Scale-free unique continuation estimates and Logvinenko-Sereda Theorems on the torus

TL;DR

This paper establishes scale-free uncertainty principles and Logvinenko-Sereda type theorems for functions on the torus, linking spectral inequalities, observability estimates, and unique continuation, with results independent of the torus size.

Contribution

It introduces scale-free unique continuation estimates on the torus for functions with Fourier support in parallelepipeds, extending previous results with new techniques.

Findings

Estimates depend only on size and number of spectral supports, not on torus size.

Connections established between spectral inequalities, observability, and Logvinenko-Sereda theorems.

Proves a spectral subspace energy estimate for Schrödinger operators.

Abstract

We study uncertainty principles for function classes on the torus. The classes are defined in terms of spectral subspaces of the energy or the momentum, respectively. In our main theorems, the support of the Fourier transform of the considered functions is allowed to be supported in a (finite number of) parallelepipeds. The estimates we obtain do not depend on the size of the torus and the position of the parallelepipeds, but only on their size and number, and the density and scale of the observability set. Our results are on the one hand closely related to unique continuation for linear combinations of eigenfunctions (aka spectral inequalities) which can be obtained by Carleman estimates, on the other hand to observability estimates for the time-dependent Schroedinger and for the heat equation, and finally to the Logvinenko & Sereda theorem. In fact, they are based on the methods developed by Kovrijkine to refine and generalize the results of Logvinenko & Sereda and Kacnel'son. Furthermore, relying on completely different techniques associated with the time-dependent Schroedinger equation, we prove a companion theorem where the energy of the considered functions is allowed to be in a spectral subspace of a Schroedinger operator.