Monday, January 30, 2012

1201.5573 (Jean-Francois Bony et al.)

Semiclassical estimates of the cut-off resolvent for trapping

Jean-Francois Bony, Vesselin Petkov
This paper is devoted to the study of a semiclassical "black box" operator
$P$. We estimate the norm of its resolvent truncated near the trapped set by
the norm of its resolvent truncated on rings far away from the origin. For $z$
in the unphysical sheet with $- h |ln h| < Im z < 0$, we prove that this
estimate holds with a constant $h |Im z|^{-1} e^{C|Im z|/h}$. We also obtain
analogous bounds for the resonances states of $P$. These results hold without
any assumption on the trapped set neither any assumption on the multiplicity of
the resonances.
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