Resolvent estimates for a sum of tensor products with applications to the spectral theory of differential operators
Let H be a self-adjoint operator in a Hilbert space \({\mathcal H}\), \(\sigma\) (H) its spectrum and \(R(z)=(H-z)^{-1}\), Im \(z\neq 0\), its resolvent. Let \(K\subseteq \sigma (H)\). H is said to satisfy a limiting absorption principle (l.a.p. for short) in K, if the limits \(R^{\pm}(\lambda)=\lim_{\epsilon \to 0^+}R(\lambda \pm i\epsilon)\), \(\lambda\in K\) exist in some suitable operator topology. (*) \(H=H_ 1\otimes I_ 2+I_ 1\otimes H_ 2\) where \(H_ 1\) and \(H_ 2\) are self-adjoint operators on Hilbert spaces \({\mathcal H}_ 1\) and \({\mathcal H}_ 2\), and \(I_ 1\) and \(I_ 2\) are the identity operators of these Hilbert spaces, respectively. Clearly H acts on \({\mathcal H}={\mathcal H}_ 1\otimes {\mathcal H}_ 2.\) The paper revolves around the question: Given that \(H_ 1\) and \(H_ 2\) satisfy a l.a.p. in \({\mathcal H}_ 1,{\mathcal H}_ 2\) respectively. What are the conditions needed to guarantee that H defined by (*), satisfies a l.a.p. in \({\mathcal H}?\) After developing some elementary expressions for the resolvent of H, the authors relate the validity of the l.a.p. for H with the Hölder continuity properties of \(A_ 1(\lambda)\) i.e. \(\frac{d}{d\lambda}E_ 1(\lambda)\), where \(E_ 1(\lambda)\) is the spectral family of \(H_ 1\). The succeed in obtaining the l.a.p. for a class of operators including the Hamiltonian of a uniform electric field. Later they replace Hölder continuity properties by other assumptions on \(H_ 1,H_ 2\). Several concrete examples of differential operators are outlined. There are two appendices one containing the review of some basic properties of the spectral measure associated with a self-adjoint constant coefficient differential operator in \({\mathbb{R}}^ n\), and the l.a.p. for some second order ordinary differential operators is discussed in the second.
self- adjoint constant coefficient differential operator
second order ordinary differential operators
scientific article; zbMATH DE number 3949094
scientific article