A local trace formula for resonances of perturbed periodic Schrödinger operators. (Q1868676)

Let \(P_0 = -\Delta+V(y)\), where \(V\) is real valued and periodic with respect to the lattice \(\Gamma\) in \({\mathbb R}^n\). Assume that \(W(y)\leq C| z| ^{-n-\epsilon}\) and \(h\) is a small positive parameter. The authors prove a local trace formula for the pair \((P_0+W(hy),P_0)\). An application of this formula yields a lower bound for the number of resonances of \(P_0+W(hy)\) near any point of the analytic support of \(\int_{| x| <R} w(s-W(x))\,dx\), where \(R\) is a large constant and \(w(s)\) is the density of states of \(P_0\).