Geodesic
On $m$-sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 91-100 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We consider a Schrödinger-type differential expression $H_V=\nabla^*\nabla+V$, where $\nabla $ is a $C^{\infty}$-bounded Hermitian connection on a Hermitian vector bundle $E$ of bounded geometry over a manifold of bounded geometry $(M,g)$ with metric $g$ and positive $C^{\infty}$-bounded measure $d\mu$, and $V$ is a locally integrable section of the bundle of endomorphisms of $E$. We give a sufficient condition for $m$-sectoriality of a realization of $H_V$ in $L^2(E)$. In the proof we use generalized Kato's inequality as well as a result on the positivity of $u\in L^2(M)$ satisfying the equation $(\Delta _M+b)u=\nu $, where $\Delta _M$ is the scalar Laplacian on $M$, $b>0$ is a constant and $\nu\geq 0$ is a positive distribution on $M$.
We consider a Schrödinger-type differential expression $H_V=\nabla^*\nabla+V$, where $\nabla $ is a $C^{\infty}$-bounded Hermitian connection on a Hermitian vector bundle $E$ of bounded geometry over a manifold of bounded geometry $(M,g)$ with metric $g$ and positive $C^{\infty}$-bounded measure $d\mu$, and $V$ is a locally integrable section of the bundle of endomorphisms of $E$. We give a sufficient condition for $m$-sectoriality of a realization of $H_V$ in $L^2(E)$. In the proof we use generalized Kato's inequality as well as a result on the positivity of $u\in L^2(M)$ satisfying the equation $(\Delta _M+b)u=\nu $, where $\Delta _M$ is the scalar Laplacian on $M$, $b>0$ is a constant and $\nu\geq 0$ is a positive distribution on $M$.
Classification : 35J10, 35P05, 47B25, 58J05, 58J50, 81Q10
Keywords: Schrödinger operator; $m$-sectorial; manifold; bounded geometry; singular potential 
@article{CMUC_2004_45_1_a5,
     author = {Milatovic, Ognjen},
     title = {On $m$-sectorial {Schr\"odinger-type} operators with singular potentials on manifolds of bounded geometry},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {91--100},
     year = {2004},
     volume = {45},
     number = {1},
     mrnumber = {2076861},
     zbl = {1127.35348},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2004_45_1_a5/}
}
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Milatovic, Ognjen. On $m$-sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 1, pp. 91-100. http://geodesic.mathdoc.fr/item/CMUC_2004_45_1_a5/