On a framework of scattering for dissipative systems
Let \({\mathcal H}\) be a separable Hilbert space. Let \(A\) be the generator of a contraction semigroup \(\{V(t)\}_{t\geq 0}\) in \({\mathcal H}\) and \(\{U_0(t)\}_{t\in\mathbb{R}}\) be an unitary group in \({\mathcal H}\). Let \({\mathcal H}_b\) be the space generated by the eigenvectors of \(A\) with real eigenvalues. Under some assumptions, the author proves the existence of the wave operator \[ wf= \lim_{t\to\infty} U_0(t) V(t) f,\quad f\in{\mathcal H}^1_b. \] As a corollary, he finds scattering solutions of \(d(V(t)f)/dt= -iAV(t)f\), \(f\in D(A)\). He also obtains standard results concerning real eigenvalues of \(A\): Any non-zero eigenvalue has finite multiplicity and the possible finite accumulation point of the real eigenvalues of \(A\) is zero. As applications, one considers elastic wave equation with dissipative boundary condition in a half space of \(\mathbb{R}^3\), the Schrödinger equation with complex valued potential and acoustic wave equation with dissipative term in inhomogeneous media.
scientific article; zbMATH DE number 1894636
scientific article