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Article Dans Une Revue St. Petersburg Mathematical Journal Année : 2018

Passage through a potential barrier and multiple wells

Résumé

Consider the semiclassical limit, as the Planck constant $\hbar\ri 0$, of bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. We show that, for each eigenvalue of the Schr\"odinger operator, the Bohr-Sommerfeld quantization condition is satisfied at least for one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. We show that, at least from one side, the barrier fixes the phase of wave functions in the same way as a potential barrier of infinite width. On the other hand, it turns out that for each well there exists an eigenvalue in a small neighborhood of every point satisfying the Bohr-Sommerfeld condition.

Dates et versions

hal-01397307 , version 1 (15-11-2016)

Identifiants

Citer

Dimitri R Yafaev. Passage through a potential barrier and multiple wells. St. Petersburg Mathematical Journal, 2018, 29 (2), pp.399-422. ⟨10.1090/spmj/1499⟩. ⟨hal-01397307⟩
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