Quasi-$m$-accretivity of Schrödinger operators with singular first-order coefficients

Noboru Okazawa; Tomomi Yokota

doi:10.3934/dcds.2008.22.1081

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2008, Volume 22, Issue 4: 1081-1090. Doi: 10.3934/dcds.2008.22.1081

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Quasi-$m$-accretivity of Schrödinger operators with singular first-order coefficients

Noboru Okazawa^1, and
Tomomi Yokota^1,

1.
Department of Mathematics, Science University of Tokyo, 26 Wakamiya-cho, Shinjuku-ku, Tokyo 162-8601

Received: April 30, 2007

Revised: September 30, 2007

Published: September 2008

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Abstract

The Schrödinger operator $T = (i\nabla +b)^2+a \cdot \nabla + q$ on $\mathbb{R}^N$ is considered for $N \ge 2$. Here $a=(a_{j})$ and $b=(b_{j})$ are real-vector-valued functions on $\mathbb{R}^N$, while $q$ is a complex-scalar-valued function on $\mathbb{R}^N$. Over twenty years ago late Professor Kato proved that the minimal realization $T_{min}$ is essentially quasi-$m$-accretive in $L^2(\mathbb{R}^N)$ if, among others, $(1+|x|)^{-1}a_j \in L^4(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$. In this paper it is shown that under some additional conditions the same conclusion remains true even if $a_j \in L^4_{loc}(\mathbb{R}^N)$.

Keywords:

Schrödinger operators,
quasi-$m$-accretive operators.

Mathematics Subject Classification: Primary: 35J10; Secondary: 47B44.

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