Journal of Operator Theory

Volume 55, Issue 1, Winter 2006 pp. 185-211.

Schroedinger operators with unbounded drift

Authors: Wolfgang Arendt (1), Giorgio Metafune (2), and Diego Pallara (3)
Author institution: (1) Abteilung Angewandte Analysis, Universitaet Ulm, D-89069 Ulm, Germany
(2) Dipartimento di Matematica ``Ennio de Giorgi'', Universit\`a di Lecce, C.P. 193, 73100, Lecce, Italia
Dipartimento di Matematica
(3) ``Ennio de Giorgi'', Universit\`a di Lecce, C.P. 193, 73100, Lecce, Italia

Summary: Let $a_{ij}\in C_\mathrm b^1(\re^N)$, $i,j=1,\ldots, N$ be uniformly elliptic, and let $b\in C^1(\re^N)$, $V\in C(\re^N)$. If $\frac{\diver{b}}{p}\leqslant V$, then we construct a unique minimal positive semigroup generated by a restriction of the operator $A$ defined by the expression $$ Au=\sum_{i,j=1}^N D_i(a_{ij}D_ju) - \sum_{i=1}^N b_iD_iu - Vu $$ on $L^p(\re^N)$ with maximal domain. We give a criterion for $C_\mathrm c^\infty(\re^N)$ to be a core and we give conditions on $V$ and $b$ which imply that the semigroup is given by kernels allowing an upper Gaussian bound. By a specific example we show that our criteria are close to optimal.

Keywords: Schroedinger operators, positive contraction semigroups

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