A limiting absorption principle for Schrödinger operators with generalized von Neumann-Wigner potentials. I: Construction of approximate phase (Q1359604)

The authors generalize an Agmon result concerning the principle of limiting absorption for short range perturbations of the Laplacian acting in \(\mathcal L^{2,s}( \mathbb{R}^3)\). Let \(V\) be a short range real potential. That is to say the function \(V(x)\) is real and the map \(V: \mathcal H^{2,-s} (\mathbb{R}^3) \to \mathcal L^{2,s}( \mathbb{R}^3)\) is compact for some \(s>1/2\). Next define, at least formally, \(H=-\Delta+V\) and as usual define the resolvent by \(R(z) = (H-z)^{-1}\) for \(\text{Im }z\neq 0\). Then the result of Agmon implies the following: There is a discrete set \(\{\lambda_i\}\) in \(\mathbb{R} \setminus \{0\}\) such that the two limits, \[ R^\pm (\lambda) = \lim_{\varepsilon\to+0} R(\lambda\pm i\varepsilon), \quad \lambda\not= \lambda_i, 0, \] exist in the norm topology of bounded operators from \(\mathcal L^{2,s} (\mathbb{R}^3)\) to \(\mathcal H^{2,-s}(\mathbb{R}^3)\). [See conclusion (4.17) of Theorem 4.2 in \textit{S. Agmon}, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 2, 151-218 (1975; Zbl 0315.47007)]. Now the authors show that in the previous Agmon result the short range potential \(V\) can be replaced by a potential of the form \(c\sin b | x|/| x|^\beta+V(x)\), where \(c\) is an arbitrary real constant, \(\beta\) satisfies the inequality \(2/3 < \beta \leq 1\) and \(V\) is short range. Note that for \(c=0\) this potential yields the one of Agmon. Note also that for \(V=0\) and \(c=\beta=1\) this potential gives the oscillating part of the one of von Neumann-Wigner [Über merkwürdige diskrete Eigenwerte, Phys. Z. 30, 465-467 (1929)] or [\textit{M. Reed} and \textit{B. Simon}, Analysis of operators, Methods of Modern Mathematical Physics IV, Section XIII. 13, Example 1, Academic Press (1978; Zbl 0401.47001)]. In other words, there is a short range perturbation of this potential so that the resulting Schrödinger operator has a stricly positive eigenvalue. The novelty of the authors' result is the case \(2/3 < \beta <1\). The authors' proof hinges on an adaptation of the notion of an \textit{approximate phase} to their class of potentials. Part I gives this adaptation and constructs approximate phases. This construction, in turn, is an adaptation of the ones of [\textit{W. Jäger} and \textit{P. Rejto}, J. Math. Anal. Appl. 91, 192-228 (1983; Zbl 0533.35018)] and [\textit{A. Devinatz, R. Moeckel} and \textit{P. Rejto}, Integral Equations Oper. Theory 14, 13-68 (1991; Zbl 0731.47050)]. Then Part II shows that the existence of such approximate phases imply, under general circumstances, the principle of limiting absorption.