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Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators

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Abstract

We 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.

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Author notes

  1. Peter J. Olver

    Present address: School of Mathematics, University of Minnesota, 55455, Minneapolis, Minnesota, USA

Authors and Affiliations

  1. Department de Física Teórica II, Universidad Complutense, 28040, Madrid, Spain

    Artemio González-López

  2. Department of Mathematics, McGill University, H3A 2K6, Montréal, Quebec, Canada

    Niky Kamran

  3. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Peter J. Olver

Authors
  1. Artemio González-López
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  2. Niky Kamran
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  3. Peter J. Olver
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Additional information

Communicated by B. Simon

Supported in Part by DGICYT Grant PS 89-0011

Supported in Part by an NSERC Grant

Supported in Part by NSF Grant DMS 92-04192

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González-López, A., Kamran, N. & Olver, P.J. Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators. Commun.Math. Phys. 153, 117–146 (1993). https://doi.org/10.1007/BF02099042

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  • DOI: https://doi.org/10.1007/BF02099042

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