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 Schrödinger 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 Schrödinger equation, we prove a companion theorem where the energy of the considered functions is allowed to be in a spectral subspace of a Schrödinger operator.

我们研究圆环上函数类的不确定性原理。这些类别分别根据能量或动量的谱子空间来定义。在我们的主要定理中，所考虑函数的傅立叶变换的支持被包含在d维立方体中（有限数量）。我们获得的估计值不取决于圆环的大小和d的位置维立方体，但仅取决于其大小和数量以及可观察性集的密度和比例。我们的结果一方面与可以通过Carleman估计获得的本征函数线性组合（即谱不等式）的唯一延续密切相关，另一方面与时间相关的Schrödinger和热方程的可观性估计密切相关，并且最终得出洛格维年科和Sereda定理。实际上，它们是基于Kovrijkine开发的方法来精炼和推广Logvinenko和Sereda和Kacnel'son的结果。此外，依靠与时间相关的Schrödinger方程相关的完全不同的技术，我们证明了伴随定理，其中所考虑函数的能量被允许在Schrödinger算子的谱子空间中。