On the point spectrum of a non-self-adjoint quasiperiodic operator

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On the point spectrum of a non-self-adjoint quasiperiodic operator. / Борисов, Денис; Федотов, Александр Александрович.

In: Russian Journal of Mathematical Physics, Vol. 31, No. 3, 01.09.2024, p. 389-406.

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

BibTeX

@article{1e28356a0a864a668475d5f35916a3f8,

title = "On the point spectrum of a non-self-adjoint quasiperiodic operator",

abstract = "We consider a difference operator acting in by the formula,, where,, and are parameters. This operator was introduced by P. Sarnak in 1982. For, the operator is quasiperiodic. Previously, within the framework of a renormalization approach (monodromization method), we described the location of the spectrum of this operator. In the present work, we first establish the existence of the point spectrum for different values of parameters, and then study the eigenfunctions. To do so, using ideas of the renormalization approach, we study the difference operator on the circle obtained from the original one by the Fourier transform. This allows us, first, to obtain a new type condition guaranteeing the existence of point spectrum and, second, to describe in detail a multi-scale self-similar structure of the Fourier transforms of the eigenfunctions. DOI 10.1134/S106192082403004X",

author = "Денис Борисов and Федотов, {Александр Александрович}",

year = "2024",

month = sep,

day = "1",

doi = "10.1134/S106192082403004X",

language = "English",

volume = "31",

pages = "389--406",

journal = "Russian Journal of Mathematical Physics",

issn = "1061-9208",

publisher = "МАИК {"}Наука/Интерпериодика{"}",

number = "3",

}

RIS

TY - JOUR

T1 - On the point spectrum of a non-self-adjoint quasiperiodic operator

AU - Борисов, Денис

AU - Федотов, Александр Александрович

PY - 2024/9/1

Y1 - 2024/9/1

N2 - We consider a difference operator acting in by the formula,, where,, and are parameters. This operator was introduced by P. Sarnak in 1982. For, the operator is quasiperiodic. Previously, within the framework of a renormalization approach (monodromization method), we described the location of the spectrum of this operator. In the present work, we first establish the existence of the point spectrum for different values of parameters, and then study the eigenfunctions. To do so, using ideas of the renormalization approach, we study the difference operator on the circle obtained from the original one by the Fourier transform. This allows us, first, to obtain a new type condition guaranteeing the existence of point spectrum and, second, to describe in detail a multi-scale self-similar structure of the Fourier transforms of the eigenfunctions. DOI 10.1134/S106192082403004X

AB - We consider a difference operator acting in by the formula,, where,, and are parameters. This operator was introduced by P. Sarnak in 1982. For, the operator is quasiperiodic. Previously, within the framework of a renormalization approach (monodromization method), we described the location of the spectrum of this operator. In the present work, we first establish the existence of the point spectrum for different values of parameters, and then study the eigenfunctions. To do so, using ideas of the renormalization approach, we study the difference operator on the circle obtained from the original one by the Fourier transform. This allows us, first, to obtain a new type condition guaranteeing the existence of point spectrum and, second, to describe in detail a multi-scale self-similar structure of the Fourier transforms of the eigenfunctions. DOI 10.1134/S106192082403004X

UR - https://www.mendeley.com/catalogue/9a3901c1-9e50-32c1-94fd-8c116b379144/

U2 - 10.1134/S106192082403004X

DO - 10.1134/S106192082403004X

M3 - Article

VL - 31

SP - 389

EP - 406

JO - Russian Journal of Mathematical Physics

JF - Russian Journal of Mathematical Physics

SN - 1061-9208

IS - 3

ER -

ID: 123004193