On elements of scattering theory for abstract Schrödinger equation. Lax-Phillips approach
In a series of papers the author has developed a version of the Lax-Phillips scattering theory for an abstract wave equation \(u_{tt}=-Lu\) and Schrödinger equation \(iu_t=Lu\) where \(L\) is a positive self-adjoint operator on a Hilbert space \(H\) which is an extension of a symmetric operator \(B^2\). Here \(B\) is a simple maximal symmetric operator on a certain subspace \(H_0\subset H\). The unperturbed operator \(L_0\) satisfies the condition \((L_0u,u)=\|B^*u\|^2\) for all \(u\in D(L_0)\). If \(H=H_0\), then \(L\) is called 0-perturbed. In the paper under review the author finds an expression for the analytic continuation of the \(S\)-matrix corresponding to a 0-perturbed Hamiltonian. A description for all 0-perturbations of the operator \(-\Delta\) on \(L_2(\mathbb R^3)\) is also found.
scientific article; zbMATH DE number 1668467