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.
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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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Communicated by Jerome A. Goldstein.
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Maichine, A. Generation of semigroup for symmetric matrix Schrödinger operators in \({\varvec{L}}^{\varvec{p}}\)-spaces. Semigroup Forum 99, 245–259 (2019). https://doi.org/10.1007/s00233-018-9955-0
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DOI: https://doi.org/10.1007/s00233-018-9955-0