Literature cited
Yu. M. Berezanskii, Eigenfunction Expansion of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
M. L. Gorbachuk, “Self-adjoint boundary-value problems for a second-order differential equation with an unbounded operator coefficient,” Funktsional'. Analiz i Ego Prilozhen.,5, No. 1, 10–21 (1971).
L. I. Vainerman, “Self-adjoint boundary-value problems for strongly elliptic and hyperbolic second-order equations in Hilbert space,” Dokl. Akad. Nauk SSSR,218, No. 4, 745–748 (1974).
V. M. Bruk, “Dissipative extensions of differential operators of elliptic type,” in: Functional Analysis,3, Ul'yanovsk (1975).
L. I. Vainerman and M. L. Gorbachuk, “Direct methods of qualitative spectral analysis for a singular Sturm-Liouville equation with an unbounded operator potential,” Dop. Akad. Nauk UkrSSR, Ser. A, No. 7, 583–585 (1972).
V. I. Gorbachuk and M. L. Gorbachuk, “On the spectrum of self-adjoint extensions of a minimal operator generated by the Sturm-Liouville equation with an operator potential,” Ukrains. Matem. Zh.,24, No. 6, 726–734 (1972).
L. I. Vainerman, “Boundary-value problems for a strongly elliptic second-order equation in Hilbert space,” Kibernetika, No. 6 (1973).
V. A. Mikhailets', “Boundedness from the left and discreteness of an operator of the non-self-adjoint Schrödinger operator equation,” Dop. Akad. Nauk UkrSSR, Ser. A, No. 4, 305–307 (1974).
V. P. Maslov, “Discreteness criterion of spectrum of Sturm-Liouville equation with operator coefficient,” Funktsional'. Analiz i Ego Prilozhen.,2, No. 2, 63–67 (1968).
M. Otelbaev, “On the nature of the spectrum of one-dimensional differential operators,” Vestnik MGU, No. 5 (1972).
V. I. Gorbachuk and M. L. Gorbachuk, “On certain classes of boundary-value problems for the Sturm-Liouville equation with an operator potential,” Ukrainsk. Matem. Zh.,24, No. 3, 291–306 (1972).
B. M. Levitan and G. A. Suvorchenkova, “Sufficient conditions of discreteness of spectrum of Sturm — Liouville equation with operator coefficients,” Funktsional'. Analizi Ego Prilozhen.,2, No. 2 (1961).
I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators [in Russian], Fizmatgiz, Moscow (1963).
V. I. Gorbachuk and M. L. Gorbachuk, “Certain problems of spectral theory of a linear second-order differential equation with an unbounded operator coefficient,” Ukrainsk. Matem. Zh.,23, No. 1, 3–14 (1971).
L. I. Vainerman, “Dissipative boundary-value problems for a second-order differential equation with an unbounded operator coefficient,” Ukrainsk. Matem. Zh.,25, No. 4, 530–534 (1974).
L. I. Vainerman and M. L. Gorbachuk, “On self-adjointness of semibounded abstract differential operators,” Ukrainsk. Matem. Zh.,22, No. 6, 806–808 (1970).
M. M. Gekhtman, “On self-adjointness of abstract differential operators,” Matem. Zametki,6, No. 1, 65–72 (1069).
F. S. Rofe-Beketov, “Self-adjoint extensions of differential operators in a space of vector functions,” Dokl. Akad. Nauk SSSR,184, No. 5, 1034–1037 (1969).
M. L. Gorbachuk, A. N. Kochubei, and M. A. Rybak, “Dissipative extensions of differential operators in a space of vector functions,” Dokl. Akad. Nauk SSSR,205, No. 5, 1029–1032 (1972).
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford (1946).
D. R. Yafaev, “On the negative spectrum of the Schrödinger operator equation,” Matem. Zametki,7, No. 6, 753–763 (1970).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 28, No. 4, pp. 473–482, July–August, 1976.
In conclusion the author expresses his deep gratitude to M. L. Gorbachuk for posing the problem and for his direct assistance in writing this paper.
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Kutovoi, V.A. Spectrum of sturm — Liouville equation with unbounded operator coefficient. Ukr Math J 28, 365–372 (1976). https://doi.org/10.1007/BF01101656
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DOI: https://doi.org/10.1007/BF01101656