A finite dimensional trace formula

TL;DR

The paper develops a finite-dimensional trace formula by applying a Von-Neumann ergodic theorem framework to a smooth family of unitary matrices $U(k)$ and using a determinant-zero condition $f(k)=\det(U(k)-I)=0$ to express spectral sums as a distribution of Dirac masses on $k$-space. For observables $A(k)$, the result generalizes to $(1/2\pi)\sum_{m\in\mathbb{Z}}\mathrm{tr}(U^m(k)A(k)) = \sum_{k_0} \frac{\langle \lambda(k_0)|A(k_0)|\lambda(k_0)\rangle}{|\langle \lambda(k_0)|U'(k_0)|\lambda(k_0)\rangle|} \delta(k-k_0)$, with the special case $A(k)=U'(k)$ yielding a crystalline measure. The paper further provides a constructive example with $U(k)=\mathrm{e}^{ik\ell_1}\cdots\mathrm{e}^{ik\ell_n}S$ giving explicit delta-sum representations, and offers two visualization routes: a $t$-weighted trace-sum limit and a Newton–Cayley–Hamilton approach to relate trace sums to $h(1,k)=\det(I-U(k))$. This framework extends Poisson summation to higher dimensions and links spectral data of unitary families to crystalline measures with potential implications for spectral analysis and zero-set dynamics.

Abstract

We take the trace of Von-Neumann's ergodic theorem and get a trace formula of a unitary matrix family. It is an extension of Poisson summation formula in higher dimension. We also construct a family of crystalline measure with complex coefficient.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5