2023/04/24 by Alix Deleporte, Deleporte, Alix, Gaultier Lambert +1 · 2 citations
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2304.12275
openalex publication_date 2023/04/24 · openalex created_date 2023/04/27 · openalex updated_date 2026/07/28
We consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on ℝ, with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő-type central limit theorem (CLT) for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalization to a Gaussian limit with a H1/2-type variance, which depends on the potential. In the one-well (one-cut) case, using the quantum action-angle theorem and additional micro-local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi-cut case, we show that for generic potentials, a similar result holds and the contributions of each wells are independent in the limit.