Hilbert bundles with ends

TL;DR

This paper introduces the concept of Hilbert bundles with ends, a new structure that generalizes traditional bundles, and explores their properties, invariants, and appearances in spectral theory and infinite coverings.

Contribution

It defines Hilbert bundles with ends, introduces characteristic classes for them, and demonstrates their relevance in infinite coverings and spectral decompositions.

Findings

Hilbert bundles with ends are a new mathematical structure.

Characteristic classes serve as invariants for these bundles.

Applications include infinite coverings and spectral analysis of differential operators.

Abstract

Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.