An uncertainty principle and lower bounds for the Dirichlet Laplacian on graphs

Abstract

This paper is dedicated to W. Kirsch and B. Simon as part of the celebration of their recent birthdays. We are grateful for their inspiration. AbstractWe prove a quantitative uncertainty principle at low energies for the Laplacian on fairly general weighted graphs with a uniform explicit control of the constants in terms of geometric quantities. A major step consists in establishing lower bounds for Dirichlet eigenvalues in terms of the geometry.

Sufficient criteria for stabilization properties in Banach spaces

“…This leads to a plain condition for stabilizability in Banach spaces which does not involve assumptions on the constant in the uncertainty principle. Let us stress that the latter improvement allows to apply our result to models where an uncertainty principle is avaible only for some spectral parameters as in [LSS20]. We will pursue this application in a forthcoming paper.…”

Lower bounds for Dirichlet Laplacians and uncertainty principles

“…In comparison with the discrete case, [29], this is probably the most tricky part of the present analysis.…”

“…This has provided ample motivation for more thorough studies of the geometric properties required for subsets of configuration space to guarantee that these subsets carry a "substantial" part of the mass of low energy states of the Laplacian, both in the continuous setting and for discrete Laplacians on graphs. Our goal here is to establish a result in the continuous case, similar to work in the discrete setting in [29], and we refer to the literature cited in that paper. We mention that from a harmonic analysis point of view, our results are close in spirit to Logvinenko-Sereda theorems (see [27]), with the important difference that we have to restrict ourselves to spectral projectors with energy intervals close to the ground state energy.…”

“…Our proof of Theorem 1.1 consists of three parts, covered in the remaining three sections of this paper. The same general strategy has been used in [29] to prove corresponding results for Laplacians on graphs. The continuous setting considered here leads to some additional complications.…”