## Pure and Applied Analysis

### 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 $λ 2$ in a compact interval in $ℝ +$ and for any smooth cut-off function $χ$ supported in a ball near the support of the potential $V$, for some constant $C > 0$, one has

$∥ χ ( − h 2 Δ + V − λ 2 ) − 1 χ ∥ L 2 → H 1 ≤ C e C h − 4 ∕ 3 log 1 ∕ h .$

This bound shows in particular an upper bound on the imaginary parts of the resonances $λ$, defined as a pole of the meromorphic continuation of the resolvent $( − h 2 Δ + V − λ 2 ) − 1$ as an operator $L comp 2 → H loc 2$: any resonance $λ$ with real part in a compact interval away from $0$ has imaginary part at most

$Im λ ≤ − C − 1 e C h − 4 ∕ 3 log 1 ∕ h .$

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 $− Δ u = V u$ with $0 ≢ V ∈ L ∞ ( ℝ d )$. We show that there exists a constant $M > 0$ such that for any such $u$, for $R > 0$ sufficiently large, one has

$∫ B ( 0 , R + 1 ) ∖ B ( 0 , R ) ¯ | u ( x ) | 2 d x ≥ M − 1 R − 4 ∕ 3 e − M ∥ V ∥ ∞ 2 ∕ 3 R 4 ∕ 3 ∥ u ∥ 2 2 .$

#### Article information

Source
Pure Appl. Anal., Volume 1, Number 1 (2019), 1-25.

Dates
Revised: 10 July 2018
Accepted: 24 August 2018
First available in Project Euclid: 4 February 2019

https://projecteuclid.org/euclid.paa/1549297977

Digital Object Identifier
doi:10.2140/paa.2019.1.1

Mathematical Reviews number (MathSciNet)
MR3900028

Zentralblatt MATH identifier
07027484

#### Citation

Klopp, Frédéric; Vogel, Martin. Semiclassical resolvent estimates for bounded potentials. Pure Appl. Anal. 1 (2019), no. 1, 1--25. doi:10.2140/paa.2019.1.1. https://projecteuclid.org/euclid.paa/1549297977

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