Pure and Applied Analysis

Semiclassical resolvent estimates for bounded potentials

Frédéric Klopp and Martin Vogel

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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 .

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J10: Schrödinger operator [See also 35Pxx] 35P25: Scattering theory [See also 47A40] 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)

spectral theory resolvent estimates resonances semiclassical analysis


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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