Fredholm Theory on Twisted Hilbert Scales: A Frame-Theoretic Approach to Half-Integer Fourier Modes
Anik Chakraborty, Varinder Kumar
TL;DR
We introduce a half-integer twisted Hilbert scale on $L^2([0,1])$ built from half-integer Fourier modes via a unitary twist that maps the standard basis to $psi_k = e^{2π i (k+1/2) x}$. A diagonal operator with eigenvalues $lambda_k = k+1/2$ is self-adjoint with compact resolvent, and between adjacent scale levels defines a Fredholm map of index zero; we solve an explicit antiperiodic boundary-value problem and compute the zeta-regularized determinant $det_zeta(|~A|)=2$; stability under bounded perturbations and a well-defined spectral flow follow. The framework encodes twisted boundary conditions in a purely operator-theoretic setting, motivated by twisted spinor boundary conditions on non-orientable manifolds, and offers a frame-theoretic perspective that avoids differential operators.
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 $\det_ζ(|\widetilde{A}|) = 2$ 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.