Absorption semigroups, their generators, and Schrödinger semigroups
type: Item
instance of: scholarly article
title: Absorption semigroups, their generators, and Schrödinger semigroups
zbMATH Open document ID: 0628.47027
DOI: 10.1016/0022-1236(86)90036-4
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 4021980
zbMATH Keywords: absorption semigroup
zbMATH Keywords: general solution for the heat equation with absorption-excitation
zbMATH Keywords: semigroups
zbMATH Keywords: realization of the Schrödinger operator
zbMATH Keywords: initial value problem for the Schrödinger equation
author: Jürgen Voigt
reviewed by: Heinrich Hering
MaRDI profile type: Publication
full work available at URL: 0022-1236(86)90036-4
OpenAlex ID: W2010769307
cites work: Brownian motion and harnack inequality for Schrödinger operators
cites work: Regularity properties of Schrödinger and Dirichlet semigroups
cites work: Q3951111
cites work: Q3906660
cites work: On the theory of semigroups of operators on locally convex spaces
cites work: Q3897567
cites work: The Trotter product formula for perturbations of semibounded operators
cites work: Q5526484
cites work: Q3923136
cites work: Q3244654
cites work: Uniqueness of Solutions to
cites work: Q5526485
cites work: Q4094022
cites work: SOME PROBLEMS ON EXPANSION IN GENERALIZED EIGENFUNCTIONS OF THE SCHRÖDINGER OPERATOR WITH STRONGLY SINGULAR POTENTIALS
cites work: Schrödinger operators with singular magnetic vector potentials
cites work: On perturbation theory for semi-groups of operators
cites work: An \(L^ p\) theory for Schrödinger operators with nonnegative potentials
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q3909512
cites work: Q4066072
cites work: Cut-off potentials and forms extensions
cites work: Schrödinger operators with \(L^p_{loc}\)-potentials
cites work: A canonical decomposition for quadratic forms with applications to monotone convergence theorems
cites work: Q3872705
cites work: Schrödinger semigroups
cites work: Störungstheoretische Regularitätsuntersuchungen
cites work: Q5607484
cites work: A regular potential which is nowhere in \(L_ 1\)
cites work: Q3662132
cites work: On the perturbation theory for strongly continuous semigroups
cites work: Q5531981
description: scientific article; zbMATH DE number 4021980