On extensions of matrix-valued Hahn–Sturm–Liouville operators
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 2, pp. 1-12.

Voir la notice de l'article provenant de la source Library of Science

In this paper, we study matrix-valued Hahn–Sturm–Liouville equations. We give an existence and uniqueness result. We introduce the corresponding maximal and minimal operators for this system, and some properties of these operators are investigated. Finally, we characterize extensions (maximal dissipative, maximal accumulative and self-adjoint) of the minimal symmetric operator.
Keywords: Hahn–Sturm–Liouville equation, minimal and maximal operators, maximal dissipative, accumulative and self-adjoint extensions
@article{AUM_2021_75_2_a0,
     author = {Allahverdiev, Bilender and Tuna, Huseyin},
     title = {On extensions of matrix-valued {Hahn{\textendash}Sturm{\textendash}Liouville} operators},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     pages = {1--12},
     publisher = {mathdoc},
     volume = {75},
     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2021_75_2_a0/}
}
TY  - JOUR
AU  - Allahverdiev, Bilender
AU  - Tuna, Huseyin
TI  - On extensions of matrix-valued Hahn–Sturm–Liouville operators
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2021
SP  - 1
EP  - 12
VL  - 75
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2021_75_2_a0/
LA  - en
ID  - AUM_2021_75_2_a0
ER  - 
%0 Journal Article
%A Allahverdiev, Bilender
%A Tuna, Huseyin
%T On extensions of matrix-valued Hahn–Sturm–Liouville operators
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2021
%P 1-12
%V 75
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2021_75_2_a0/
%G en
%F AUM_2021_75_2_a0
Allahverdiev, Bilender; Tuna, Huseyin. On extensions of matrix-valued Hahn–Sturm–Liouville operators. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 2, pp. 1-12. http://geodesic.mathdoc.fr/item/AUM_2021_75_2_a0/

[1] Allakhverdiev, B. P., On extensions of symmetric Schrodinger operators with a matrix potential, Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), 19–54 (Russian); English translation Izv. Math. 59 (1995), 45–62.

[2] Allahverdiev, B. P., Tuna, H., A spectral expansion for Hahn–Sturm–Liouville equation on the whole line, Folia Math. 23 (1) (2021), 3–19.

[3] Allahverdiev, B. P., Tuna, H., Spectral theory of singular Hahn difference equation of the Sturm–Liouville type, Commun. Math. 28 (1) (2020), 13–25.

[4] Allahverdiev, B. P., Tuna, H., A representation of the resolvent operator of singular Hahn–Sturm–Liouville problem, Numer. Funct. Anal. Optim. 41 (4) (2020), 413–431.

[5] Allahverdiev, B. P., Tuna, H., Extensions of the matrix-valued q-Sturm–Liouville operators, Turkish J. Math. 45 (2021), 1479–1494.

[6] Annaby, M. H., Hamza, A. E., Aldwoah, K. A., Hahn difference operator and associated Jackson–Norlund integrals, J. Optim. Theory Appl. 154 (2012), 133–153.

[7] Annaby, M. H., Hamza, A. E., Makharesh, S. D., A Sturm–Liouville theory for Hahn difference operator, in: Xin Li, Zuhair Nashed (Eds.), Frontiers of Orthogonal Polynomials and q-Series, World Scientific, Singapore, 2018, 35–84.

[8] Atkinson, F. V., Discrete and Continuous Boundary Problems, Acad. Press Inc., New York, 1964.

[9] Aygar, Y., Bohner, M., Spectral analysis of a matrix-valued quantum-difference operator, Dynam. Syst. Appl. 25 (2016), 1–9.

[10] Bairamov, E., Cebesoy, S¸., Spectral singularities of the matrix Schrodinger equations, Hacettepe J. Math. Stat. 45 (4) (2016), 1007–1014.

[11] Bastard, G., Brum, J. A., Electronic states in semi conductor heterostructures, IEEE J. Quantum Electron. 22 (1986), 1625–1644.

[12] Bastard, G., Wave Mechanics Applied to Semi Conductor Hetero Structures, Paris, ´Editions de Physique, 1989.

[13] Beals, R., Henkin, G. M., Novikova, N. N., The inverse boundary problem for the Rayleigh system, J. Math. Phys. 36 (12) (1965), 6688–6708.

