Proving superintegrability in $\beta$-deformed eigenvalue models

TL;DR

This paper proves superintegrability properties in $eta$-deformed eigenvalue models, confirming several conjectured expectation value formulas for symmetric polynomials with various potentials.

Contribution

It provides rigorous proofs of superintegrability in $eta$-deformed eigenvalue models, completing previous conjectures in the literature.

Findings

Derived explicit expectation value formulas for symmetric polynomials.

Confirmed superintegrability conjectures for models with different potentials.

Completed the mathematical proof framework for superintegrability in these models.

Abstract

In this note we provide proofs of various expressions for expectation values of symmetric polynomials in $\beta$-deformed eigenvalue models with quadratic, linear, and logarithmic potentials. The relations we derive are also referred to as superintegrability. Our work completes proofs of superintegrability statements conjectured earlier in literature.