Passage through a potential barrier and multiple wells

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Passage through a potential barrier and multiple wells. / Яфаев, Дмитрий Рауэльевич.

In: St. Petersburg Mathematical Journal, Vol. 29, No. 2, 2018, p. 399-422.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{9768b5afbed947af8ea4deca00ad3685,

title = "Passage through a potential barrier and multiple wells",

abstract = "The semiclassical limit as the Planck constant h 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 Schr{\"o}dinger 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.",

keywords = "Bohr-Sommerfeld quantization conditions, Fixing conditions, Multiple potential wells, Schr{\"o}dinger equation",

author = "Яфаев, {Дмитрий Рауэльевич}",

year = "2018",

doi = "10.1090/spmj/1499",

language = "English",

volume = "29",

pages = "399--422",

journal = "St. Petersburg Mathematical Journal",

issn = "1061-0022",

publisher = "American Mathematical Society",

number = "2",

}

RIS

TY - JOUR

T1 - Passage through a potential barrier and multiple wells

AU - Яфаев, Дмитрий Рауэльевич

PY - 2018

Y1 - 2018

N2 - The semiclassical limit as the Planck constant h 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.

AB - The semiclassical limit as the Planck constant h 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.

KW - Bohr-Sommerfeld quantization conditions

KW - Fixing conditions

KW - Multiple potential wells

KW - Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=85043536045&partnerID=8YFLogxK

U2 - 10.1090/spmj/1499

DO - 10.1090/spmj/1499

M3 - Article

VL - 29

SP - 399

EP - 422

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 2

ER -

ID: 36667950