Trace estimates and invariance of the essential spectrum (Q2431755)

The purpose of this paper is to estimate the resolvent difference of two elliptic differential operators of order \(2m\) in terms of \(L^{p}\) norm of the difference of the coefficients. Let \(a=(a_{\alpha\beta}(x))_{|\alpha|=|\beta|=m}\) be a complex matrix-valued function which is symmetric and positive definite for a.e. \(x\in{\mathbb R}^{N}\). Assume that \(a,\,a^{-1}\in L^{1}_{\text{loc}}({\mathbb R}^{N})\). Suppose that the quadratic form \[ Q:C_{c}^{\infty}({\mathbb R}^{N})\owns u\mapsto\int_{\Omega}\sum_{_{\substack{ |\alpha|=m\\|\beta|=m}}}a_{\alpha\beta}(x)D^{\alpha}u(x)D^{\beta}\bar{u}(x)\, dx\in{\mathbb R} \] in \(L^{2}({\mathbb R}^{N})\) is classifyable. Let \(H\) be the selfadjoint operator associated with the closure of \(Q\). This operator is formally expressed as \[ Hu(x)=(-1)^{m}\sum_{_{\substack{ |\alpha|=m\\|\beta|=m}}}D^{\alpha}\{a_{\alpha\beta}(x)D^{\beta}u(x)\},\quad x\in{\mathbb R}^{N}. \] In a similar way we introduce a selfadjoint elliptic operator \(\widetilde{H}\). The main results of this paper are the following two theorems. Theorem 1: Let \(H\) and \(\widetilde{H}\) be uniformly elliptic self-adjoint operators of order \(2m\) and let \(a\) and \(\widetilde{a}\) be the respective coefficient matrices. Assume that \(H\) has constant coefficients. Then for any \(p\in(N/m,\infty)\) there exists a positive constant \(c=c(p,H)\) such that \[ \| (\widetilde{H}+1)^{-1}-(H+1)^{-1}\|_{{\mathcal C}^{p}}\leq c\| \widetilde{a}^{-1/2}(\widetilde{a}-a)a^{-1/2}\|_{L^{p}}, \] where \(C^{p}=C^{p}(L^{2}({\mathbb R}^{N}))\) is the Schatten class. Theorem 2: Let \(H\) and \(\widetilde{H}\) be selfadjoint elliptic operators of order \(2m\) and let \(a\) and \(\widetilde{a}\) be the respective coefficient matrices. Assume that \(H\) has constant coefficients. If \(\widetilde{a}^{-1/2}(\widetilde{a}-a)\in L^{p}\) for some \(p\in(N/m,\infty)\) then there exists a positive constant \(c=c(p,H)\) such that \[ \| (\widetilde{H}+1)^{-1}-(H+1)^{-1}\|\leq c\| \widetilde{a}^{-1/2}(\widetilde{a}-a)a^{-1/2}\|_{L^{p}}, \] where the norm on the left-hand side is the operator norm on \(L^{2}({\mathbb R}^{N})\).