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Zbl 1090.35065
Dimassi, Mouez; Zerzeri, Maher
A local trace formula for resonances of perturbed periodic Schrödinger operators. (English)
[J] J. Funct. Anal. 198, No. 1, 142-159 (2003). ISSN 0022-1236

Let $P_0 = -\Delta+V(y)$, where $V$ is real valued and periodic with respect to the lattice $\Gamma$ in ${\Bbb R}^n$. Assume that $W(y)\leq C\vert z\vert ^{-n-\epsilon}$ 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 $\int_{\vert x\vert <R} w(s-W(x))\,dx$, where $R$ is a large constant and $w(s)$ is the density of states of $P_0$.
[Michael Perelmuter (Ky{\"\i}v)]

MSC 2000:: *35J10 Schroedinger operator
35B10 Periodic solutions of PDE
35B20 Perturbations (PDE)
35P25 Scattering theory (PDE)
47F05 Partial differential operators
47N50 Appl. of operator theory in quantum physics
81Q10 Selfadjoint operator theory in quantum theory

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