Extensions of the matrix-valued \(q\)-Sturm-Liouville operators. (English) Zbl 1493.34081
Summary: In this paper, we investigate the matrix-valued \(q\)-Sturm-Liouville problems. We establish an existence and uniqueness result. Later, we introduce the corresponding maximal and minimal operators for this system. Moreover, we give a criterion under which these operators are self-adjoint. Finally, we characterize extensions (maximal dissipative, maximal accumulative, and self-adjoint) of the minimal symmetric operator.
MSC:
| 34B20 | Weyl theory and its generalizations for ordinary differential equations |
| 34B24 | Sturm-Liouville theory |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
Keywords:
boundary value space; boundary condition; dissipative extensions; accretive extensions; self-adjoint extensionsReferences:
| [1] | Adıvar M, Bohner M. Spectral analysis of q− difference equations with spectral singularities. Mathematical and Computer Modelling 2006; 43 (7-8): 69 5-703. doi: 10.1016/j.mcm.2005.04.014 · Zbl 1134.39018 · doi:10.1016/j.mcm.2005.04.014 |
| [2] | Allahverdiev BP, Tuna H. An expansion theorem for q− Sturm-Liouville operators on the whole line. Turkish Journal of Mathematics 2018; 42: 1060-1071. doi: 10.3906/mat-1705-22 · Zbl 1424.39018 · doi:10.3906/mat-1705-22 |
| [3] | Allahverdiev BP, Tuna H. Limit-point criteria for q− Sturm-Liouville equations. Quaestiones Mathematicae 2019; 42 (10): 1291-1299. doi: 10.2989/16073606.2018.1514541 · Zbl 1427.39003 · doi:10.2989/16073606.2018.1514541 |
| [4] | Allahverdiev BP, Tuna H. Qualitative spectral analysis of singular q− Sturm-Liouville operators. Bulletin of the Malaysian Mathematical Sciences Society 2020; 43: 1391. doi: 10.1007/s40840-019-00747-3 · Zbl 1496.34126 · doi:10.1007/s40840-019-00747-3 |
| [5] | Allahverdiev BP, Tuna H. Eigenfunction expansion in the singular case for q− Sturm-Liouville operators. Caspian Journal of Mathematical Sciences 2019; 8 (2): 91-102. doi: 10.22080/CJMS.2018.13943.1339 · Zbl 1463.39008 · doi:10.22080/CJMS.2018.13943.1339 |
| [6] | Allakhverdiev BP. On extensions of symmetric Schrödinger operators with a matrix potential, Izvestiya Rossiĭskoĭ Akademii Nauk. Seriya Matematicheskaya 1995; 59: 19-54; |
| [7] | English transl. Izvestiya: Mathematics 1995; 59: 45-62. doi: 10.1070/IM1995v059n01ABEH000002 · Zbl 0839.47016 · doi:10.1070/IM1995v059n01ABEH000002 |
| [8] | Annaby MH, Mansour ZS. q− Fractional Calculus and Equations. Lecture Notes in Mathematics. vol. 2056, Berlin, Germany: Springer-Verlag, 2012. · Zbl 1267.26001 |
| [9] | Annaby MH, Mansour ZS. Basic Sturm-Liouville problems. Journal of Physics. A. Mathematical and Theoretical 2005; 38: 3775-3797. doi: 10.1088/0305-4470/38/17/005 · Zbl 1073.33012 · doi:10.1088/0305-4470/38/17/005 |
| [10] | Aygar Y, Bairamov E. Jost solution and the spectral properties of the matrix-valued difference operators. Applied Mathematics and Computation 2012; 218 (3): 9676-9681. doi: 10.1016/j.amc.2012.02.081 · Zbl 1246.39003 · doi:10.1016/j.amc.2012.02.081 |
| [11] | Aygar Y, Bohner M. Spectral analysis of a matrix-valued quantum-difference operator. Dynamic Systems and Applications 2016; 25: 1-9. · Zbl 1355.39010 |
| [12] | Aygar Y, Bohner M. A Polynomial-Type Jost Solution and spectral properties of a self-adjoint quantum-difference operator. Complex Analysis and Operator Theory 2016; 10 (6): 1171-1180. doi: 10.1007/s11785-015-0463-x · Zbl 1354.39012 · doi:10.1007/s11785-015-0463-x |
| [13] | Aygar Y. A research on spectral analysis of a matrix quantum difference equations with spectral singularities. Quaestiones Mathematicae 2017; 40 (2): 245-249. doi: 10.2989/16073606.2017.1284911 · Zbl 1461.39005 · doi:10.2989/16073606.2017.1284911 |
