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Semigroup Forum

, Volume 99, Issue 2, pp 245–259 | Cite as

Generation of semigroup for symmetric matrix Schrödinger operators in \({\varvec{L}}^{\varvec{p}}\)-spaces

  • A. MaichineEmail author
Research Article

Abstract

In this paper we prove that the symmetric matrix Schrödinger operator \(\mathrm {div}(Q\nabla u)-Vu\) generates an analytic semigroup when, for every \(x\in \mathbb {R}^d\), \(V(x)=(v_{ij}(x))\) is a semi-definite positive and symmetric matrix, the diffusion matrix \(Q(\cdot )\) is supposed to be strongly elliptic and bounded and the potential V satisfies the weak condition \(v_{ij}\in L^1_\textit{loc}(\mathbb {R}^d)\), for all \(i,j\in \{1,\ldots ,m\}\). We also determine the positivity and compactness of the semigroup.

Keywords

Schrödinger operator Matrix potential Sesquilinear forms Beurling–Deny criterion 

Notes

Acknowledgements

The author would like to thank Markus Kunze for suggesting the reference [9] which contains the vectorial Deny–Beurling criterion of \(L^\infty \)-contractivity. He is also grateful to the referee for valuable comments and suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di SalernoFiscianoItaly

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