The Spectrum of the Berezin transform for Gelfand pairs

TL;DR

This paper analyzes the spectrum of the Berezin transform associated with orbit POVMs on Gelfand pairs, providing explicit formulas, case studies on $SU(2)$, and insights into eigenvalue oscillations and quantization properties.

Contribution

It derives an explicit spectrum formula for the Berezin transform on Gelfand pairs and applies it to $SU(2)$, connecting to geometric quantization and eigenvalue behavior.

Findings

Explicit spectrum formula for Berezin transform on Gelfand pairs.

Spectral analysis of orbit POVMs on $S^2$ confirming previous results.

Conjectures and proofs regarding eigenvalue oscillations and quantization violations.

Abstract

We discuss the Berezin transform, a Markov operator associated to positive-operator valued measures (POVMs). We consider the class of so-called orbit POVMs, constructed on the quotient space $\Omega = G/K$ of a compact group $G$ by its subgroup $K$. We restrict attention to the case where $(G, K)$ is a Gelfand pair and derive an explicit formula for the spectrum of the Berezin transform in terms of the characters of the irreducible unitary representations of $G$. We then specialize our results to the case study $G = \text{SU}(2)$ and $K \simeq S^1$, and find the spectra of orbit POVMs on $S^2$. We confirm previous calculations by Zhang and Donaldson of the spectrum of the standard quantization of $S^2$ coming from K\"ahler geometry. Then, we make a couple of conjectures about the oscillations in the sequence of eigenvalues, and prove them in the simplest case of second-highest weight vector. Finally, for low weights, we prove that the corresponding orbit POVMs on $S^2$ violate the axioms of a Berezin-Toeplitz quantization.