Fredholm Theory on Twisted Hilbert Scales: A Frame-Theoretic Approach to Half-Integer Fourier Modes

TL;DR

This paper develops a new Hilbert scale using a twist operator on $L^2([0,1])$, enabling analysis of a Fredholm operator with explicit spectral properties, stability, and zeta-regularized determinants, motivated by twisted boundary conditions.

Contribution

It introduces a novel Hilbert scale via a unitary twist, analyzes a diagonal operator with explicit spectrum, and applies Fredholm theory in an abstract, boundary-condition-free framework.

Findings

Constructed a Hilbert scale with weighted norms and explicit spectrum.

Proved the Fredholm property and index zero of the operator between scale levels.

Computed the zeta-regularized determinant as 2.

Abstract

We construct a Hilbert scale on $L^2([0,1])$ via a unitary twist operator that maps the standard Fourier basis to half-integer frequency exponentials. The resulting weighted spaces, equipped with norms indexed by $(1+|k+\tfrac{1}{2}|^2)^s$, admit a canonical diagonal operator with the compact resolvent and spectrum $\{k+\tfrac{1}{2}\}_{k\in\mathbb{Z}}$. We prove that this operator defines a Fredholm mapping between adjacent scale levels with index zero, provide an explicit solution to an antiperiodic boundary value problem illustrating the framework, and compute the zeta-regularized determinant $\det_\zeta(|\widetilde{A}|) = 2$ using the Hurwitz zeta function. We establish stability under bounded perturbations and verify the well-definedness of spectral flow. The framework is developed entirely through functional-analytic methods without differential operators or boundary value problems. The construction is motivated by twisted spinor boundary conditions on non-orientable manifolds, though the present work is formulated abstractly in operator-theoretic language.