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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.

MSC:

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
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References:

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