[14] Bondarenko, N., Spectral analysis for the matrix Sturm–Liouville operator on a finite interval, Tamkang J. Math. 42 (3) (2011), 305–327.

[15] Boutet de Monvel, A., Shepelsky, D., Inverse scattering problem for anisotropic media, J. Math. Phys. 36 (7) (1995), 3443–3453.

[16] Bruk, V. M., On a class of boundary-value problems with a spectral parameter in the boundary conditions, Mat. Sb. (N.S.) 100 (1976), 210–216 (Russian).

[17] Calkin, J. W., Abstract symmetric boundary conditions, Trans. Amer. Math. Soc. 45 (3) (1939), 369–442.

[18] Chabanov, V. M., Recovering the M-channel Sturm–Liouville operator from M + 1 spectra, J. Math. Phys. 45 (11) (2004), 4255–4260.

[19] Gorbachuk, M. L., On spectral functions of a second order differential operator with operator coefficients, Ukrain. Mat. Zhurnal 18 (2) (1966), 3–21 (Russian); English translation American Mathematical Society Translations: Ser. 2 72 (1968), 177–202.

[20] Gorbachuk, M. L., Gorbachuk, V. I., Kochubei, A. N., The theory of extensions of symmetric operators and boundary-value problems for differential equations, Ukrain. Mat. Zhurnal 41 (1989), 1299–1312 (Russian); English translation Ukrain. Math. J. 41 (1989), 1117–1129.

[21] Gorbachuk, M. L., Gorbachuk, V. I., Boundary Value Problems for Operator Differential Equations, Naukova Dumka, Kiev, 1984 (Russian); English translation Birkhauser Verlag, 1991.

[22] Hahn, W., Uber Orthogonalpolynome, die q-Differenzengleichungen genugen, Math. Nachr. 2 (1949), 4–34 (German).

[23] Hahn, W., Ein Beitrag zur Theorie der Orthogonalpolynome, Monatsh. Math. 95 (1983), 19–24 (German).

[24] Kochubei, A. N., Extensions of symmetric operators and symmetric binary relations, Mat. Zametki 17 (1975), 41–48 (Russian); English translation Math. Notes 17 (1975), 25–28.

[25] Krall, A. M., Hilbert Space, Boundary Value Problems and Orthogonal Polynomials, Birkhauser Verlag, Basel–Boston–Berlin, 2002.

[26] Krein, M. G., On the indeterminate case of the Sturm–Liouville boundary value problem in the interval (0;1), Izvest. Akad. Nauk SSSR. Ser. Mat. 16 (1952), 292–324 (Russian).

[27] Naimark, M. A., Linear Differential Operators, 2nd ed., Nauka, Moscow, 1969 (Russian); English translation of 1st. ed., Part I, Part II, Frederick Ungar Publishing Co., New York, 1967, 1968.

[28] Maksudov, F. G., Allakhverdiev, B. P., On the extensions of Schr¨odinger operators with a matrix potentials, Doklady Akad. Nauk 332 (1) (1993), 18–20 (Russian); English translation Russian Acad. Sci. Doklady Math. 48 (2) (1994), 240–243.

[29] Malamud, M. M., Mogilevskii, V. I., On extensions of dual pairs of operators, Dopov. Nats. Akad. Nauk Ukr. 11 (1997), 30–37.

[30] Mogilevskii, V. I., On proper extensions of a singular differential operator in a space of vector functions, Dopov. Akad. Nauk Ukraini 9 (1994), 29–33 (Russian).

[31] Rofe-Beketov, F. S., Self-adjoint extensions of differential operators in a space of vector valued functions, Dokl. Akad. Nauk SSSR 184 (1969), 1034–1037 (Russian); English translation Soviet Math. Dokl. 10 (1969), 188–192.

[32] Shi, Y., Weyl–Titchmarsh theory for a class of discrete linear Hamiltonian systems, Linear Algebra Appl. 416 (2006), 452–519.

[33] von Neumann, J., Allgemeine Eigenwertheorie Hermitischer Functionaloperatoren, Math. Annal. 102 (1929), 49–131 (German).

[34] Yurko, V., Inverse problems for the matrix Sturm–Liouville equation on a finite interval, Inverse Problems 22 (2006), 1139–1149.

[35] Zettl, A., Sturm–Liouville Theory, American Mathematical Society, Providence, RI, 2005.