Vector-valued Schrödinger operators in Lp-spaces

Markus Kunze; Abdallah Maichine; Abdelaziz Rhandi

doi:10.3934/dcdss.2020086

Article Contents

2020, Volume 13, Issue 5: 1529-1541. Doi: 10.3934/dcdss.2020086

This issue Previous Article A generalized Cox-Ingersoll-Ross equation with growing initial conditions Next Article Mean periodic solutions of a inhomogeneous heat equation with random coefficients

Vector-valued Schrödinger operators in L^p-spaces

1.
Fachbereich Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany
2.
Dipartimento di Matematica, Università degli Studi di Salerno, via Giovanni Paolo Ⅱ, 132, 84084, Fisciano (Sa), Italy
3.
Dipartimento di Ingegneria dell'Informazione, Ingegneria Elettrica e Matematica Applicata, Università degli Studi di Salerno, via Giovanni Paolo Ⅱ, 132, 84084, Fisciano (Sa), Italy

Received: January 31, 2018

Revised: October 31, 2018

Published: March 2020

This work has been supported by the M.I.U.R. research project Prin 2015233N54 "Deterministic and Stochastic Evolution Equations". The third author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

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Abstract

In this paper we consider the vector-valued operator div$ (Q\nabla u)-Vu $ of Schrödinger type. Here $ V = (v_{ij}) $ is a nonnegative, locally bounded, matrix-valued function and $ Q $ is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential $ V $, we assume an that it is pointwise accretive and that its entries are in $ L^\infty_{{\rm loc}}( \mathbb{R}^d) $. Under these assumptions, we prove that a realization of the vector-valued Schrödinger operator generates a $ C_0 $-semigroup of contractions in $ L^p( \mathbb{R}^d; \mathbb{C}^m) $. Further properties are also investigated.

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Mathematics Subject Classification: Primary: 35K40, 47D08; Secondary: 47D06.

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