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2019 Semiclassical resolvent estimates for bounded potentials
Frédéric Klopp, Martin Vogel
Pure Appl. Anal. 1(1): 1-25 (2019). DOI: 10.2140/paa.2019.1.1

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 λ 2 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 ( 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 ( 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 .

Citation

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Frédéric Klopp. Martin Vogel. "Semiclassical resolvent estimates for bounded potentials." Pure Appl. Anal. 1 (1) 1 - 25, 2019. https://doi.org/10.2140/paa.2019.1.1

Information

Received: 1 April 2018; Revised: 10 July 2018; Accepted: 24 August 2018; Published: 2019
First available in Project Euclid: 4 February 2019

zbMATH: 07027484
MathSciNet: MR3900028
Digital Object Identifier: 10.2140/paa.2019.1.1

Subjects:
Primary: 35J10, 35P25, 47F05

Rights: Copyright © 2019 Mathematical Sciences Publishers

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