On the spectrum of the superposition of separated potentials (Q1943297)

Suppose that \(L\) is a constant-coefficient linear differential operator of order \(s\geq 1\) which acts on functions \(f:\mathbb R^n\to \mathbb R^d\), \(L:(W^{s,p}(\mathbb R^n))^d \to(L^p(\mathbb R^n))^d\). It is assumed that \(L\) satisfies the following estimate: \[ C^{-1}\|u\|_{(W^{s',p}(\mathbb R^n))^d}\leq\|u\|_{(L^p(\mathbb R^n))^d}+ \|Lu\|_{(L^p(\mathbb R^n))^d}\leq C\|u\|_{(W^{s,p}(\mathbb R^n))^d}, \] where \(1\leq s'\leq s\). Further, it is assumed in the paper that, for \(j=1,\dots,N\), the matrix valued ``potential'' functions \(V_j:\mathbb R^n\to\mathbb R^d\) are \(C^{\infty}\) and are ``exponentially localized at \(0\)'', i.e., there exist \(\beta_0,\Omega>0\) such that, for \(j=1,\dots,N\), \(\sup_{x\in\mathbb R^n}|\cosh(\beta_0|x|)V_j(x)|\leq\Omega\). Set \(A:=L+\sum^N_{j=1}V_j(x-x_j)\). The main goal of the paper is to determine the spectrum of \(A\) if one knows the spectra of all \(A_j:=L+V_j(x)\).