Homogeneous Schrödinger operators on half-line. (English) Zbl 1226.47049

This article is devoted to a thorough study of the differential expression \(L_m:=-\partial_x^2+(m^2-1/4)x^{-2}\) on \(C^\infty_{c}(0,\infty)\), and of the operators it induces in the complex space \(L^2(0,\infty)\). These operators appear in the decomposition of the Aharonov-Bohm Hamiltonian. They are also of interest due to their role in the theory of special functions.
The results state essentially that there is a unique holomorphic family \(\{H_m\}_{\operatorname{Re}\,m>-1}\) of closed operators in \(L^2\) such that \(H_m\) coincides with the closure of \(L_m\) for \(m\geq1\). The operators \(H_m\) are homogeneous of degree \(-2\) with respect to the group of dilations in \(L^2\). The spectrum and essential spectrum of \(H_m\) is \([0,\infty)\), independently of \(m\). The numerical range of \(H_m\) is calculated explicitly. If \(\operatorname{Re}\,m>-1\), \(\operatorname{Re}\,k>-1\) and \(\lambda\in\mathbb{C}\setminus[0,\infty)\), then \((H_m-\lambda)^{-1}- (H_k-\lambda)^{-1}\) is a compact operator. There are also results on the scattering theory for \(H_m\), e.g., an explicit expression for the wave operators.
The proofs rest on an abstract study of operators that are homogeneous with respect to a strongly continuous group of unitary operators in a Hilbert space. Moreover, explicit formulas and estimates involving Bessel functions and the Hankel transform are employed.


47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L25 Scattering theory, inverse scattering involving ordinary differential operators
47A55 Perturbation theory of linear operators
47A40 Scattering theory of linear operators
47A20 Dilations, extensions, compressions of linear operators
47A12 Numerical range, numerical radius
81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text: DOI arXiv


[1] Everitt, W. N.; Kalf, H., The Bessel differential equation and the Hankel transform, J. Comput. Appl. Math., 208, 3-19 (2007) · Zbl 1144.34013 · doi:10.1016/j.cam.2006.10.029
[2] Kato, T.: Perturbation theory for linear operators · Zbl 0148.12601
[3] Kellendonk, J., Richard, S.: Weber-Schafheitlin type integrals with exponent 1. Int. Transforms Special Funct. 20 (2009) · Zbl 1163.33313
[4] Le Yaouanc, A.; Oliver, L.; Raynal, J.-C., The Hamiltonian (p^2 + m^2)^1/2 − α/r near the critical value α_c = 2/π, J. Math. Phys., 38, 3997-4012 (1997) · Zbl 0883.47084 · doi:10.1063/1.532106
[5] Pankrashkin, K., Richard, S.: Spectral and scattering theory for the Aharonov-Bohm operators (preprint) (2009). arXiv:0911.4715 · Zbl 1230.81054
[6] Stone, M.H.: Linear transformations in Hilbert space. Am. Math. Soc. (1932) · JFM 58.0420.02
[7] Watson, G. N., A Treatise on the Theory of Bessel Functions (1995), Cambridge: Cambridge University Press, Cambridge · Zbl 0849.33001
[8] Zhu, K.: Operator theory in functions spaces, 2nd edn. Mathematical Surveys and Monographs, vol. 138, ISSN 0076-5376 (2007) · Zbl 1123.47001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.