Homogeneous Schrödinger operators on half-line
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.
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