| [14] | Baiṙamov E, Cebesoy Ş. Spectral singularities of the matrix Schrödinger equations. Hacettepe Journal of Mathe-matics and Statistics 2016; 45 (4): 1007-1014. doi: 10.15672/HJMS.20164514275 · Zbl 1357.34133 · doi:10.15672/HJMS.20164514275 |
| [15] | Bairamov E, Aygar Y, Cebesoy S. Spectral analysis of a self-adjoint matrix-valued discrete operator on the whole axis. Journal of Nonlinear Sciences and Applications 2016; 9 (6): 4257-4262. · Zbl 1382.39026 |
| [16] | Bastard G, Brum JA. Electronic states in semi conductor heterostructures. IEEE Journal of Quantum Electronics 1986; 22: 1625-1644. doi: 10.1109/JQE.1986.1073186 · doi:10.1109/JQE.1986.1073186 |
| [17] | Bastard G. Wave mechanics applied to semi conductor hetero structures. Paris, Éditions de Physique: 1989. |
| [18] | Beals R, Henkin GM, Novikova NN. The inverse boundary problem for the Rayleigh system. Journal of Mathematical Physics 1965; 36 (12): 6688-6708. doi: 10.1063/1.531182 · Zbl 0857.73025 · doi:10.1063/1.531182 |
| [19] | Bondarenko N. Spectral analysis for the matrix Sturm-Liouville operator on a finite interval. Tamkang Journal of Mathematics 2011; 42 (3): 305-327. doi: 10.5556/j.tkjm.42.2011.305-327 · Zbl 1242.34026 · doi:10.5556/j.tkjm.42.2011.305-327 |
| [20] | Bondarenko N. Matrix Sturm-Liouville equation with a Bessel-type singularity on a finite interval. Analysis and Mathematical Physics 2017; 7 (1): 77-92. doi: 10.1007/s13324-016-0131-y · Zbl 1370.34037 · doi:10.1007/s13324-016-0131-y |
| [21] | Boutet de Monvel A, Shepelsky D. Inverse scattering problem for anisotropic media. Journal of Mathematical Physics 1995; 36 (7): 3443-3453. doi: 10.1063/1.530971 · Zbl 0842.34084 · doi:10.1063/1.530971 |
| [22] | Bruk VM. On a class of boundary -value problemswith a spectral parameter in the boundary conditions, Matem-aticheskiĭ Sbornik 1976; 100: 210-216. doi: 10.1070/SM1976v029n02ABEH003662 · Zbl 0376.47007 · doi:10.1070/SM1976v029n02ABEH003662 |
| [23] | Calkin JW. Abstract symmetric boundary conditions. Transactions of the American Mathematical Society 1939; 45 (3): 369-442. · JFM 65.0511.01 |
| [24] | Chabanov VM. Recovering the M-channel Sturm-Liouville operator from M + 1 spectra. Journal of Mathematical Physics 2004; 45 (11): 4255-4260. doi: 10.1063/1.1794844 · Zbl 1064.34006 · doi:10.1063/1.1794844 |
| [25] | Coskun C, Olgun M. Principal functions of non-selfadjoint matrix Sturm-Liouville equations. Journal of Compu-tational and Applied Mathematics 2011; 235 (16): 4834-4838. doi: 10.1016/j.cam.2010.12.004 · Zbl 1236.34034 · doi:10.1016/j.cam.2010.12.004 |
| [26] | Ernst T. The History of q− Calculus and a New Method. U. U. D. M. Report (2000): 16, ISSN1101-3591, Department of Mathematics, Uppsala University, 2000. |
| [27] | Eryılmaz A, Tuna H. Spectral theory of dissipative q− Sturm-Liouville problems. Studia Scientiarum Mathemati-carum Hungarica 2014; 51 (3): 366-383. doi: 10.1556/SScMath.51.2014.3.1289 · Zbl 1324.47018 · doi:10.1556/SScMath.51.2014.3.1289 |
| [28] | Gorbachuk ML. On spectral functions of a second order differential operator with operator coefficients. Ukrains’kyi Matematychnyi Zhurnal 1966; 18 (2): 3-21; · Zbl 0166.40802 |
| [29] | English transl . American Mathematical Society Translations: Series 2 1968; 72: 177-202 . |
| [30] | Gorbachuk ML, Gorbachuk VI, Kochubei AN. The theory of extensions of symmetric operators and boundary-value problems for differential equations. Ukrains’kyi Matematychnyi Zhurnal 1989; 41: 1299-1312; |
| [31] | English transl. in Ukrainian Mathematical Journal 1989; 41: 1117-1129. doi: 10.1007/BF01057246 · Zbl 0706.47001 · doi:10.1007/BF01057246 |
| [32] | Gorbachuk ML, Gorbachuk VI. Boundary Value Problems for Operator Differential Equations, Naukova Dumka, Kiev, 1984; English transl. Birkhauser Verlag, 1991. · Zbl 0567.47041 |
