Spectral shift function and resonances for slowly varying perturbations of periodic Schrödinger operators

AbstractWe study the spectral shift function s(λ,h) and the resonances of the operator P(h)=-Δ+V(x)+W(hx). Here V is a periodic potential, W a decreasing perturbation and h a small positive constant. We give a representation of the derivative of s(λ,h) related to the resonances of P(h), and we obtain a Weyl-type asymptotics of s(λ,h). We establish an upper bound O(h-n+1) for the number of the resonances of P(h) lying in a disk of radius h

Spectral Shift Function for the Perturbations of Schrödinger Operators at High Energy

2000 Mathematics Subject Classification: 35P20, 35J10, 35Q40.We give a complete pointwise asymptotic expansion for the Spectral Shift Function for Schrödinger operators that are perturbations of the Laplacian on Rn with slowly decaying potentials

Resonance free regions for systems of semiclassical Schrodinger operators and applications (Spectral and Scattering Theory and Related Topics)

We consider an N × N system of semiclassical differential operators with N Schrödinger operators in the diagonal part and small interactions of order h^ν, where h is a semiclassical parameter and ν is a constant larger than one. We study the absence of resonance near a non-trapping energy for each Schrödinger operators. The width of resonances is estimated from below by Mh log(1/h) and the coefficient M can be taken propotional to ν - 1