Mouez Dimassi and Vesselin Petkov
Spectral shift function and resonances for non semi-bounded and Stark hamiltonians
(440K, postscript)
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 $V$.