
SIGMA 17 (2021), 077, 13 pages arXiv:2105.08641
https://doi.org/10.3842/SIGMA.2021.077
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan
SecondOrder Differential Operators in the Limit Circle Case
Dmitri R. Yafaev ^{abc}
^{a)} Université de Rennes, CNRS, IRMARUMR 6625, F35000 Rennes, France
^{b)} St. Petersburg University, 7/9 Universitetskaya Emb., St. Petersburg, 199034, Russia
^{c)} Sirius University of Science and Technology, 1 Olympiysky Ave., Sochi, 354340, Russia
Received May 20, 2021, in final form August 14, 2021; Published online August 16, 2021
Abstract
We consider symmetric secondorder 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 selfadjoint 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 selfadjoint realizations. In particular, this yields a simple representation for the CauchyStieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.
Key words: secondorder differential equations; minimal and maximal differential operators; selfadjoint extensions; quasiresolvents; resolvents.
pdf (343 kb)
tex (17 kb)
References
 Coddington E.A., Levinson N., Theory of ordinary differential equations, McGrawHill Book Company, Inc., New York  Toronto  London, 1955.
 Naimark M.A., Linear differential operators, Ungar, New York, 1968.
 Nevanlinna R., Asymptotische Entwickelungen beschränkter Funktionen und das Stieltjessche Momentenproblem, Ann. Acad. Sci. Fenn. A 18 (1922), no. 5, 152.
 Reed M., Simon B., Methods of modern mathematical physics. II. Fourier analysis, selfadjointness, Academic Press, New York  London, 1975.
 Schmüdgen K., Unbounded selfadjoint operators on Hilbert space, Graduate Texts in Mathematics, Vol. 265, Springer, Dordrecht, 2012.
 Simon B., The classical moment problem as a selfadjoint finite difference operator, Adv. Math. 137 (1998), 82203, arXiv:mathph/9906008.
 Yafaev D.R., Analytic scattering theory for Jacobi operators and BernsteinSzegö asymptotics of orthogonal polynomials, Rev. Math. Phys. 30 (2018), 1840019, 47 pages, arXiv:1711.05029.
 Yafaev D.R., Semiclassical asymptotic behavior of orthogonal polynomials, Lett. Math. Phys. 110 (2020), 28572891, arXiv:1811.09254.
 Yafaev D.R., Selfadjoint Jacobi operators in the limit circle case, J. Operator Theory, to appear, arXiv:2104.13609.
 Zettl A., SturmLiouville theory, Mathematical Surveys and Monographs, Vol. 121, Amer. Math. Soc., Providence, RI, 2005.

