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Spectral properties for discontinuous Dirac system with eigenparameter‐dependent boundary condition

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Jiajia Zheng
Jiajia Zheng
Jiajia Zheng
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Kun Li at Qufu Normal University
Kun Li
Zhaowen Zheng at Qufu Normal University
Zhaowen Zheng
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Abstract

In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems.
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Received: 7 January 2024 Revised: 25 May 2024 Accepted: 11 July 2024
DOI: 10.1002/mma.10364
RESEARCH ARTICLE
Spectral properties for discontinuous Dirac system with
eigenparameter-dependent boundary condition
Jiajia Zheng1Kun Li1Zhaowen Zheng2
1School of Mathematical Sciences, Qufu
Normal University, Qufu, China
2School of Food Science and Biology,
Guangdong Polytechnic of Science and
Trade, Guangzhou, China
Correspondence
Kun Li, School of Mathematical Sciences,
Qufu Normal University, Qufu, 273100,
China.
Email: qslikun@163.com
Communicated by: S. Nicaise
Funding information
National Nature Science Foundation of
China, Grant/Award Number: 12401160;
Natural Science Foundation of Shandong
Province, Grant/Award Number:
ZR2023MA023, ZR2021MA047 and
ZR2021QF041; Youth Innovation Team of
Universities of Shandong Province,
Grant/Award Number: 2022KJ314;
National Natural Science Foundation of
China, Grant/Award Number: 61973183
In this paper, Dirac system with interface conditions and spectral parameter
dependent boundary conditions is investigated. By introducing a new Hilbert
space, the original problem is transformed into an operator problem. Then the
continuity and differentiability of the eigenvalues with respect to the parameters
in the problem are showed. In particular, the differential expressions of eigen-
values for each parameter are given. These results would provide theoretical
support for the calculation of eigenvalues of the corresponding problems.
KEYWORDS
dependence of eigenvalues, differential expressions, Dirac system, eigenparameter-dependent
boundary condition, interface condition
MSC CLASSIFICATION
34L15, 34L05
1INTRODUCTION
As an important part of spectral theory, the continuous dependence of eigenvalues of operators occupies vital importance
position in the numerical calculation of eigenvalues and in the theory of differential operators. Most recent years, there
have been extensive study of the continuous dependence of eigenvalues of the Sturm–Liouville (S-L) problems; see [1, 2]
and the survey paper [3]; the authors obtained that the eigenvalues of regular S-L problem depend not only continuously
but also smoothly on the parameters (the coefficient of equation, the boundary conditions, the endpoints); moreover, the
differential expressions of the eigenvalues with respect to these parameters are given. Such problem has obtained various
generalizations; for singular S-L problems, see [4, 5]; for higher-order differential operators, see [6–8]; and for differential
operators with interface conditions, see [9–12] and so on. The authors in these papers considered the continuity and
differentiability of eigenvalues with respect to coefficient functions, boundary conditions, interval endpoints, and so on.
Moreover, they presented the differential expressions of each eigenvalue about the parameters. These results played an
important role in the numerical calculation of eigenvalues of differential operators [13].
In general, the eigenparameter only appears in the differential equations. However, in many practical applications
such as mechanics and acoustic scattering theory, the spectral parameter is required to appear not only in the differ-
ential equations, but also in the boundary conditions [14–19]. For the continuous dependence of the eigenvalues of
eigen-dependent S-L problems, see [20]. On this basis, Ao et al. extended the results of the dependence of eigenvalues to the
case of eigenparameter-dependent S-L problems with interface conditions [11] and third-order discontinuous differential
operators [21]. The authors gave the continuity and differentiability of the eigenvalues of the corresponding problems.
Math. Meth. Appl. Sci. 2025;48:870–889.wileyonlinelibrary.com/journal/mma© 2024 John Wiley & Sons, Ltd.
870
... For applications to problems in physics and engineering, see [4,5]. For the latest works in that direction, refer to [6][7][8][9][10][11][12][13][14]. ...
... with 0 , ′ 0 defined by (13) are maximal dissipative or maximal accumulative. Conversely, any maximal dissipative (maximal accumulative) extensions on  of the operator 0 are generated by the expressioñ0 and condition (14) or (15). Maximal symmetric extensions are completely defined by (14) or (15) with isometric operator K. (14) In work [20] by M.Gorbachuk given description of extension operators in space 2 (0, 1; ), so not in larger direct sum space , that is why they are defined by boundary conditions at both end points of interval. ...
... Conversely, any maximal dissipative (maximal accumulative) extensions on  of the operator 0 are generated by the expressioñ0 and condition (14) or (15). Maximal symmetric extensions are completely defined by (14) or (15) with isometric operator K. (14) In work [20] by M.Gorbachuk given description of extension operators in space 2 (0, 1; ), so not in larger direct sum space , that is why they are defined by boundary conditions at both end points of interval. But in our study, it is enough to provide boundary condition only at zero. ...
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