## A local trace formula for resonances of perturbed periodic Schrödinger operators.(English)Zbl 1090.35065

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 $$P_0$$.

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35B10 Periodic solutions to PDEs 35B20 Perturbations in context of PDEs 35P25 Scattering theory for PDEs 47F05 General theory of partial differential operators 47N50 Applications of operator theory in the physical sciences 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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