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Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 2, Pages 113–139 (Mi izv8623)  

On the basis property of the system of eigenfunctions and associated functions of a one-dimensional Dirac operator

A. M. Savchuk

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study a one-dimensional Dirac system on a finite interval. The potential (a $2\times 2$ matrix) is assumed to be complex-valued and integrable. The boundary conditions are assumed to be regular in the sense of Birkhoff. It is known that such an operator has a discrete spectrum and the system $\{\mathbf{y}_n\}_1^\infty$ of its eigenfunctions and associated functions is a Riesz basis (possibly with brackets) in $L_2\oplus L_2$. Our results concern the basis property of this system in the spaces $L_\mu\oplus L_\mu$ for $\mu\ne2$, the Sobolev spaces ${W_2^\theta\oplus W_2^\theta}$ for $\theta\in[0,1]$, and the Besov spaces $B^\theta_{p,q}\oplus B^\theta_{p,q}$.

Keywords: Dirac operator, eigenfunctions and associated functions, conditional basis, Riesz basis.

Funding Agency Grant Number
Russian Science Foundation 17-11-01215
This work is supported by the Russian Science Foundation under grant 17-11-01215.


DOI: https://doi.org/10.4213/im8623

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English version:
Izvestiya: Mathematics, 2018, 82:2, 351–376

Bibliographic databases:

UDC: 517.984.52
MSC: 34L10, 34L40, 47E05
Received: 26.10.2016
Revised: 19.08.2017

Citation: A. M. Savchuk, “On the basis property of the system of eigenfunctions and associated functions of a one-dimensional Dirac operator”, Izv. RAN. Ser. Mat., 82:2 (2018), 113–139; Izv. Math., 82:2 (2018), 351–376

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