| [33] | Kac V, Cheung P. Quantum Calculus. Berlin, Germany: Springer-Verlag, 2002. · Zbl 0986.05001 |
| [34] | Karahan D, Mamedov KhR. Sampling theory associated with q− Sturm-Liouville operator with discontinuity conditions. Journal of Contemporary Applied Mathematics 2020; 10 (2): 1-9. · Zbl 1461.34098 |
| [35] | Kochubei AN. Extensions of symmetric operators and symmetric binary relations. Matematicheskie Zametki 1975; 17: 41-48; English transl. in Mathematical Notes 1975; 17: 25-28. doi: 10.1007/BF01093837 · Zbl 0322.47006 · doi:10.1007/BF01093837 |
| [36] | Krall AM. Hilbert Space, Boundary Value Problems and Orthogonal Polynomials. Berlin, Germany: Birkhäuser Verlag, 2002. · Zbl 1033.34080 |
| [37] | Krein MG. On the indeterminate case of the Sturm-Liouville boundaryvalue problem in the interval (0, ∞) , Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 1952; 16: 292-324. · Zbl 0048.32602 |
| [38] | Maksudov FG, Allahverdiev BP. On the extensions of Schrödinger operators with a matrix potentials, Doklady Akademii Nauk 1993; 332(1):18-20;English transl. Russian Academy of Sciences. Doklady. Mathematics 1994; 48 (2):240-243. doi: 10.1070/IM1995v059n01ABEH000002 · Zbl 0839.47016 · doi:10.1070/IM1995v059n01ABEH000002 |
| [39] | Malamud MM, Mogilevskiy VI. On extensions of dual pairs of operators. Dopovidi Natsional’noï Akademiï Nauk Ukraïny 1997; 1: 30-37. · Zbl 0887.47008 |
| [40] | Mogilevskiy VI. On proper extensions of a singular differential operator in a space of vector functions. Dopovidi Nat-sional’noï Akademiï Nauk Ukraïny 1994; 9: 29-33 (in Russian with an English abstract) . doi: 10.1007/BF01059050 · doi:10.1007/BF01059050 |
| [41] | Naimark MA. Linear Differential Operators. 2nd ed. Moscow, USSR: Nauka, 1969 (in Russian). · Zbl 0193.04101 |
| [42] | Rofe-Beketov FS. Self-adjoint extensions of differential operators in a space of vector valued functions, Doklady Akademii Nauk SSSR 1969; 184: 1034-1037; English transl. in Soviet Mathematics. Doklady 1969; 10:188-192. · Zbl 0181.15401 |
| [43] | Shi YM. Spectral theory of discrete linear Hamiltonian systems. Journal of Mathematical Analysis and Applications 2004; 289 (2): 554-570. doi: 10.1016/j.jmaa.2003.08.039 · Zbl 1047.39016 · doi:10.1016/j.jmaa.2003.08.039 |
| [44] | Tuna H, Eryılmaz A. Completeness of the system of root functions of q− Sturm-Liouville operators. Mathematical Communications 2014; 19 (1): 65-73. · Zbl 1296.39005 |
| [45] | Tuna H, Eryılmaz A. On q− Sturm-Liouville operators with eigenvalue parameter contained in the boundary conditions. Dynamic Systems and Applications 2015; 24 (4): 491-501. · Zbl 1338.39019 |
| [46] | Tuna H, Eryılmaz A. Livšic’s theorem for q− Sturm-Liouville operators. Studia Scientiarum Mathematicarum Hungarica 2016; 53 (4): 512-524. doi: 10.1556/012.2016.53.4.1348 · Zbl 1399.34275 · doi:10.1556/012.2016.53.4.1348 |
| [47] | von Neumann J. Allgemeine Eigenwertheorie Hermitischer Functionaloperatoren. Mathematische Annalen 1929; 102: 49-131 (in German). doi: 10.1007/BF01782338 · JFM 55.0824.02 · doi:10.1007/BF01782338 |
| [48] | Yardimci S. A note on the spectral singularities of non-selfadjoint matrix-valued difference operators. Journal of Computational and Applied Mathematics 2010; 234: 3039-3042. doi: 10.1016/j.cam.2010.04.017 · Zbl 1194.47033 · doi:10.1016/j.cam.2010.04.017 |
| [49] | Yurko V. Inverse problems for the matrix Sturm-Liouville equation on a finite interval. Inverse Problems 2006; 22: 1139-1149. doi: 10.1088/0266-5611/22/4/002 · Zbl 1107.34005 · doi:10.1088/0266-5611/22/4/002 |
| [50] | Zettl A. Sturm-Liouville Theory. Mathematical Surveys and Monographs. vol. 121. Providence, Rhode Island, USA: American Mathematical Society, 2005. · Zbl 1103.34001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
