Scale-free Unique Continuation Estimates and Logvinenko–Sereda Theorems on the Torus

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 contained in (a finite number of) d-dimensional cubes. The estimates we obtain do not depend on the size of the torus and the position of the d-dimensional cubes, 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 Schrodinger and for the heat equation, and finally to the Logvinenko and Sereda theorem. In fact, they are based on the methods developed by Kovrijkine to refine and generalize the results of Logvinenko and Sereda and Kacnel’son. Furthermore, relying on completely different techniques associated with the time-dependent Schrodinger equation, we prove a companion theorem where the energy of the considered functions is allowed to be in a spectral subspace of a Schrodinger operator.