2023/05/14 by Han, Yi
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2305.08205
Consider the random Schrödinger operator Hn defined on \0,1,⋯,n\⊂ℤ (Hnψ)_ℓ=ψℓ-1,n+ψℓ+1,n+σ\fracω_ℓaℓ,nψℓ,n, ψ0=ψn+1=0, where σ>0, ω_ℓ are i.i.d. random variables and aℓ,n typically has order √(n) for ℓ∈[εn,(1-ε)n] and any ε>0. Two important cases: the vanishing case aℓ,n=√(n) and the decaying case aℓ,n=√(ℓ) were studied before in \citekritchevski2011scaling. In this paper we consider more general decaying profiles that lie in between these two extreme cases. We characterize the scaling limit of transfer matrices and determine the point process limit of eigenvalues near a fixed energy in the bulk, in terms of solutions to coupled SDEs. We obtain new point processes that share similar properties to the Sechτ process. We determine the shape profile of eigenfunctions after a suitable re-scaling, that correspond to a uniformly chosen eigenvalue of Hn. We also give more description of the new point processes we just defined, including the probability of small and large gaps and a variance estimate.