Abstract
We establish theorems about the expansion of nonuniformly right-definite multiparameter problems in generalized eigenfunctions. We also give an abstract criterion of existence of commuting self-adjoint extensions of a family of symmetric operators.
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References
F. V. Atkinson,Multiparameter Eigenvalue Problems. Vol. 1: Matrices and Compact Operators, Academic Press, New York (1972).
B. D. Sleeman,Multiparameter Spectral Theory in Hilbert Space, Pitman Press, London (1978).
H. Volkmer, “Multiparameter eigenvalue problems and expansion theorems,”Springer Lect. Notes Math.,1356 (1988).
P. J. Browne, “A singular multi-parameter eigenvalue problem in second order ordinary differential equations,”J. Different. Equal.,12, 81–94 (1972).
G. A. Isaev, “On singular multiparameter differential operators. Expansion theorems,”Mat. Sb.,131, No. 1 (9), 52–72 (1986).
G. Sh. Guseinov, “On the spectral theory of multiparameter second-order difference equations,”Izv. Akad. Nauk SSSR, Ser. Mat.,51, No. 4, 785–811 (1987).
Yu. M. Berezanskii and A. Yu. Konstantinov, “Expansion in eigenvectors of multiparameter spectral problems,”Usp. Mat. Nauk,46, No. 6, 158–159 (1991).
Yu. M. Berezanskii and A. Yu. Konstantinov, “On expansion in eigenvectors of multiparameter spectral problems,”Funkts. Anal. Prilozhen.,26, No. 1, 81–83 (1992).
Yu. M. Berezanskii and A. Yu. Konstantinov, “Expansion in eigenvectors of multiparameter spectral problems,”Ukr. Mat. Zh.,44, No. 7, 901–913 (1992).
P. Binding, “Nonuniform right definiteness,”J. Math. Anal. Appl.,102, 233–243 (1984).
A. Yu. Konstantinov, “Nonuniformly definite multiparameter problems,”Usp. Mat. Nauk,49, No. 4, 87 (1994).
A. Yu. Konstantinov, “Commuting extensions of Hermitian operators and multiparameter spectral problems,”Usp. Mat. Nauk,48, No. 4, 205 (1993).
A. Yu. Konstantinov, “Eigenfunction expansion of right definite multiparameter problems,”Operator Theory: Advances and Applications,70, 343–346 (1993).
A. Yu. Konstantinov, “Multiparameter spectral problems and commuting extensions of Hermitian operators,”Funkts. Anal. Prilozhen.,28, No. 2, 55–57 (1994).
H. Volkmer, “On multiparameter theory,”J. Math. Anal. AppL,86, 44–53 (1982).
Yu. M. Berezanskii,Expansion of Self-Adjoint Operators in Eigenfunctions [in Russian], Naukova Dumka, Kiev (1965).
Yu. M. Berezanskii, G. F. Us, and Z. G. Sheftel'Functional Analysis [in Russian], Vyshcha Shkola, Kiev (1990).
A. Yu. Konstantinov and A. G. Stadnyuk, “Expansion in eigenfunctions of symmetric multiparameter problems,”Ukr. Mat. Zh.,48, No. 8, 1166–1169 (1993).
Ya. A. Roitberg,Elliptic Boundary-Value Problems in Generalized Functions. III [in Russian], Preprint, Chernigov Pedagogical Institute, Chernigov (1991).
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To Yurii Makarovich Berezanskii
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 659–670, May, 1995.
This research was supported by the Ukrainian State Committee on Science and Technology.
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Konstantinov, A.Y. Nonuniformly definite multiparameter spectral problems. Ukr Math J 47, 760–772 (1995). https://doi.org/10.1007/BF01059049
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DOI: https://doi.org/10.1007/BF01059049