On polynomials in spectral projections of spin operators

TL;DR

This paper investigates the asymptotic behavior of polynomial evaluations on spectral projections of spin operators, revealing a convergence to a maximum value and contrasting it with random projections, highlighting a spectral concentration phenomenon.

Contribution

It introduces a new analysis of polynomial norms on spectral projections of spin operators and identifies a spectral concentration effect in the semiclassical limit.

Findings

Operator norm converges to maximum value in the semiclassical limit.

Contrasts behavior with random projections showing spectral concentration.

Proves cases of Slepian spectral concentration phenomenon.

Abstract

We show that the operator norm of an arbitrary bivariate polynomial, evaluated on certain spectral projections of spin operators, converges to the maximal value in the semiclassical limit. We contrast this limiting behavior with that of the polynomial when evaluated on random pairs of projections. The discrepancy is a consequence of a type of Slepian spectral concentration phenomenon, which we prove in some cases.