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


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.


Schrödinger operator Matrix potential Sesquilinear forms Beurling–Deny criterion 



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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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di SalernoFiscianoItaly

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