03-220 Mouez Dimassi and Vesselin Petkov

Spectral shift function and resonances for non semi-bounded and Stark hamiltonians (440K, postscript) May 13, 03

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Abstract. We generalize for non semi-bounded and Stark hamiltonians the results of Bruneau-Petkov proving a representation of the derivative of the spectral shift function $\xi(\lambda, h)$ related to the semi-classical resonances. For Stark hamiltonians $P_2(h) = -h^2 \Delta + \beta x_1 + V(x), \beta > 0$ we obtain a local trace formula and we establish an upper bound for the number of the resonances in a compact domain $\Omega \subset C_{-}$. For potentials $V \in C_0^{\infty}(R^n)$ with ${\rm supp}_{x_1} V \subset [R_0, + \infty[$, we obtain a Weyl-type asymptotics of $\xi(\lambda, h)$ and we establish the existence of resonances in every h-independent complex neighborhood of $E_0$ if $E_0$ is an analytic singularity of a suitable measure related to $V$.

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