2022/08/05 by N. S. Witte, L. Wei, Witte, N. S. +1
Chemistry · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Cauchy distribution #Classical Analysis and ODEs (math.CA) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Initial value problem #Laguerre polynomials #Mathematical Physics (math-ph) #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Orthogonal polynomials #Polynomial #Pure mathematics #math-ph #math.CA #math.MP #nlin.SI
paper · pdf · doi:10.48550/arxiv.2208.03278
published in arXiv (Cornell University) (Cornell University)
arxiv created 2022/08/05 · openalex publication_date 2022/08/05 · arxiv updated 2022/08/08 · openalex created_date 2022/10/03 · openalex updated_date 2026/08/06
The Bures metric and the associated Bures-Hall measure is arguably the best choice for studying the spectrum of the quantum mechanical density matrix with no apriori knowledge of the system. We investigate the probability of a gap in the spectrum of this model, either at the bottom [0,s) or at the top (s,1] , utilising the connection of this Pfaffian point-process with the allied problem in the determinantal point-process of the two-dimensional Cauchy-Laguerre bi-orthogonal polynomial system, now deformed with two variables s,t. To this end we develop new general results about Cauchy bi-orthogonal polynomial system for a more general class of weights than the Laguerre densities: in particular a new Christoffel-Darboux formula, reproducing kernels and differential equations for the polynomials and their associated functions. This system is most simply expressed as rank-3 matrix variables and possesses an associated cubic bilinear form. Furthermore under specialisation to truncated Laguerre type densities for the weight, of direct relevance to the Cauchy-Laguerre system, we construct a closed system of constrained, nonlinear differential equations in two deformation variables s,t, and observe that the recurrence, spectral and deformation derivative structures form a compatible and integrable triplet of Lax equations.