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Abstract

We study the cutoff resolvent of semiclassical Schrödinger operators on
${\mathbb{R}}^{d}$
with bounded compactly supported potentials
$V$. We prove that for real
energies
${\lambda}^{2}$ in a compact
interval in
${\mathbb{R}}_{+}$ and for any
smooth cutoff function
$\chi $
supported in a ball near the support of the potential
$V$, for some
constant
$C>0$,
one has
$$\parallel \chi {\left({h}^{2}\Delta +V{\lambda}^{2}\right)}^{1}\chi {\parallel}_{{L}^{2}\to {H}^{1}}\le C\phantom{\rule{0.3em}{0ex}}{e}^{C{h}^{4\u22153}log1\u2215h}.$$
This bound shows in particular an upper bound on the imaginary parts of the resonances
$\lambda $,
defined as a pole of the meromorphic continuation of the resolvent
${\left({h}^{2}\Delta +V{\lambda}^{2}\right)}^{1}$ as an operator
${L}_{comp}^{2}\to {H}_{loc}^{2}$: any resonance
$\lambda $ with real part in a compact
interval away from
$0$
has imaginary part at most
$$Im\lambda \le {C}^{1}\phantom{\rule{0.3em}{0ex}}{e}^{C{h}^{4\u22153}log1\u2215h}.$$
This is related to a conjecture by Landis: The principal Carleman
estimate in our proof provides as well a lower bound on the decay rate of
${L}^{2}$ solutions
$u$ to
$\Delta u=Vu$ with
$0\not\equiv V\in {L}^{\infty}\left({\mathbb{R}}^{d}\right)$. We show that there
exists a constant
$M>0$ such
that for any such
$u$,
for
$R>0$
sufficiently large, one has
$${\int}_{B\left(0,R+1\right)\setminus \overline{B\left(0,R\right)}}u\left(x\right){}^{2}\phantom{\rule{0.3em}{0ex}}dx\ge {M}^{1}{R}^{4\u22153}{e}^{M\parallel V{\parallel}_{\infty}^{2\u22153}{R}^{4\u22153}}\parallel u{\parallel}_{2}^{2}.$$

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Keywords
spectral theory, resolvent estimates, resonances,
semiclassical analysis

Mathematical Subject Classification 2010
Primary: 35J10, 35P25, 47F05

Milestones
Received: 1 April 2018
Revised: 10 July 2018
Accepted: 24 August 2018
Published: 21 November 2018

