Semiclassical resolvent estimates for bounded potentials

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

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

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

$$Im\lambda \le -C^{-1}{e}^{C{h}^{-4∕3}log1∕h}.$$

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

$${\int }_{B\left(0,R+1\right)\setminus \overline{B\left(0,R\right)}}|u\left(x\right){|}^{2}dx\ge M^{-1}{R}^{-4∕3}{e}^{-M\parallel V{\parallel }_{\infty }^{2∕3}{R}^{4∕3}}\parallel u{\parallel }_2^2.$$