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Pankratova, T. F.

Hermite polynomial ansatz for a multidimensional well. (English. Russian original) Zbl 0930.35049

J. Math. Sci., New York 86, No. 3, 2755-2765 (1997); translation from Zap. Nauchn. Semin. POMI 218, 149-165 (1994).
Summary: The Schrödinger operator in \(\mathbb{R}^d\) with analytic potential that has a nondegenerate minimum (a well) at the origin is considered. Under the additional Diophantine condition on the frequencies, the full asymptotic expansions (as Planck’s constant \(h\) tends to zero) of a set of eigenfunctions (Ansatz with Hermite polynomials) and of eigenvalues with given quantum numbers (\(n\in\mathbb{N}^d\), \(|n|= 0,1,2,\dots\)) located at the bottom of the potential well are constructed in a neighborhood of the origin, which is independent of \(h\). The asymptotics obtained can be prolonged onto a larger domain by using the ray method. A method of approximately describing the zero-sets of eigenfunctions (and of their intersections) is discussed. Some simple examples in the two-dimensional case are considered.

MSC:

35J10 Schrödinger operator, Schrödinger equation
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35C20 Asymptotic expansions of solutions to PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory

Keywords:

analytic potential; nondegenerate minimum; Diophantine condition; ray method

Citations:

Zbl 0924.00023
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Full Text: DOI EuDML

References:

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