By the Heisenberg uncertainty principle, a bandlimited function cannot be compactly supported. However, one can ask: to what extent can a bandlimited function concentrate most of its energy within a given bounded domain?
In this talk, I will show that this question can be understood through the behaviour of the eigenvalues of a certain compact, and positive-definite localization operator. Specifically, the eigenfunctions corresponding to the largest eigenvalues of this operator (along with any function in the subspace they span) are bandlimited and have optimal energy concentration within the predetermined domain. Therefore, understanding the distribution of the eigenvalues and determining the number of large eigenvalues are central to addressing the concentration problem.
Specifically, we aim to provide a non-asymptotic estimate for the number of large eigenvalues in higher dimensions. This extends and improves the classical asymptotic results of Landau et al. in the one-dimensional case. Our approach is based on the construction of $\epsilon$-concentrated wave packets, which we develop and use as key tools in establishing our bounds and results.
I will conclude the talk by highlighting some open problems and some directions for future research.