Criteria For The Absence And Existence Of Bounded Solutions At The Threshold Frequency In A Junction Of Quantum Waveguides

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Criteria For The Absence And Existence Of Bounded Solutions At The Threshold Frequency In A Junction Of Quantum Waveguides. / Bakharev, F. L.; Nazarov, S. A.

In: St. Petersburg Mathematical Journal, Vol. 32, No. 6, 2021, p. 955-973.

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

BibTeX

@article{7c396490c9e840a987bedd7300640206,

title = "Criteria For The Absence And Existence Of Bounded Solutions At The Threshold Frequency In A Junction Of Quantum Waveguides",

abstract = "In the junction Ω of several semi-infinite cylindrical waveguides, the Dirichlet Laplacian is treated whose continuous spectrum is the ray [λ†,+∞) with a positive cutoff value λ†. Two different criteria are presented for the threshold resonance generated by nontrivial bounded solutions to the Dirichlet problem for the Helmholtz equation −Δu = λ†u in Ω. The first criterion is quite simple andis convenient to disprove the existence of bounded solutions. The second criterion is rather involved but can help to detect concrete shapes supporting the resonance. Moreover, the latter distinguishes in a natural way between stabilizing, i.e., bounded but nondecaying solutions, and trapped modes with exponential decay at infinity",

keywords = "criteria for threshold resonances, Junction of quantum waveguides, stabilizing solutions, trapped waves, CONVERGENCE, SPECTRA",

author = "Bakharev, {F. L.} and Nazarov, {S. A.}",

note = "Publisher Copyright: {\textcopyright} 2021. American Mathematical Society",

year = "2021",

doi = "10.1090/spmj/1679",

language = "English",

volume = "32",

pages = "955--973",

journal = "St. Petersburg Mathematical Journal",

issn = "1061-0022",

publisher = "American Mathematical Society",

number = "6",

}

RIS

TY - JOUR

T1 - Criteria For The Absence And Existence Of Bounded Solutions At The Threshold Frequency In A Junction Of Quantum Waveguides

AU - Bakharev, F. L.

AU - Nazarov, S. A.

PY - 2021

Y1 - 2021

N2 - In the junction Ω of several semi-infinite cylindrical waveguides, the Dirichlet Laplacian is treated whose continuous spectrum is the ray [λ†,+∞) with a positive cutoff value λ†. Two different criteria are presented for the threshold resonance generated by nontrivial bounded solutions to the Dirichlet problem for the Helmholtz equation −Δu = λ†u in Ω. The first criterion is quite simple andis convenient to disprove the existence of bounded solutions. The second criterion is rather involved but can help to detect concrete shapes supporting the resonance. Moreover, the latter distinguishes in a natural way between stabilizing, i.e., bounded but nondecaying solutions, and trapped modes with exponential decay at infinity

AB - In the junction Ω of several semi-infinite cylindrical waveguides, the Dirichlet Laplacian is treated whose continuous spectrum is the ray [λ†,+∞) with a positive cutoff value λ†. Two different criteria are presented for the threshold resonance generated by nontrivial bounded solutions to the Dirichlet problem for the Helmholtz equation −Δu = λ†u in Ω. The first criterion is quite simple andis convenient to disprove the existence of bounded solutions. The second criterion is rather involved but can help to detect concrete shapes supporting the resonance. Moreover, the latter distinguishes in a natural way between stabilizing, i.e., bounded but nondecaying solutions, and trapped modes with exponential decay at infinity

KW - criteria for threshold resonances

KW - Junction of quantum waveguides

KW - stabilizing solutions

KW - trapped waves

KW - CONVERGENCE

KW - SPECTRA

UR - http://www.scopus.com/inward/record.url?scp=85118508592&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/2414cc71-e912-39c8-9b46-ab0da9eb40bc/

U2 - 10.1090/spmj/1679

DO - 10.1090/spmj/1679

M3 - Article

AN - SCOPUS:85118508592

VL - 32

SP - 955

EP - 973

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 6

ER -

ID: 88365537