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Vector-valued Schrödinger operators in Lp-spaces

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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  • 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.

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

    Citation:

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