Spectral problem for the mean field Hamiltonian

We consider the mean field Hamiltonian̄ HV = κ̄V + ξ(·) in l2(V ), whereV = {x} is a finite set. Characteristicequations for eigenvaluesand expressions for eigenfunctions of H̄V are obtained. Using this result, the spectral representa tion of the solution of the corresponding ("heat transition") differential equation is derived.


Introduction
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 formulā where¯ V ψ = N −1 x∈V ψ(x), κ is a positive constant and the potential ξ(·) = {ξ(x): x ∈ V } consists of real scalars. In this paper, we obtain equations for eigenvalues and derive formulas for eigenfunctions of the HamiltonianH V (Theorem 1). Theorem 1 is applied to obtain the spectral representation of the solution u(t, x) to the ("heat transition") differential equation (2) (Theorem 6). The Feynman-Kac formula for u(t, x) is also discussed (Theorem 5). The Hamiltonian (1) is a simplified modification of the lattice Schrödinger operator H V = κ V + ξ(·) in l 2 (V ), V ⊂ Z ν , with the lattice Laplacian V (cf. [3]).
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 (·)). We note that, for a continuous F (·), the variables ξ(x), x ∈ V , are all distinct with probability one. In this paper, we consider the general case of scalars ξ(x), x ∈ V .
In Section 1, we study the eigenvalue problem forH V . Section 2 is devoted to the discussion of representations of the function u(·, ·).
be the variational series of the scalars ξ(x), x ∈ V , and assume that there are exactly L 1 strict inequalities in (4): where, without loss of generality, Then the HamiltonianH V has N real eigenvalues λ 1,V λ 2,V · · · λ N,V which are specified as follows: and the corresponding (normed) eigenfunction has the form , then λ = ξ k i ,V is an eigenvalue with multiplicity m i − 1 and the set of (normed) eigenfunctions associated with λ can be chosen as an orthonormal basis of the Theorem 1 is proved below. Remark 2. With notation of part (ii) of Theorem 1, we have that 2, 3, . . . , N).

Remark 4. From Theorem 1 we obtain that
Proof of Theorem 1. We rewrite (3) in the following form To prove part (i) of the theorem, assume that λ = ξ(x) for each x. From (7) we obtain that i.e., Assume now that x∈V ψ(x) = 0. Since λ = ξ(x) for each x, from (7) we see that ψ(·) = 0, i.e., the eigenfunction ψ(·) associated with eigenvalue λ is necessarily zero. This contradicts the definition of the eigenfunction, therefore,¯ V ψ = 0.

Application to evolution systems
Let us consider the evolution system described by the equation (2). Recall that the oper-ator¯ V ψ − κψ(·) (ψ(·): V → R) is a generator of the random walkx · = {x t : t 0} in V with continuous time which stays at any site during the time, distributed exponentially with parameter κ > 0 and then takes a jump to one of sites in V with probability 1/N (a mean field random walk, a totally symmetric random walk). In other words, the local transition probabilities of the homogeneous random walkx · are given by the formula as t → 0, for all x ∈ V and y ∈ V . The equation (2) is to describe the evolution of the system of noninteracting particles in V . Each particle moves according to the random walkx · (diffusion of the system). Additionally, each particle situated at x ∈ V splits into two particles at the same x with probability max (ξ (x), 0) t + o( t) and disappears with probability max (−ξ(x), 0) t + o( t) during time interval (t; t + t) (branching mechanism of the system). Then u(t, x) is the mean number of particles at site x at time t (cf. [6]).
THEOREM 5 (Feynman-Kac formula). Equation (2) has a unique nonnegative solution u(·, ·) represented as an integral over paths: where the expectation E x is taken with respect to the mean field random walkx · which starts at x ∈ V .
Proof. Using a strong Markovian property and local transition probabilities (9) of the random walkx · , we have that as t → 0. We take u(t, x) to the left-hand side and divide both sides by t. Passing to the limit as t → 0, we obtain the assertion of Theorem 5.
for each t 0. Substituting this into (2) and noting that the eigenfunctions ψ k (·) are linearly independent, we obtain the equations for a k (·): da k (t) dt = (λ k,V − κ)a k (t), t 0; here 1 k N . This equation has the solution a k (t) = c k exp{tλ k,V − tκ}, where c k = (ψ k , u 0 ) V by calculating the inner product of ψ k (·) and u 0 (·) = N l=1 c l ψ l (·). Summarizing, we obtain the assertion of the theorem.
For the theory of linear differential equations we refer to, e.g., [5].