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Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators. (English) Zbl 0767.35052
Summary: The authors completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi-exactly solvable Schrödinger operator on the line. Methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi-exactly solvable Hamiltonians and explicit normalizability conditions in general coordinate systems.

35P20 Asymptotic distributions of eigenvalues in context of PDEs
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
35Q40 PDEs in connection with quantum mechanics
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[1] Alhassid, Y., Engel, J., Wu, J.: Algebraic approach to the scattering matrix. Phys. Lett.53, 17–20 (1984) · doi:10.1103/PhysRevLett.53.17
[2] Alhassid, Y., Gürsey, F., Iachello, F.: Group theory approach to scattering. Ann. Phys.148, 346–380 (1983) · Zbl 0526.22018 · doi:10.1016/0003-4916(83)90244-0
[3] Alhassid, Y., Gürsey, F., Iachello, F.: Group theory approach to scattering. II. The euclidean connection. Ann. Phys.167, 181–200 (1986) · Zbl 0601.22018 · doi:10.1016/S0003-4916(86)80011-2
[4] Galindo, A., Pascual, P.: Quantum Mechanics I. Berlin, Heidelberg, New York: Springer 1990 · Zbl 0824.00008
[5] González-López, A., Kamran, N., Olver, P.J.: Lie algebras of differential operators in two complex variables. Am. J. Math. to appear · Zbl 0781.17011
[6] González-López, A., Kamran, N., Olver, P.J.: Quasi-exactly solvable Lie algebras of first order differential operators in two complex variables. J. Phys. A24, 3995–4008 (1991) · Zbl 0760.17021 · doi:10.1088/0305-4470/24/17/016
[7] González-López, A., Kamran, N., Olver, P.J.: New quasi-exactly solvable Hamiltonians in two dimensions. Preprint, Univ. of Minnesota 1992 · Zbl 0811.35106
[8] Gorsky, A.: Relationship between exactly solvable and quasi-exactly solvable versions of quantum mechanics with conformal block equations in 2D theories. JETP Lett.54, 289–292 (1991)
[9] Grace, J.H., Young, A.: The Algebra of Invariants. Cambridge: Cambridge Univ. Press 1903 · JFM 34.0114.01
[10] Gradshteyn, I.S., Ryzhik, I.W.: Table of Integrals, Series and Products. New York: Academic Press 1965 · Zbl 0918.65002
[11] Gurevich, G.B.: Foundations of the Theory of Algebraic Invariants. Groningen, Holland: P. Noordhoff 1964 · Zbl 0128.24601
[12] Kamran, N., Olver, P.J.: Lie algebras of differential operators and Lie-algebraic potentials. J. Math. Anal. Appl.145, 342–356 (1990) · Zbl 0693.34021 · doi:10.1016/0022-247X(90)90404-4
[13] Landau, L.D., Lifshitz, E.M.: Quantum Mechanics (Non-relativistic Theory). Course of Theoretical Physics, vol3. New York: Pergamon Press 1977
[14] Levine, R.D.: Lie algebraic approach to molecular structure and dynamics. In: Mathematical Frontiers in Computational Chemical Physics, D.G. Truhlar (ed.) IMA Volumes in Mathematics and its Applications, vol.15. Berlin Heidelberg, New York: Springer 1988, pp. 245–261
[15] Littlejohn, L.L.: On the classification of differential equations having orthogonal polynomial solutions. Ann. di Mat.138, 35–53 (1984) · Zbl 0571.34003 · doi:10.1007/BF01762538
[16] Miller, W., Jr.: Lie Theory and Special Functions. New York: Academic Press 1968 · Zbl 0174.10502
[17] Morozov, A.Y., Perelomov, A.M., Rosly, A.A., Shifman, M.A., Turbiner, A.V.: Quasi-exactly solvable quantal problems: one-dimensional analogue of rational conformal field theories. Int. J. Mod. Phys.5, 803–832 (1990) · Zbl 0709.58048 · doi:10.1142/S0217751X90000374
[18] Reed, M., Simon, B.: Functional Analysis, vol.2. New York: Academic Press 1975 · Zbl 0308.47002
[19] Shifman, M.A.: New findings in quantum mechanics (partial algebraization of the spectral problem). Int. J. Mod. Phys. A126, 2897–2952 (1989) · doi:10.1142/S0217751X89001151
[20] Shifman, M.A.: Quasi-exactly solvable spectral problems and conformal field theory. preprint, Theoretical Physics Inst., Univ. of Minnesota 1992.
[21] Shitman, M.A., Turbiner, A.V.: Quantal problems with partial algebraization of the spectrum. Commun. Math. Phys.126, 347–365 (1989) · Zbl 0696.35183 · doi:10.1007/BF02125129
[22] Turbiner, A.V.: Quasi-exactly solvable problems andsl(2) algebra. Commun. Math. Phys.118, 467–474 (1988) · Zbl 0683.35063 · doi:10.1007/BF01466727
[23] Turbiner, A.V.: Lie algebras and polynomials in one variable. J. Phys. A25, L1087-L1093 (1992) · Zbl 0764.34005 · doi:10.1088/0305-4470/25/18/001
[24] Ushveridze, A.G.: Quasi-exactly solvable models in quantum mechanics. Sov. J. Part. Nucl.20, 504–528 (1989)
[25] Wilczynski, E.J.: Projective Differential Geometry of Curves and Ruled Surfaces. Leipzig: B.G. Teubner 1906 · JFM 37.0620.02
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