Absorption semigroups, their generators, and Schrödinger semigroups
To find a general solution for the heat equation with absorption- excitation, \[ u_ t=()\Delta u-Vu,\quad u(0)=u_ 0,\quad in\quad L_ p({\mathbb{R}}^{\nu}) \] with a ``bad'' \(V: {\mathbb{R}}^{\nu}\to {\mathbb{R}}\), it seems natural to try to approximate V by \(L_{\infty}\)-functions and find out whether the corresponding semigroups converge. It helps that there is a natural order in \(L_ p({\mathbb{R}}^{\nu})\) and the generated semigroups are order-preserving. If V is suitable for this procedure, the negative generator of the corresponding semigroup should be considered as the natural realization of the Schrödinger operator \(()\Delta +V\). In particular, if \(p=2\), this realization is selfadjoint and leads to the solution of the usual initial value problem for the Schrödinger equation, \(u_ t=(1/i)(-\Delta u+Vu)\). The author realizes this program, suitably defining the admissability of V, giving criteria for admissability, and arriving at in parts unexpected results.
general solution for the heat equation with absorption-excitation
initial value problem for the Schrödinger equation
scientific article; zbMATH DE number 4021980
scientific article