Geodesic
Second-Order Differential Operators in the Limit Circle Case
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy–Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.
Keywords: second-order differential equations, minimal and maximal differential operators, self-adjoint extensions, quasiresolvents, resolvents. 
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     author = {Dmitri R. Yafaev},
     title = {Second-Order {Differential} {Operators} in the {Limit} {Circle} {Case}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a76/}
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Dmitri R. Yafaev. Second-Order Differential Operators in the Limit Circle Case. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a76/

[1] Coddington E. A., Levinson N., Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York – Toronto – London, 1955

[2] Naimark M. A., Linear differential operators, Ungar, New York, 1968

[3] Nevanlinna R., “Asymptotische Entwickelungen beschränkter Funktionen und das Stieltjessche Momentenproblem”, Ann. Acad. Sci. Fenn. A, 18:5 (1922), 1–52

[4] Reed M., Simon B., Methods of modern mathematical physics, v. II, Fourier analysis, self-adjointness, Academic Press, New York – London, 1975

[5] Schmüdgen K., Unbounded self-adjoint operators on Hilbert space, Graduate Texts in Mathematics, 265, Springer, Dordrecht, 2012 | DOI

[6] Simon B., “The classical moment problem as a self-adjoint finite difference operator”, Adv. Math., 137 (1998), 82–203 | DOI

[7] Yafaev D. R., “Analytic scattering theory for Jacobi operators and Bernstein–Szegö asymptotics of orthogonal polynomials”, Rev. Math. Phys., 30 (2018), 1840019, 47 pp. | DOI

[8] Yafaev D. R., “Semiclassical asymptotic behavior of orthogonal polynomials”, Lett. Math. Phys., 110 (2020), 2857–2891, arXiv: 1811.09254 | DOI

[9] Yafaev D. R., “Self-adjoint Jacobi operators in the limit circle case”, J. Operator Theory (to appear) , arXiv: 2104.13609

[10] Zettl A., Sturm–Liouville theory, Mathematical Surveys and Monographs, 121, Amer. Math. Soc., Providence, RI, 2005 | DOI