The degenerate-elliptic operator \(Au=-\partial_ ja_{jk}(x)\partial_ ku+a_ j(x)\partial_ ju+a(x)u\) is considered in \(L^ p(R^ n)\), \(1<p<\infty\). The coefficients are real-valued functions, \(a_{jk}\in C^ 2(R^ n)\), \(a_ j\in C^ 1(R^ n)\), \(a\in L^{\infty}(R^ n)\), the \(n\times n\) matrix \((a_{jk}(x))\) is symmetric and \(a_{jk}\xi_ j\xi_ k\geq 0\) for all \((\xi_ 1,...,\xi_ n)\in R^ n\), \(x\in R^ n\). The author proves that the operator \(-A^ p_{\min}\) given by -A and restricted to \(C_ 0^{\infty}(R^ n)\) is essentially quasi-m- dispersive in \(L^ p(R^ n) (A^ p_{\min}\) is essentially quasi-m- accretive) and that its closure consider with the maximal realization of -A.