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In this talk, I will discuss eigenvalue estimates for operators related to simultaneous localization in space and frequency. This problem is connected to phase-space localization, the uncertainty principle, and spectral concentration. It also appears naturally in harmonic analysis, signal representation, approximation theory, and mathematical physics.

The main objects are space-frequency limiting operators. Their eigenvalues measure how well a bandlimited function can be concentrated on a given spatial region. Usually, most eigenvalues are close to either 0 or 1, and only a smaller number are in the middle transition region, which is often called the plunge region. Estimating this region gives quantitative information about the effective number of degrees of freedom. This is similar in spirit to counting states in phase space.

The method is based on a wave packet decomposition adapted to the geometry of the spatial and frequency domains. These wave packets are constructed using smooth Gevrey cutoffs. This gives strong Fourier localization and also useful spatial control. The wave packets behave like approximate eigenfunctions for the localization operator.

As a result, we obtain explicit bounds on the number of eigenvalues in (ε,1−ε). I will explain how phase-space localization and geometric decomposition can lead to quantitative spectral estimates in higher-dimensional settings.