Frédéric Klopp, Martin Vogel

Pure Appl. Anal. 1 (1), 1-25, (2019) DOI: 10.2140/paa.2019.1.1
KEYWORDS: Spectral theory, resolvent estimates, resonances, semiclassical analysis, 35J10, 35P25, 47F05

We study the cut-off 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 cut-off function $\chi $ supported in a ball near the support of the potential $V$, for some constant $C>0$, one has

$$\parallel \chi {\left(-{h}^{2}\mathrm{\Delta}+V-{\lambda}^{2}\right)}^{-1}\chi {\parallel}_{{L}^{2}\to {H}^{1}}\le C{e}^{C{h}^{-4\u22153}log1\u2215h}.$$

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 ${\left(-{h}^{2}\mathrm{\Delta}+V-{\lambda}^{2}\right)}^{-1}$ 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

$$Im\lambda \le -{C}^{-1}{e}^{C{h}^{-4\u22153}log1\u2215h}.$$

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 $-\mathrm{\Delta}u=Vu$ 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

$${\int}_{B\left(0,R+1\right)\setminus \overline{B\left(0,R\right)}}|u\left(x\right){|}^{2}dx\ge {M}^{-1}{R}^{-4\u22153}{e}^{-M\parallel V{\parallel}_{\infty}^{2\u22153}{R}^{4\u22153}}\parallel u{\parallel}_{2}^{2}.$$