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Semiclassical eigenstates in a multidimensional well. (English) Zbl 0937.35511
Séminaire de théorie spectrale et géométrie. Année 1992-1993. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Math. Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 11, 147-155 (1993).
Summary: The two-dimensional Schrödinger operator with an analytic potential, having a non-degenerated minimum (well) at the origin, is considered. Under the Diophantine condition on the frequencies, the full asymptotic series (the Planck constant $$\hbar$$ tending to zero) for eigenfunctions with given quantum numbers $$(n_1, n_2)$$, concentrated at the bottom of the well, is constructed, the Gaussian-like asymptotics being valid in a neighbourhood of the origin which is independent of $$\hbar$$. For small quantum numbers the second approximation to the eigenvalues is written in terms of the derivatives of the potential.
For the entire collection see [Zbl 0812.00008].
##### MSC:
 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs 35J10 Schrödinger operator, Schrödinger equation
##### Keywords:
Gaussian-like asymptotics
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