Perturbation theory for convolution semigroups
The paper under review is an extension in various directions of the results from the author's previous paper [J.~Funct.\ Anal.\ 256, No.\,6, 1998--2025 (2009; Zbl 1168.47035)] devoted to Schrödinger operators \(-\Delta+V\) with singular negative potentials \(V\). In the present paper, the author deals with the generators \(T_p\) of convolution semigroups (not necessarily symmetric) \[ S_p(t): L^p ({\mathbb R}^N)\to L^p ({\mathbb R}^N),\;f\mapsto \int f(x-y) m_t(dy), \] where \(\{m_t\}_{t\geq 0}\) are Borel sub-probability measures on \({\mathbb R}^N\) such that \(m_0 = \delta_0\) (Dirac measure), \(m_t \star m_s = m_{t+s}\) and \(m_t\to m_0\) (\(t\to 0_+\)). The typical examples of such functions are Gaussian semigroups, \(\alpha\)-stable semigroups, relativistic Schrödinger semigroups, etc. The author provides a general perturbation theory of their generators by real potentials \(V = V_+ - V_-\), where \(V_+, V_-\) are singular nonnegative measurable functions. In the investigation of \(L^1\)-perturbation theory, the smallness of the Kato norm with respect to \(T_1^+=T_1+V_+\) is substituted by the weaker assumption that \(V_-\) is \(T_1^+\)-weakly compact. This assumption implies the following spectral radius \(r_\sigma\) inequality \[ \lim_{\lambda\to\infty} r_\sigma[V_-(\lambda-T_1^+)^{-1}]<1. \] The proof of the generation result is based on Desch's theorem. Feynman-Kac formulas are also proved in this framework. In the \(L^2\) setting, Dirichlet form-perturbation theory is related to \(L^1\)-theory. Using interpolation arguments, it is shown (even in the non-symmetric case) how to extend corresponding semigroups to \(L^p\) spaces as a \(C_0\)-semigroup. Finally, various perturbation problems for subordinate semigroups and measure perturbations of the Hamiltonian \(T_2\) are considered.
\(L^{1}\)-relative weakly compact perturbation
scientific article; zbMATH DE number 5731618
scientific article