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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.

MSC:
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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