Speaker
Description
Given a measurable set $U\subset \mathbb{R}$, we define the projection onto $U$ as $P_U:L^2(\mathbb{R})\to L^2(\mathbb{R})$ given by $(P_U f)(x) = f(x)\chi_U(x)$. Similarly, for the set $V\subset \mathbb{R}$ we define the Fourier projection onto $V$ as $Q_V = \mathcal{F}^{-1}P_V \mathcal{F}$, where $\mathcal{F}$ is a Fourier transform. The operator $S_{U, V} = P_UQ_VP_U$ is called a time-frequency localization operator, associated with $U$ and $V$.
It is easy to see that $S_{U, V}$ is a non-negative definite operator of norm at most $1$. If both $U$ and $V$ have finite measure it turns out that $S_{U, V}$ is a Hilbert--Schmidt operator with $||S_{U, V}||_{HS}^2 = |U| |V|$. In particular, it is a compact operator and as such it has a sequence of eigenvalues $1 \ge \lambda_1(U, V) \ge \lambda_2(U, V) \ge \ldots > 0$.
In this talk, we will focus on the case when both $U$ and $V$ are intervals. In this case the eigenvalues depend only on the product of length of the intervals $c = |U| |V|$, so we have a sequence $1 > \lambda_1(c) > \lambda_2(c) > \ldots > 0$. It turns out that these eigenvalues exhibit a phase transition: first $\approx c$ of them are very close to $1$, then there are $\approx \log c$ intermediate ones and the remaining eigenvalues decay to zero extremely fast. We will discuss the behaviour of eigenvalues in these regimes, with focus on the most interesting, intermediate region. If time permits we will also mention a new exponential lower bound for the eigenvalues $\lambda_n(c)$ when $n < (1-\varepsilon)c, \varepsilon > 0$.
