Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 077, 13 pages      arXiv:2105.08641
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

Second-Order Differential Operators in the Limit Circle Case

Dmitri R. Yafaev abc
a) Université de Rennes, CNRS, IRMAR-UMR 6625, F-35000 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

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.

Key words: second-order differential equations; minimal and maximal differential operators; self-adjoint extensions; quasiresolvents; resolvents.

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