Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-05-09T09:00:15.841Z Has data issue: false hasContentIssue false
  • English
  • Français

On classification of singular matrix difference equations of mixed order

Part of: Boundary value problems

Published online by Cambridge University Press:  11 August 2023

Li Zhu ,
Huaqing Sun  and
Bing Xie
Li Zhu
Affiliation:
College of Sciences, Northeastern University, Shenyang, Liaoning 110819, P. R. China (zhulimathematics@163.com) Department of Mathematics, Shandong University at Weihai, Weihai, Shandong 264209, P. R. China
Huaqing Sun*
Affiliation:
College of Sciences, Northeastern University, Shenyang, Liaoning 110819, P. R. China (sunhuaqing@mail.neu.edu.cn)
Bing Xie
Affiliation:
Department of Mathematics, Shandong University at Weihai, Weihai, Shandong 264209, P. R. China (xiebing@sdu.edu.cn)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

Keywords

block operator matrixmatrix differential equationmatrix difference equationlimit point caselimit circle case

MSC classification

Primary: 34B20: Weyl theory and its generalizations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 24
Check for updates
Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adamyan, V., Langer, H. and Langer, M.. A spectral theory for a $\lambda$-rational Sturm–Liouville problem. J. Differ. Equ. 171 (2001), 315–345.CrossRefGoogle Scholar
Ammar, A., Ezzadam, A. and Jeribi, A.. A characterization of the essential spectra of $2 \times 2$ block matrices of linear relations. Bull. Iran. Math. Soc. 48 (2022), 2463–2485.CrossRefGoogle Scholar
Atkinson, F. V.. Discrete and Continuous Boundary Problems (New York: Academic Press, 1964).Google Scholar
Atkinson, F. V., Langer, H., Mennicken, R. and Shkalikov, A. A.. The essential spectrum of some matrix operators. Math. Nachr. 167 (1994), 5–22.CrossRefGoogle Scholar
Atkinson, F. V., Langer, H. and Mennicken, R.. Sturm–Liouville problems with coefficients which depend analytically on the eigenvalue parameter. Acta Sci. Math. 57 (1993), 25–44.Google Scholar
Behrndt, J., Gesztesy, F., Holden, H. and Nichols, R.. Dirichlet-to-Neumann maps, abstract Weyl–Titchmarsh M-functions, and a generalized index of unbounded meromorphic operator-valued functions. J. Differ. Equ. 261 (2016), 3551–3587.CrossRefGoogle Scholar
Behrndt, J. and Rohleder, J.. Titchmarsh–Weyl theory for Schrödinger operators on unbounded domains. J. Spectr. Theory 6 (2016), 67–87.CrossRefGoogle Scholar
Behrndt, J., Hassi, S. and de Snoo, H., Boundary Value Problems, Weyl Functions, and Differential Operators, Monographs in Mathematics, Vol. 108 (Birkhäuser, Cham, 2020).CrossRefGoogle Scholar
Brown, B. M., Marletta, M., Naboko, S. N. and Wood, I. G.. Detectable subspaces and inverse problems for Hain-Lüst-type operators. Math. Nachr. 289 (2016), 2108–2132.CrossRefGoogle Scholar
Brown, B. M. and Marletta, M.. Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. J. Lond. Math. Soc. 77 (2008), 700–718.CrossRefGoogle Scholar
Chen, J. and Shi, Y.. The limit-circle and limit-point criteria for second-order linear difference equations. Comput. Math. Appl. 47 (2004), 967–976.CrossRefGoogle Scholar
Clark, S. L. and Zemánek, P.. On a Weyl–Titchmarsh theory for discrete symplectic systems on a half line. Appl. Math. Comput. 217 (2010), 2952–2976.Google Scholar
Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
Everitt, W., Knowles, I. and Read, T.. Limit-point and limit-circle criteria for Sturm–Liouville equations with intermittently negative principal coefficients. Proc. Roy. Soc. Edinburgh Sect. A 103 (1986), 215–228.CrossRefGoogle Scholar
