Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 4 (2018), 1049-1081.

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

Ivica Nakić, Matthias Täufer, Martin Tautenhahn, and Ivan Veselić

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Abstract

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector χ ( , E ] ( H L ) of a Schrödinger operator H L on a cube of side L , with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors χ ( E γ , E ] ( H L ) with small γ . Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. Our main application of such an estimate is to find lower bounds for the lifting of eigenvalues under semidefinite positive perturbations, which in turn can be applied to derive a Wegner estimate for random Schrödinger operators with nonlinear parameter-dependence. Another application is an estimate of the control cost for the heat equation in a multiscale domain in terms of geometric model parameters. Let us emphasize that previous uncertainty principles for individual eigenfunctions or spectral projectors onto small intervals were not sufficient to study such applications.

Article information

Source
Anal. PDE, Volume 11, Number 4 (2018), 1049-1081.

Dates
Received: 26 May 2017
Revised: 11 August 2017
Accepted: 16 October 2017
First available in Project Euclid: 1 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.apde/1517454163

Digital Object Identifier
doi:10.2140/apde.2018.11.1049

Mathematical Reviews number (MathSciNet)
MR3749376

Zentralblatt MATH identifier
1383.35068

Subjects
Primary: 35J10: Schrödinger operator [See also 35Pxx] 35P15: Estimation of eigenvalues, upper and lower bounds 35Q82: PDEs in connection with statistical mechanics 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis
Secondary: 81Q15: Perturbation theories for operators and differential equations

Keywords
uncertainty relation spectral inequality Wegner estimate control of heat equation random Schroedinger operator

Citation

Nakić, Ivica; Täufer, Matthias; Tautenhahn, Martin; Veselić, Ivan. Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators. Anal. PDE 11 (2018), no. 4, 1049--1081. doi:10.2140/apde.2018.11.1049. https://projecteuclid.org/euclid.apde/1517454163


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