 01131 A. Pushnitski, M. Ruzhansky
 Spectral shift function of the Schrodinger operator
in the large coupling constant limit, II. Positive perturbations.
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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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