Evans, W. D.. On the limit point, limit circle classification of a second order differential equation with a complex potential. J. London Math. Soc. 4 (1971), 245–256.CrossRefGoogle Scholar
Faierman, M., Mennicken, R. and Möller, M.. The essential spectrum of a system of singular ordinary differential operators of mixed order. Part I: The general problem and an almost regular case. Math. Nachr. 208 (1999), 101–115.CrossRefGoogle Scholar
Faierman, M., Mennicken, R. and Möller, M.. The essential spectrum of a system of singular ordinary differential operators of mixed order. II. The generalization of Kako's problem. Math. Nachr. 209 (2000), 55–81.3.0.CO;2-W>CrossRefGoogle Scholar
Giribet, J., Langer, M., Pería, F. M., Philipp, F. and Trunk, C.. Spectral enclosures for a class of block operator matrices. J. Funct. Anal. 278 (2020), 108455.CrossRefGoogle Scholar
Hardt, V., Mennicken, R. and Naboko, S.. Systems of singular differential operators of mixed order and applications to 1-dimensional MHD problems. Math. Nachr. 206 (1999), 19–68.CrossRefGoogle Scholar
Hassi, S., Möller, M. and de Snoo, H.. Singular Sturm–Liouville problems whose coefficients depend rationally on the eigenvalue parameter. J. Math. Anal. Appl. 295 (2004), 258–275.CrossRefGoogle Scholar
Hassi, S., Möller, M. and de Snoo, H.. Limit-point/limit-circle classification for Hain-Lüst type equations. Math. Nachr. 291 (2018), 652–668.CrossRefGoogle Scholar
Hinton, D. B. and Lewis, R. T.. Spectral analysis of second order difference equations. J. Math. Anal. Appl. 63 (1978), 421–438.CrossRefGoogle Scholar
Hinton, D. B. and Shaw, J. K.. On Titchmarsh–Weyl $M(\lambda )$-functions for linear Hamiltonian systems. J. Differ. Equ. 40 (1981), 316–342.CrossRefGoogle Scholar
Ibrogimov, O. O., Siegl, P. and Tretter, C.. Analysis of the essential spectrum of singular matrix differential operators. J. Differ. Equ. 260 (2016), 3881–3926.CrossRefGoogle Scholar
Ibrogimov, O. O.. Essential spectrum of non-self-adjoint singular matrix differential operators. J. Math. Anal. Appl. 451 (2017), 473–496.CrossRefGoogle Scholar
Jeribi, A.. Spectral Theory and Applications of Linear Operators and Block Operator Matrices (Cham: Springer, 2015).CrossRefGoogle Scholar
Jeribi, A., Krichen, B. and Zitouni, A.. Properties of demicompact operators, essential spectra and some perturbation results for block operator matrices with applications. Linear Multilinear Algebra 68 (2020), 2506–2522.CrossRefGoogle Scholar
Jirari, A.. Second-order Sturm–Liouville difference equations and orthogonal polynomimals. Mem. Am. Math. Soc. 113 (1995), no. 542, x+138 pp.Google Scholar
Kako, T.. Essential spectrum of linearized operator for MHD plasma in cylindrical region. Z. Angew. Math. Phys. 38 (1987), 433–449.CrossRefGoogle Scholar
Konstantinov, A. and Mennicken, R.. On the Friedrichs extension of some block operator matrices. Integr. Equ. Oper. Theory 42 (2002), 472–481.CrossRefGoogle Scholar
Kurasov, P. and Naboko, S.. On the essential spectrum of a class of singular matrix differential operators. I. Quasiregularity conditions and essential self-adjointness. Math. Phys. Anal. Geom. 5 (2002), 243–286.CrossRefGoogle Scholar
Kurasov, P., Lelyavin, I. and Naboko, S.. On the essential spectrum of a class of singular matrix differential operators. II. Weyl's limit circles for the Hain-Lüst operator whenever quasi-regularity conditions are not satisfied. Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 109–138.CrossRefGoogle Scholar
