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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
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