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.


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
Full Text: DOI arXiv


[1] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz, I. A. Stegun, Dover, New York, 1974.
[2] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, New York, 1993, Vol. 2. · Zbl 0874.47001
[3] Alekseeva, VS; Ananieva, AY, On extensions of the Bessel operator on a finite interval and a half-line, J. of Math. Sci., 187, 1-8, (2012) · Zbl 1393.47020
[4] Bruneau, L; Dereziński, J; Georgescu, V, Homogeneous Schrödinger operators on half-line, Ann. H. Poincaré, 12, 547-590, (2011) · Zbl 1226.47049
[5] S. Clark, F. Gesztesy, and R. Nichols, “Principal Solutions Revisited,” arXiv.org:1401.1285. · Zbl 0406.34029
[6] Derkach, VA; Malamud, MM, Generalized rezolvent and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal., 95, 1-95, (1991) · Zbl 0748.47004
[7] Derkach, VA; Malamud, MM, The extension theory of Hermitian operators and the moment problem, J. Math. Sci., 73, 141-242, (1995) · Zbl 0848.47004
[8] Eckhardt, J; Gesztesy, F; Nichols, R; Teschl, G, Weyl-titchmarsh theory for Sturm-Liouville operators with distributional potentials, Opuscula Math., 33, 467-563, (2013) · Zbl 1283.34022
[9] W. N. Everitt and H. Kalf, “The Bessel differential equation and the Hankel transform,” J. of Comp. Appl. Math., No. 208, 3-19 (2007). · Zbl 1144.34013
[10] Fulton, C, Titchmarsh-Weyl \(m\)-functions for second-order Sturm-Liouville problems with two singular endpoints, Math. Nachr., 281, 1418-1475, (2008) · Zbl 1165.34011
[11] Fulton, C; Langer, H, Sturm-Liouville operators with singularities and generalized Nevanlinna functions, Compl. Anal. Operat. Theory, 4, 179-243, (2010) · Zbl 1214.34022
[12] V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations, Kluwer, Dordrecht, 1991. · Zbl 0751.47025
[13] C. R. de Oliveira, Intermediate Spectral Theory and Quantum Dynamics, Birkhäuser, Basel, 2000.
[14] Kalf, H, A characterization of the Friedrichs extension of Sturm-Liouville operators, J. London Math. Soc., 17, 511-521, (1978) · Zbl 0406.34029
[15] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995. · Zbl 0836.47009
[16] Kochybei, AN, Self-adjoint extensions of the Schrödinger operator with singular potential, Sibir. Mat. Zh., 32, 60-69, (1991)
[17] A. Kostenko, A. Sakhnovich, and G. Teschl, Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials, arXiv:1007.0136v3. · Zbl 1248.34027
[18] Malamud, MM, On some classes of extensions of the Hermitian operators with gaps, Ukr. Mat. Zh., 44, 215-233, (1992) · Zbl 0804.47011
[19] M. A. Naimark, Linear Differential Operators, Ungar, New York, 1967. · Zbl 0219.34001
[20] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, New York, 1972. · Zbl 0242.46001
[21] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.