Pure and Applied Analysis
- Pure Appl. Anal.
- Volume 1, Number 1 (2019), 1-25.
Semiclassical resolvent estimates for bounded potentials
We study the cut-off resolvent of semiclassical Schrödinger operators on with bounded compactly supported potentials . We prove that for real energies in a compact interval in and for any smooth cut-off function supported in a ball near the support of the potential , for some constant , one has
This bound shows in particular an upper bound on the imaginary parts of the resonances , defined as a pole of the meromorphic continuation of the resolvent as an operator : any resonance with real part in a compact interval away from has imaginary part at most
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 solutions to with . We show that there exists a constant such that for any such , for sufficiently large, one has
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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Digital Object Identifier
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)
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