Resonance eigenfunctions of a dilation-analytic Schrödinger operator, based on the Mellin transform (Q1081021)

Let \(H=H_ 0+V\), \(H_ 0=-\Delta\) with domain \(D(H_ 0)\), a Schrödinger operator on \(L^ 2({\mathbb{R}}^ 3)\) where the potential V satisfies for some \(0<a<\pi /2\) the conditions (i), (ii), (iii). (i) V is symmetric and \(H_ 0\)-compact; (ii) V is a-dilation-analytic; (iii) \(V(\theta)=U(\theta)VU(\theta)\) has a continuous extension from \(\{\theta_ 1|| Im \theta_ 1| <a\}\) to \(\{\theta_ 1|| Im \theta_ 1| \leq a\}\) as a function with values in \(B(D(H_ 0)\), \(L^ 2({\mathbb{R}}^ 3))\) (bounded linear operators from \(D(H_ 0)\) to \(L^ 2({\mathbb{R}}^ 3))\), \(\{U(\theta)\}_{\theta \in {\mathbb{R}}}\) being the dilation group. A notion of resonance eigenfunctions is introduced and an isomorphic connection to the space \(N(H_ M(\theta)-z_ 0)\), Im \(\theta\) \(>-(1/2)\arg z_ 0\), is given, where \(N(H_ M(\theta)-z_ 0)\) is the space of eigenfunctions associated with a resonance \(z_ 0\) and the \(\theta\)-dilated operator \(H_ m(\theta)\). - For a multiplicative, radial and analytic potential see \textit{R. G. Newton} [''Scattering theory of waves and particles'' (1982; Zbl 0496.47011)].