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Bao, Qinglan; Wei, Guangsheng; Zettl, Anton

The Friedrichs extension of regular symmetric differential operators. (English) Zbl 1486.34067

Oper. Matrices 16, No. 1, 213-237 (2022).
Summary: We increase the class of regular symmetric differential operators andfind, explicitly, the boundary conditions which determine the Friedrichs extension of each one of these symmetric differential operators.

Cited in 2 Documents

MSC:

34B24 Sturm-Liouville theory
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B08 Parameter dependent boundary value problems for ordinary differential equations
34L05 General spectral theory of ordinary differential operators

Keywords:

Friedrichs extension; symmetric differential expressions; boundary matrix
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Full Text: DOI

References:

[1] Q. BAO, J. SUN, X. HAO, A. ZETTL,Characterization of self-adjoint domains for regular even order C -symmetric differential operators, Electronic Journal of Qualitative Theory of Differential Equations, · Zbl 1449.47082
[2] J. BEHRNDT, S. HASSI, H.DESNOO,Boundary Value Problems, Weyl Functions, and Differential Operators, Monographs in Mathematics, vol. 108, 2020. · Zbl 1457.47001
[3] J. BARRETT,Oscillation theory of ordinary linear differential equations, Advances in mathematics, 1969, 3 (4): 415-509. · Zbl 0213.10801
[4] E. CODDINGTON, N. LEVENSON,Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. · Zbl 0064.33002
[5] N. DUNFORD, J. SCHWARTZ,Linear Operators, part II, Interscience Publishers, New York, 1963. · Zbl 0128.34803
[6] K. FRIEDRICHS,Spektraltheorie halbbeschrkter operatoren und anwendung auf die spektralzerlegung von differentialoperatoren, Mathematische Annalen, 1934, 109 (1): 465-487 (in German). · Zbl 0008.39203
[7] K. FRIEDRICHS,Uber die ausgezeichnete randbedingung in der spektraltheorie der halbbeschr¨ankten¨ gew¨ohnlichen differentialoperatoren zweiter ordnung, Mathematische Annalen, 1936, 112 (1): 1-23 (in German). · JFM 61.1198.02
[8] I. GLAZMAN,On the theory of singular differential operators, Uspehi Molekularnoj Onkologii, 1950, 6 (40): 102-135; English translation Transactions of the American Mathematical Society, 1962, 4 (1): 331-372.
[9] D. HINTON,Disconjugate properties of a system of differential equations, Journal of Differential Equations, 1966, 2 (4): 420-437. · Zbl 0161.27904
[10] I. HALPERIN,Closures and adjoints of linear differential operators, Annals of Mathematics (Second Series), 1937, 38 (4): 880-919. · JFM 63.0363.04
[11] V. KOGAN, F. ROFE-BEKETOV,On square-integrable solutions of symmetric systems of differential equations of arbitrary order, Proceedings of the Royal Society of Edinburgh Section A Mathematics, 1976, 74 (1): 5-40. · Zbl 0333.34021
[12] M. M ¨OLLER, A. ZETTL,Weighted norm inequalities for the quasi-derivatives of ordinary differential operators, Results in Mathematics, 1993, 24 (1-2): 153-160. · Zbl 0787.34016
[13] M. M ¨OLLER, A. ZETTL,Semi-boundedness of ordinary differential operators, Journal of Differential Equations, 1995, 115 (1): 24-49. · Zbl 0817.34047
[14] M. M ¨OLLER, A. ZETTL,Symmetrical differential operators and their Friedrichs extension, Journal of Differential Equations, 1995, 115 (1): 50-69. · Zbl 0817.34048
[15] M. NAIMARK,Linear Differential Operators, Ungar, New York, 1968. · Zbl 0227.34020
[16] H. NIESSEN, A. ZETTL,The Friedrichs extension of regular ordinary differential operators, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1990, 114 (3): 229-236. · Zbl 0712.34020
[17] W. REID,Oscillation criteria for self-adjoint differential systems, Transactions of the American Mathematical Society, 1961, 101 (1): 91-106. · Zbl 0114.29202
[18] D. SHIN,Existence theorems for quasi-differential equations of order n, Doklady Akademii Nauk SSSR 1938, 18 (1938): 515-518. · JFM 64.0427.01
[19] D. SHIN,On quasi-differential operators in Hilbert space, Doklady Akademii Nauk SSSR, 1938, 18 (1938): 523-526. · JFM 64.0382.02
[20] D. SHIN,Uber die L¨osungen einer quasi-differentialgleichung der n -ten Ordnung¨, (Russian. German summary) Rec. Math. [Mat. Sbornik] N. S. 1940, 7 (49): 479-532. · Zbl 0023.31701
[21] M. STONE,Linear Transformations in Hilbert Space and Their Applications to Analysis, American Mathematical Society Colloquium Publications, vol. XV, 1932. · JFM 58.0420.02
[22] E. TITCHMARSH,Eigenfunction expansions associated with second order differential equations, Part I, Oxford University Press, 1962. · Zbl 0099.05201
[23] P. WALKER,A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable square, Journal of the London Mathematical Society, 1974, s2-9 (1): 151-159. · Zbl 0308.34011
[24] H. WEYL,Uber gew¨ohnliche differentialgleichungen mit singularit¨aten und die zugeh¨origen¨ entwicklungen willk¨urlicher funktionen, Mathematische Annalen, 1910, 68 (2): 220-269. · JFM 41.0343.01
[25] J. WEIDMANN,Linear Operators in Hilbet Space, Springer-Verlag, 1976.
[26] A. WANG, A. ZETTL,Symmetric differential operators, https://www.ams.org/publications/authors/books/postpub/surv-245-wz0107.pdf, 2020. · Zbl 1495.47046
[27] A. WANG, J. SUN, A. ZETTL,Characterrization of domains of self-adjoint ordinary differential operators, Journal of Differential Equations, 2009, 246 (4): 1600-1622. · Zbl 1169.47033
[28] J. WEIDMANN,Spectral Theory of Ordinary Differential Operators, Springer-Verlag, 1987. · Zbl 0647.47052
[29] A. ZETTL,Formally self-adjoint quasi-differential operators, Rocky Mountain Journal of Mathematics, 1975, 5 (3): 453-474 · Zbl 0443.34019
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