Matthias Täufer

Scale-free uncertainty principles and Wegner estimates for random breather potentials ☆

Abstract We present new scale-free quantitative unique continuation principles for Schrodinger operators. They apply to linear combinations of eigenfunctions corresponding to eigenvalues below a prescribed energy, and can be formulated as an uncertainty principle for spectral projectors. This extends recent results of Rojas-Molina & Veselic [15] , and Klein [10] . We apply the scale-free unique continuation principle to obtain a Wegner estimate for a random Schrodinger operator of breather type. It holds for arbitrarily high energies. Schrodinger operators with random breather potentials have a non-linear dependence on random variables. We explain the challenges arising from this non-linear dependence.

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

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector χ ( − ∞ , E ] ( H L ) χ(−∞, E](HL) of a Schrodinger operator H L HL on a cube of side L ∈ N L∈ℕ, with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors χ ( E − γ , E ] ( H L ) χ(E−γ, E](HL) with small γ 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 Schrodinger 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.

Conditional Wegner estimate for the standard random breather potential

We prove a conditional Wegner estimate for Schrödinger operators with random potentials of breather type. More precisely, we reduce the proof of the Wegner estimate to a scale free unique continuation principle. The relevance of such unique continuation principles has been emphasized in previous papers, in particular in recent years. We consider the standard breather model, meaning that the single site potential is the characteristic function of a ball or a cube. While our methods work for a substantially larger class of random breather potentials, we discuss in this particular paper only the standard model in order to make the arguments and ideas easily accessible.

Harmonic Analysis and Random Schrödinger Operators

This survey is based on a series of lectures given during the School on Random Schrodinger Operators and the International Conference on Spectral Theory and Mathematical Physics at the Pontificia Universidad Catolica de Chile, held in Santiago in November 2014. As the title suggests, the presented material has two foci: Harmonic analysis, more precisely, unique continuation properties of several natural function classes and Schrodinger operators, more precisely properties of their eigenvalues, eigenfunctions and solutions of associated differential equations. It mixes topics from (rather) pure to (rather) applied mathematics, as well as classical questions and results dating back a whole century to very recent and even unpublished ones. The selection of material covered is based on the selection made for the minicourse, and is certainly a personal choice corresponding to the research interests of the authors.

Scale-free and quantitative unique continuation for infinite dimensional spectral subspaces of Schrödinger operators

We prove a quantitative unique continuation principle for infinite dimensional spectral subspaces of Schrodinger operators. Let \begin{document}

Wegner estimate for Landau-breather Hamiltonians

Λ_L = (-L/2, L/2)^d

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space.

\end{document} and \begin{document}

Spectral localization for quantum Hamiltonians with weak random delta interaction

H_L = -Δ_L + V_L

Wegner Estimate and Disorder Dependence for Alloy-Type Hamiltonians with Bounded Magnetic Potential

\end{document} be a Schrodinger operator on \begin{document}

Unique continuation and lifting of spectral band edges of Schr\"odinger operators on unbounded domains (With an Appendix by Albrecht Seelmann)

L^2 (Λ_L)