#### Vol. 1, No. 1, 2019

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Semiclassical resolvent estimates for bounded potentials

### Frédéric Klopp and Martin Vogel

Vol. 1 (2019), No. 1, 1–25
##### Abstract

We study the cut-off resolvent of semiclassical Schrödinger operators on ${ℝ}^{d}$ with bounded compactly supported potentials $V$. We prove that for real energies ${\lambda }^{2}$ in a compact interval in ${ℝ}_{+}$ and for any smooth cut-off function $\chi$ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has

$\parallel \chi {\left(-{h}^{2}\Delta +V-{\lambda }^{2}\right)}^{-1}\chi {\parallel }_{{L}^{2}\to {H}^{1}}\le C\phantom{\rule{0.3em}{0ex}}{e}^{C{h}^{-4∕3}log1∕h}.$

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

$Im\lambda \le -{C}^{-1}\phantom{\rule{0.3em}{0ex}}{e}^{C{h}^{-4∕3}log1∕h}.$

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 $0\not\equiv V\in {L}^{\infty }\left({ℝ}^{d}\right)$. We show that there exists a constant $M>0$ such that for any such $u$, for $R>0$ sufficiently large, one has

${\int }_{B\left(0,R+1\right)\setminus \overline{B\left(0,R\right)}}|u\left(x\right){|}^{2}\phantom{\rule{0.3em}{0ex}}dx\ge {M}^{-1}{R}^{-4∕3}{e}^{-M\parallel V{\parallel }_{\infty }^{2∕3}{R}^{4∕3}}\parallel u{\parallel }_{2}^{2}.$

##### Keywords
spectral theory, resolvent estimates, resonances, semiclassical analysis
##### Mathematical Subject Classification 2010
Primary: 35J10, 35P25, 47F05