Giacomo Gavelli | Searcharxiv

arXiv2026

A Categorical Framework for the Direct Integration of Banach Spaces

We construct a categorical framework in which we can view the direct integral as a functor taking as input objects in a certain category of Abstract Banach bundles and outputting a Banach space. We do this by constructing a quasi-abelian category of abstract Banach bundles, and we prove certain foundational and theoretical results about this category and its objects. As a consequence, we are able to define a notion of direct integral of sheaves.

math.CT↗

arXiv2026

Spectral convergence of hyperbolic surfaces and quantum ergodicity

We extend the notion of Plancherel convergence, introduced in the study of compact quotients of locally compact groups, to finite-area hyperbolic surfaces and establish a quantum ergodicity theorem for Plancherel sequences. We show that the expectation values of integral operators whose kernels are uniformly bounded and have uniformly bounded propagation become asymptotically equidistributed, on average, for eigenvalues in a fixed compact interval. Our result recovers the case of multiplication operators studied by Le Masson and Sahlsten, corresponding to the degenerate case of propagation bound zero and kernels supported on the diagonal. Moreover, our result allows the systole to shrink without a uniform lower bound, with the allowed degeneration constrained by Plancherel convergence. This identifies Plancherel convergence as a natural spectral framework for quantum ergodicity on degenerating hyperbolic surfaces.

math.SP↗

arXiv2024

Benjamini-Schramm vs Plancherel convergence

We show that Plancherel convergence is strictly stronger than Benjamini-Schramm convergence. In order to do so, we introduce two criteria for Plancherel and Benjamini-Schramm convergence in terms of counting functions of the length spectrum.

math.GR↗