Abstract
We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years.
Highlights
A system of N bosons in the mean-field regime can be described by the Hamilton operator HN = N −Δxj +1 N v(xi − xj) j=1 i
On F, for any f ∈ L2(R3), we introduce the usual creation and annihilation
We show, for bounded interactions, a large deviation principle for the fluctuations of the many-body quantum evolution around the limiting Hartree dynamics
Summary
A system of N bosons in the mean-field regime can be described by the Hamilton operator. In order to factor out the condensate, described at time t ∈ R, by the solution φt of (1.2), we observe that every ψ ∈ L2s(R3N ) can be uniquely written as ψ = η0φ⊗t N + η1 ⊗s φ⊗t (N−1) + · · · + ηN with ηj ∈ L2⊥φt (R3)⊗sj, where L2⊥φt (R3) denotes the orthogonal complement in L2(R3) of the condensate wave function φt This remark allows us to define, for every t ∈ R, a unitary operator. (This line of research started in [17] and was further explored in [11,16,21]; recently, an expansion of the many-body dynamics in powers of N −1 was obtained in [6].) Notice that L∞(t2) acts on (a dense subspace of) the Fock space F⊥φt , constructed on the orthogonal complement of φt , with no restriction on the number of particles. This step makes use of the choice of product initial data (which implies that the expectation is taken in the vacuum); at the expenses of a longer proof, we could have proven Theorem 1.1 to a larger and physically more interesting class of initial data
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