On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
DOI:
https://doi.org/10.15407/mag10.01.044Анотація
Для рівняння Штурма-Ліувіля з блочно-трикутним матричним потенціалом, що росте на нескінченності, побудовано спадні і зростаючі на нескінченності матричні розв'язки. Встановлено структуру спектра диференціального оператора з такими коефіцієнтами.
Mathematics Subject Classification: 34K11, 47A10.
Ключові слова:
диференціальний оператор, спектр, блочно-трикутні матричні коефіцієнтиПосилання
A.M. Kholkin and F.S. Rofe-Beketov, Sturm Type Oscillation Theorems for Equations with Block-Triangular Matrix Coefficients. — Methods Funct. Anal. Topology 18 (2012), No. 2, 176–188.
E.I. Bondarenko and F.S. Rofe-Beketov, Inverse Scattering Problem on the Semiaxis for a System with a Triangular Matrix Potential. — Math. Phys., Anal., Geom. 10, (2003), No. 3, 412–424. (Russian)
E.I. Bondarenko and F.S. Rofe-Beketov, Phase Equivalent Matrix Potential. — Electromagnetic waves and electronic systems 5 (2000), No. 3, 6–24. (Russian) (Engl. transl.: Telecommun. Radio Eng. 56 (2001), Nos. 8 and 9, 4–29). https://doi.org/10.1615/TelecomRadEng.v55.i1.80
F.S. Rofe-Beketov and E.I. Zubkova, Inverse Scattering Problem on the Axis for the Triangular 2 × 2 Matrix Potential with or without a Virtual Level. — Azerbaijan J. Math. 1 (2011), No. 2, 3–69.
V.A. Marchenko, Spectral Theory of Sturm–Liouville Operators. Naukova Dumka, Kiev, 1972. (Russian)
V.A. Marchenko, Sturm–Liouville Operators and its Applications. Naukova Dumka, Kiev, 1977. (Russian) (Engl. transl.: Oper. Theory Adv. Appl. 22 (1986), Birkhauser Verlag, Basel, xii+367pp.; revised edition AMS Chelsea Publishing, Providence R.I., 2011, xiv+396 pp.). https://doi.org/10.1007/978-3-0348-5485-6
V.A. Marchenko and F.S. Rofe-Beketov, Expansion in Eigenfunctions of Nonselfadjoint Singular Differential Operators. — Dokl. Akad. Nauk. SSSR 120 (1958), No. 5, 963–966. (Russian)
F.S. Rofe-Beketov, Expansion in Eigenfunctions of Infinite Systems of Differential Equations in the Non-selfadjoint and Selfadjoint Cases. — Mat. Sb. 51 (1960), No. 3, 293–342. (Russian)
Z.S. Agranovich and V.A. Marchenko, The Inverse Problem of Scattering Theory. Kharkov State University, Kharkov, 1960. (Russian) (Engl. transl.: New YorkLondon, Gordon & Breach, 1963).
F.S. Rofe-Beketov and A.M. Kholkin, Spectral Analysis of Differential Operators. Interplay between Spectral and Oscillatory Properties, With Foreword by V.A. Marchenko, World Scientific, New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2005. https://doi.org/10.1142/5788
A. Eremenko and A. Gabrielov, Spectral Loci of Sturm–Liouville Operators with Polynomial Potentials, in print.
E.Ch. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Vol. 2, Clarendon Press, Oxford, 1958. (Russian transl.: Vol. 2, Izdat. Inostran. Lit., Moscow, 1961).
F.S. Rofe-Beketov and E.Kh. Khristov, Asymptotic and Analytic Questions Connected with Scattering by a Highly Singular Potential, Collection of Proceeding of Fiz. Tekh. Inst. Niz. Temp. Akad. Nauk Ukrain. SSR. Mathematics, Physics and Functional Analysis Series, Kharkov, 1971, No. 2, 122–168. (Russian)
A.N. Tichonov and A.A. Samarsky, Equations of Mathematical Physics. Nauka, Moscow, 1972. (Russian)
M.A. Naimark, Research of Spectrum and Expansion in Eigenfunctions of Nonselfadjoint Differential Operators of Second Order on the Semiaxis. — Tr. Mosk. Mat. Obs., 3 (1954), 181–270. (Russian)
J.T. Schwartz, Some Non-selfadjoint Operators. — Comm. Pure Appl. Math. XIII (1960), 609–639. https://doi.org/10.1002/cpa.3160130405
M.A. Naimark, Linear Differential Operators. Nauka, Moscow, 1969. (Russian) (Engl. transl.: Frederic Ungar Publishing Co., New York, Part I 1967, Part II 1968.)
V.E. Ljance, Non-selfadjoint Differential Operators of Second Order on the Semiaxis. Supplement I to the book [17].
S.A. Orlov, Nested Matrix Discs that Depend Analytically on a Parameter, and Theorems on the Invariance of the Ranks of the Radii of the Limit Matrix Discs. — Izv. Akad. Nauk SSSR 40 (1976), No. 3, 593–644. (Russian) (Engl. transl.: Math. USSR-Izv. 10 (1976), 565–613).
A.M. Kholkin, Description of Selfadjoint Extensions of Differential Operators of Arbitrary Order on an Infinite Interval in the Absolutely Indeterminate Case. — Teor. Funkts., Funkts. Anal. Prilozhen. 1985, No. 44, 112–122. (Russian) (Engl. transl.: J. Sov. Math. 48 (1990), No. 3, 337–345). https://doi.org/10.1007/BF01101256
J.F. Brasche, M. Malamud, and H. Neidhardt, Weyl Function and Spectral Properties of Selfadjoint Extensions. — Integral Equations Operator 43 (1990), No. 3, 264–289.
V.A. Derkach, S. Hassi, M.M. Malamud, and H.S.V. Snoo, Weyl Functions and Intermediate Extensions of Symmetric Operators. — Dopov. Nats. Akad. Nauk Ukr. (Mat. Prirodozn. Tekh. Nauki) (2001), No. 10, 33–39.
V.I. Khrabustovskiy, On the Characteristic Matrix of Weyl–Titchmarch Type for Differential-Operator Equations with the Spectral Parameter Entering Linearly or by Nevanlinna Type. – Mat. Fiz. Anal. Geom. 10 (2003), No. 2, 205–227. (Russian)
M. Lesch and M.M. Malamud, On the Deficiency Indices and Selfadjointness ofSymmetric Hamiltonian Systems. — J. Diff. Equations 10 (2003), 556–615. https://doi.org/10.1016/S0022-0396(02)00099-2