Commun. Math. Phys. 275, 1–36 (2007) Digital Object Identifier (DOI) 10.1007/s00220-007-0293-4
Communications in
Mathematical Physics
Exponential Times in the One-Dimensional Gross–Pitaevskii Equation with Multiple Well Potential Dario Bambusi1 , Andrea Sacchetti2 1 Dipartimento di Matematica, Universitá degli studi di Milano, Via Saldini 50, Milano 20133, Italy.
E-mail:
[email protected]
2 Dipartimento di Matematica Pura ed Applicata, Universitá degli studi di Modena e Reggio Emilia,
Via Campi 213/B, Modena 41100, Italy. E-mail:
[email protected] Received: 1 August 2006 / Accepted: 15 February 2007 Published online: 20 July 2007 – © Springer-Verlag 2007
Abstract: We consider the Gross-Pitaevskii equation in 1 space dimension with a N -well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest N eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting on M, or close to it, will remain close to M for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on M is a perturbation of a discrete nonlinear Schrödinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to M their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required.
1. Introduction In this paper we study the dynamics of low energy states of the one-dimensional GrossPitaevskii equation (hereafter also called nonlinear Schrödinger equation, NLS)
2σ t iψ˙ t = H0 ψ t + ψ t ψ t , ψ˙ t = ∂ψ ∂t , ψ t (x)t=0 = ψ 0 (x) ∈ L 2 (R), ψ 0 L 2 = 1,
(1)
This work is partially supported by the INdAM project Mathematical modeling and numerical analysis of quantum systems with applications to nanosciences. DB was also supported by MIUR under the project COFIN2005 Sistemi dinamici nonlineari ed applicazioni fisiche. AS was also supported by MIUR under the project COFIN2005 Sistemi dinamici classici, quantistici e stocastici.
2
D. Bambusi, A. Sacchetti
Fig. 1. Plot of a trapping potential with N wells
where σ = 1, 2, . . . , is a positive integer number and H0 = −2
d2 + V, x ∈ R dx2
(2)
is the linear Hamiltonian operator and V (x) a N –well potential. By this we mean that V has N nondegenerate distinct minima x1 , ..., x N , where the potential has essentially the same behavior (e.g. one can assume that its first r derivatives are equal at all the minima, for some positive integer r ≥ 4). We also assume that the potential is trapping, i.e. V tends to infinity as |x| → ∞ (see Fig. 1). Equation (1) with a multiple well potential describes particular phenomena associated with wave propagation in nonlinear multiple quantum well waveguides consisting of N unit cells where each of them is formed by linear films sandwiched between two nonlinear ones [26, 28]. Another situation we have in mind is that of a weakly interacting Bose Einstein condensate trapped in N cells of an