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On the spectral theory of the Bessel operator on a finite interval and the half-line. (To the spectral theory of the Bessel operator on finite interval and half-line.) (English. Ukrainian original) Zbl 1354.47031

J. Math. Sci., New York 211, No. 5, 624-645 (2015); translation from Ukr. Mat. Visn. 12, No. 2, 160-169 (2015).
In this work, the authors consider the Bessel operator \[ \tau_{\nu}=-\frac{d^{2}}{dx^{2}}+\frac{\nu^{2}-\frac{1}{4}}{x^{2}}, \quad \nu \in \left[0,1\right) - \left\{\frac{1}{2}\right\}. \] They study the minimal and maximal Bessel operators on a finite interval and a half-line. They prove that the domain of the minimal operator \(A\left(\nu, \infty\right)_{\min}\) associated with \(\tau_{\nu}\) in \(L^{2}\left(\mathbb{R}_{+}\right)\) is given by \[ \text{dom}\left(A\left(\nu, \infty\right)_{\min}\right)=H_{0}^{2}\left(\mathbb{R}_{+}\right) \] and they prove a similar formula for the operator on a finite interval.
Moreover, they investigate spectral properties of the Bessel operator by applying the technique of boundary triplets and the corresponding Weyl functions. They construct a boundary triplet for the maximal operator in \(L^{2}\left(\mathbb{R}_{+}\right)\) and \(L^{2}(0,b)\) and compute the corresponding Weyl functions. They also determine the domains of Friedrichs and Krein extensions.
In addition, the authors describe all self-adjoint and all nonnegative self adjoint extensions of the minimal Bessel operator. Besides, they obtain the Weyl functions on a half-line as the limit of the corresponding Weyl functions of the operator considered in a finite interval.

MSC:

47E05 General theory of ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
34L05 General spectral theory of ordinary differential operators
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