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 d with bounded compactly supported potentials V . We prove that for real energies λ2 in a compact interval in + 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(h2Δ + V λ2)1χ L2H1 CeCh43 log 1h .

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 (h2Δ + V λ2)1 as an operator Lcomp2 Hloc2: any resonance λ with real part in a compact interval away from 0 has imaginary part at most

Imλ C1 eCh43 log 1h .

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 L2 solutions u to Δu = V u with 0V L(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)|2dx M1R43 eMV 23R43 u22.

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
Authors
Frédéric Klopp
Sorbonne Université, Université Paris Diderot, CNRS
Institut de Mathématiques Jussieu – Paris Rive Gauche
Paris
France
Martin Vogel
Mathematics Department
University of California
Berkeley, CA
United States