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A local trace formula for resonances of perturbed periodic Schrödinger operators. (English) Zbl 1090.35065
Let P 0 =-Δ+V(y), where V is real valued and periodic with respect to the lattice Γ in n . Assume that W(y)C|z| -n-ϵ 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 |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:
35J10Schrödinger operator
35B10Periodic solutions of PDE
35B20Perturbations (PDE)
35P25Scattering theory (PDE)
47F05Partial differential operators
47N50Applications of operator theory in quantum physics
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis