01-131 A. Pushnitski, M. Ruzhansky

Spectral shift function of the Schrodinger operator in the large coupling constant limit, II. Positive perturbations. (86K, LaTeX2e ) Apr 3, 01

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Abstract. For the operator $-\bigtriangleup$ in $L^2({\mathbb R}^d)$, $d\geq1$, a perturbation potential $V(x)\geq0$, and a coupling constant $\alpha>0$, the spectral shift function $\xi(\lambda;-\bigtriangleup+\alpha V,-\bigtriangleup)$ is considered. Assuming that $V(x)$ decays as $|x|^{-l}$, $l>d$, for large $x$, we prove that the leading term of the asymptotics of the spectral shift function as $\alpha\to\infty$ has the form $C\alpha^{d/l}$, where the coefficient $C=C(d,l)$ is explicitly computed. The asymptotics is in agreement with the phase space volume considerations. A generalization of this result is obtained by replacing $-\bigtriangleup$ by $h(-\ bigtriangleup)$ for a class of functions $h$.

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