Spectral problem for the mean field Hamiltonian

Arvydas Astrauskas

doi:10.15388/lmr.2007.24252

Abstract

We consider the mean field Hamiltonian HV = κ ΔV + ξ(·) in l2(V ), where V = {x} is a finite set. Characteristic equations for eigenvalues and expressions for eigenfunctions of HV are obtained. Using this result, the spectral representation of the solution of the corresponding ("head transition'') differential equation is derived.

Highlights

Let V = {x} be a finite set, and let N be the number of elements of V
The mean field (Curie-Weiss) model in V is given by the symmetric operator (N -square matrix) H V, acting on functions (N -dimensional vectors) ψ(·): V → R according to the formula
Theorem 1 is applied to obtain the spectral representation of the solution u(t, x) to the

Summary

Introduction

We obtain equations for eigenvalues and derive formulas for eigenfunctions of the Hamiltonian H V (Theorem 1). Theorem 1 is applied to obtain the spectral representation of the solution u(t, x) to the ("heat transition") differential equation. The Feynman-Kac formula for u(t, x) is discussed (Theorem 5). In the case of independent identically distributed random variables ξ(x), x ∈ V , with distribution function F (s) = P(ξ(x) s), the spectral problem for the operator (1) and the asymptotic behavior (as N → ∞ and t → ∞) of the extreme eigenvalues and the solution u(t, x) of equation (2) were discussed in [2] (Gaussian distributions), [4] (exponential distributions) and [1] (continuous distribution functions F (·)).

Spectral problem

Application to evolution systems