Kusche, T., Mennicken, R. and Möller, M.. Friedrichs extension and essential spectrum of systems of differential operators of mixed order. Math. Nachr. 278 (2005), 1591–1606.CrossRefGoogle Scholar
Langer, H. and Tretter, C.. Spectral decomposition of some non-self-adjoint block operator matrices. J. Operat. Theor. 39 (1998), 339–359.Google Scholar
Levinson, N.. Criteria for the limit-point case for second order linear differential operators. Čascpis Pěst. Mat. 74 (1949), 17–20.CrossRefGoogle Scholar
Krall, A. M.. $M(\lambda )$ theory for singular Hamiltonian systems with one singular point. SIAM J. Math. Anal. 20 (1989), 664–700.CrossRefGoogle Scholar
Krall, A. M.. $M(\lambda )$ theory for singular Hamiltonian systems with two singular points. SIAM J. Math. Anal. 20 (1989), 701–715.CrossRefGoogle Scholar
Kogan, V. I. and Rofe-Beketov, F. S.. On square-integrable solutions of symmetric systems of differential equations of arbitrary order. Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 5–40.CrossRefGoogle Scholar
Langer, H., Markus, A. S., Matsaev, V. I. and Tretter, C.. A new concept for block operator matrices: the quadratic numerical range. Linear Algebra Appl. 330 (2001), 89–112.CrossRefGoogle Scholar
Mennicken, R., Naboko, S. and Tretter, C.. Essential spectrum of a system of singular differential operators and the asymptotic Hain-Lüst operator. Proc. Amer. Math. Soc. 130 (2002), 1699–1710.CrossRefGoogle Scholar
Mingarelli, A. B.. A limit point criterion for a three-term recurrence relation. Math. Rep. Acad. Sci. Canada 3 (1981), 171–175.Google Scholar
Monaquel, J. and Schmidt, M.. On M-functions and operator theory for non-self-adjoint discrete Hamiltonian systems. J. Comput. Appl. Math. 208 (2007), 82–101.CrossRefGoogle Scholar
Muzzulini, M.. Titchmarsh–Sims–Weyl theory for complex Hamiltonian systems of arbitrary order. J. Lond. Math. Soc. 2 (2011), 159–182.CrossRefGoogle Scholar
Möller, M.. The essential spectrum of a system of singular ordinary differential operators of mixed order. Part III: A strongly singular case. Math. Nachr. 272 (2004), 104–112.CrossRefGoogle Scholar
Qi, J. and Chen, S.. Essential spectra of singular matrix differential operators of mixed order in the limit circle case. Math. Nachr. 284 (2011), 342–354.CrossRefGoogle Scholar
Qi, J. and Chen, S.. Essential spectra of singular matrix differential operators of mixed order. J. Differ. Equ. 250 (2011), 4219–4235.CrossRefGoogle Scholar
Shi, Y.. Weyl–Titchmarsh theory for a class of discrete linear Hamiltonian systems. Linear Algebra Appl. 416 (2006), 452–519.CrossRefGoogle Scholar
Sun, H. and Shi, Y.. Limit-point and limit-circle criteria for singular second-order linear difference equations with complex coefficients. Comput. Math. Appl. 52 (2006), 539–554.CrossRefGoogle Scholar
Titchmarsh, E. C.. Eigenfunction Expansions (Oxford: Oxford University Press, 1962).Google Scholar
Tretter, C.. Spectral Theory of Block Operator Matrices and Applications (London: Imperial College Press, 2008).CrossRefGoogle Scholar
Tretter, C.. Spectral inclusion for unbounded block operator matrices. J. Funct. Anal. 256 (2009), 3806–3829.CrossRefGoogle Scholar
Weidmann, J., Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, Vol. 1258 (Springer-Verlag, Berlin, 1987).CrossRefGoogle Scholar
Weyl, H.. Übergewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen. Math. Ann. 68 (1910), 220–269.CrossRefGoogle Scholar
Zemánek, P.. Resolvent and spectrum for discrete symplectic systems in the limit point case. Linear Algebra Appl. 634 (2022), 179–209.CrossRefGoogle Scholar