form $$h=a_{\alpha \beta}J^{\alpha}J^{\beta}+b_{\alpha}J^{\alpha}$$ $$(J^{\alpha}$$ stand for the generators) allows one to generate a set of quasi-exactly- solvable problems of different types, including those that are already known. We get, in particular, problems in which the spectral Riemannian surface containing an infinite number of sheets is split off one or two finite-sheet pieces. In the general case the transition $$h\to H=-d^ 2/dx^ 2+V(x)$$ is realized with the aid of elliptic functions. All known exactly-solvable quantum problems with known spectrum and factorized Riemannian surface can be obtained in this approach.

##### MSC:
 35P05 General topics in linear spectral theory for PDEs 35J10 Schrödinger operator, Schrödinger equation 82B23 Exactly solvable models; Bethe ansatz
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##### References:
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