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Opérateurs de Schrödinger avec potentiel singulier magnétique, dans un ouvert arbitraire de \({\mathbb{R}}^ N\). (French) Zbl 0565.35024

From the author’s abstract: If \(\Omega \subset {\mathbb{R}}^ N\) is an arbitrary open set and \(\vec a=(a_ j)_{j=1,...,N}\in [L^ 2_{loc}(\Omega,{\mathbb{R}})]^ N,\) we characterize the closure Q(\(\vec a)\) of \({\mathcal D}(\Omega)\) with respect to the norm \[ \| u\|^ 2_{\vec a}=\| u\|^ 2_{L^ 2(\Omega)}+\| \vec Du\|^ 2_{[L^ 2(\Omega)]^ N}, \] where \(\vec Du=(D_ ju)_{j=1,...,N}=(\partial_ ju-ia_ ju)_{j=1,...,N}\). This allows us to give a realization A, with m-accretive closure in \(L^ 2(\Omega)\), of the operator \(-\sum^{N}_{j,k=1}D_ k(b_{jk}D_ j)+V\), \((b_{jk}\in L^{\infty}(\Omega,{\mathbb{R}})\), symmetric and elliptic, V a complex singular potential) and to prove a half-sum formula for the semigroup generated by \(\bar A.\)
We also show that the imbedding of Q(\(\vec a)\) into \(L^ 2(\Omega ')\) is compact for every relatively compact open set \(\Omega\) ’\(\subset \subset \Omega\); with some supplementary hypotheses, this allows us to prove the Trotter product formula for the same semigroup, thus generalizing a previous result of the author and R. Piraux.
Reviewer: H.Leinfelder

MSC:

35J10 Schrödinger operator, Schrödinger equation
47D03 Groups and semigroups of linear operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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