Upper bounds for Neumann-Schatten norms
Motivated by generalized Schrödinger operators, the author studies Schatten-class estimates for norms of resolvent differences of operators \(H\) and \(H^J\) where \(H^J\) is a perturbed form of \(H\). If \(H\) is a nonnegative selfadjoint operator in a Hilbert space \({\mathcal H}\) and \(J:{\mathcal D}(H^{s/2})\to{\mathcal K}\) is bounded (\({\mathcal K}\) another Hilbert space), it is shown that \(J(H+ \alpha)^{-1}\) belongs to the Schatten class \(C_p\) for \(p= 2(u- t)/(t- s/2)\) if \(s/2< t< u\), \(t- s/2< u-t\); \(J(H+ \alpha)^{-1}\) is a Hilbert-Schmidt operator. For \(s=1\) and \({\mathcal D}(J)\supseteq{\mathcal D}(H)\) there is a unique selfadjoint operator \(H^J\) in \({\mathcal H}\) such that \({\mathcal D}(H^J)\subseteq {\mathcal D}(J)\) and \[ (H^J f,g)= (H^{1/2} f, H^{1/2} g)_{\mathcal H}+ (Jf, Jg)_{\mathcal K}. \] Conditions for the compactness of \((H^J+ \alpha)^{-1}- (H+ \alpha)^{-1}\) are established and wave operators are studied. The special case \(H= (-\Delta)^r\) in \(L_2(\mathbb{R}^d)\) is investigated in detail, where \(H^J= (-\Delta)^r+ \mu\) for a suitable measure \(\mu\).
scientific article; zbMATH DE number 1565495