Generalized Fourier representation of the absolutely continuous part of a selfadjoint operator

TL;DR

This paper introduces a new method to construct a unique generalized Fourier transform for the absolutely continuous part of any selfadjoint operator, enabling a fiber decomposition approach in Hilbert spaces.

Contribution

It develops a novel technique to decompose absolutely continuous operators into a fiber direct integral, establishing existence and uniqueness of the associated Fourier transform.

Findings

Established the existence and uniqueness of the generalized Fourier transform.

Developed a new fiber decomposition method for absolutely continuous operators.

Provided a framework for analyzing selfadjoint operators via Fourier transforms.

Abstract

We formulate and prove the existence and uniqueness of the generalized Fourier transform associated with the absolutely continuous part of an arbitrary selfadjoint operator on a separable Hilbert space. To this end we develop a novel method to decompose an absolutely continuous operator into a variable fiber direct integral of selfadjoint operators.