A Class of Parameter Dependent Commuting Matrices

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.

Latex, Added References, Typos corrected

References in corpus (5)

• Poisson versus GOE statistics in integrable and non-integrable quantum hamiltonian
• Level repulsion in integrable and almost-integrable quantum spin models
• The Origin of Degeneracies and Crossings in the 1d Hubbard Model
• Dyson's Brownian Motion and Universal Dynamics of Quantum Systems
• Spectral correlations in the crossover between GUE and Poisson regularity: on the identification of scales

Cited by in corpus (18)

• Integrable time-dependent quantum Hamiltonians
• Integrals of motion for one-dimensional Anderson localized systems
• Quantum integrability in systems with finite number of levels
• The Link between Integrability, Level Crossings, and Exact Solution in Quantum Models
• Quantum integrability in the multistate Landau-Zener problem
• Integrable matrix theory: Level statistics
• Generalized microcanonical and Gibbs ensembles in classical and quantum integrable dynamics
• Classification of Parameter-Dependent Quantum Integrable Models, Their Parameterization, Exact Solution, and Other Properties
• Detuning-induced robustness of a three-state Lanzau-Zener model against dissipation
• Three-state Landau-Zener model in the presence of dissipation
• Parameter Dependent Commuting Matrices, Plücker relations and Related Quantum Glass Models
• Breathing modes, quartic nonlinearities and effective resonant systems
• Rotationally invariant ensembles of integrable matrices
• Master equation approach to the three-state open Majorana model
• Functionally independent conservations laws in a quantum integrable model
• Resonant transitions due to changing boundaries
• Open multistate Majorana model
• Weyl's Relations, Integrable Matrix Models and Quantum Computation