Abstract
The self-adjoint and m-sectorial extensions of coercive Sturm–Liouville operators are characterised, under minimal smoothness conditions on the coefficients of the differential expression.
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Alonso A., Simon B.: The Birman–Kreĭn–Vishik theory of selfadjoint extensions of semibounded operators. J. Oper. Theory 4(2), 251–270 (1980)
Alonzo, A., Simon, B.: Addenda to: “The Birman–Kreĭn-Vishik theory of selfadjoint extensions of semibounded operators” [J. Operator Theory 4 (1980), no. 2, 251–270; MR 81m:47038]. J. Oper. Theory 6(2), 407 (1981)
Arlinskii Y.: Abstract boundary conditions for maximal sectorial extensions of sectorial operators. Math. Nachr. 209, 5–36 (2000)
Arlinskiĭ, Y.: Boundary triplets and maximal accretive extensions of sectorial operators. In: Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Ser., vol. 404, pp. 35–72. Cambridge University Press, Cambridge (2012)
Ashbaugh, M.S., Gesztesy, F., Mitrea, M., Shterenberg, R., Teschl, G.: A survey on the Krein–von Neumann extension, the corresponding abstract buckling problem, and Weyl-type spectral asymptotics for perturbed Krein Laplacians in nonsmooth domains. In: Mathematical Physics, Spectral Theory and Stochastic Analysis, Oper. Theory Adv. Appl., vol. 232, pp. 1–106. Birkhäuser/Springer Basel AG, Basel (2013)
Birman M.Š: On the theory of self-adjoint extensions of positive definite operators. Mat. Sb. N.S. 38(80), 431–450 (1956)
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1987). Oxford Science Publications
Grubb G.: A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Sup. Pisa (3) 22, 425–513 (1968)
Grubb G.: Les problèmes aux limites généraux d’un opérateur elliptique, provenant de le théorie variationnelle. Bull. Sci. Math. (2) 94, 113–157 (1970)
Grubb G.: Spectral asymptotics for the “soft” selfadjoint extension of a symmetric elliptic differential operator. J. Oper. Theory 10(1), 9–20 (1983)
Kalf H.: A characterization of the Friedrichs extension of Sturm–Liouville operators. J. Lond. Math. Soc. (2) 17(3), 511–521 (1978)
Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition
Kreĭn, M.G.: The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I. Rec. Math. [Mat. Sbornik] N.S. 20(62), 431–495 (1947)
Kreĭn M.G.: The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. II. Mat. Sbornik N.S. 21(63), 365–404 (1947)
Naĭmark, M.A.: Linear Differential Operators. Part II: Linear Differential Operators in Hilbert Space. With additional material by the author, and a supplement by V. È. Ljance. Translated from the Russian by E. R. Dawson. English translation edited by W. N. Everitt. Frederick Ungar Publishing Co., New York (1968)
Rellich F.: Halbbeschränkte gewöhnliche Differentialoperatoren zweiter Ordnung. Math. Ann. 122, 343–368 (1951)
Rosenberger R.: A new characterization of the Friedrichs extension of semibounded Sturm-Liouville operators. J. Lond. Math. Soc. (2) 31(3), 501–510 (1985)
Vishik, M.: On general boundary conditions for elliptic differential operators. Trudy Moskov. Mat. Obsc (Russian) (English translation in Am. Math. Soc. Transl. 24, 107–172), pp. 187–246 (1952)
Yao S., Sun J., Zettl A.: The Sturm–Liouville Friedrichs extension. Appl. Math. 60(3), 299–320 (2015)
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The Authors are grateful to Professor Gerd Grubb for providing information on the history of the problem, to Mr. Christoph Fischbacher for reviewing the manuscript and the referee for helpful comments on the paper.
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Brown, B.M., Evans, W.D. Selfadjoint and m Sectorial Extensions of Sturm–Liouville Operators. Integr. Equ. Oper. Theory 85, 151–166 (2016). https://doi.org/10.1007/s00020-016-2296-z
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DOI: https://doi.org/10.1007/s00020-016-2296-z