December  2008, 22(4): 1081-1090. doi: 10.3934/dcds.2008.22.1081

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

1. 

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

Received  May 2007 Revised  October 2007 Published  September 2008

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 $a_j \in L^4_{loc}(\mathbb{R}^N)$.
Citation: Noboru Okazawa, Tomomi Yokota. Quasi-$m$-accretivity of Schrödinger operators with singular first-order coefficients. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1081-1090. doi: 10.3934/dcds.2008.22.1081
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