A Class of Parameter Dependent Commuting Matrices

TL;DR

This paper introduces a new class of parameter-dependent symmetric matrices that commute for all parameter values, violate the non-crossing rule, and are linked to quantum integrable systems.

Contribution

It defines a novel class of matrices with specific commutation properties and explores their connection to quantum integrable Hamiltonians.

Findings

Matrices generically violate the Wigner von Neumann non-crossing rule.

The class is connected to finite-dimensional quantum integrable systems.

Provides conditions for the solvability of related linear equations.

Abstract

We present a novel class of real symmetric matrices in arbitrary dimension $d$, linearly dependent on a parameter $x$. The matrix elements satisfy a set of nontrivial constraints that arise from asking for commutation of pairs of such matrices for all $x$, and an intuitive sufficiency condition for the solvability of certain linear equations that arise therefrom. This class of matrices generically violate the Wigner von Neumann non crossing rule, and is argued to be intimately connected with finite dimensional Hamiltonians of quantum integrable systems.