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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Passage through a potential barrier and multiple wells

Author: D. R. Yafaev
Original publication: Algebra i Analiz, tom 29 (2017), nomer 2.
Journal: St. Petersburg Math. J. 29 (2018), 399-422
MSC (2010): Primary 47A40, 81U05
Published electronically: March 12, 2018
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Abstract: The semiclassical limit as the Planck constant $ \hbar $ tends to 0 is considered for bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. It is shown that, for each eigenvalue of the Schrödinger operator, the Bohr-Sommerfeld quantization condition is satisfied for at least one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. It is shown that, at least from one side, the barrier fixes the phase of the 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.

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Additional Information

D. R. Yafaev
Affiliation: IRMAR, Université de Rennes I, Campus de Beaulieu, Rennes 35042, France; St. Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia

Keywords: Schr\"odinger equation, multiple potential wells, Bohr--Sommerfeld quantization conditions, fixing conditions
Received by editor(s): October 15, 2016
Published electronically: March 12, 2018
Dedicated: To the memory of Vladimir Savel’evich Buslaev
Article copyright: © Copyright 2018 American Mathematical Society

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