We perform a Doob h-transformation of the N-body Hamiltonian for N trapped interacting Bose particles starting from the unique real strictly positive ground state. This identifies the unitarily equivalent Markov generator describing the N interacting diffusion processes. By applying the convergence results coming from the Gross-Pitaevskii (GP) scaling limit to the energy form associated with this Markov generator we obtain the asymptotic generator uniquely corresponding to the GP Hamiltonian, which has been rigorously derived from the N-body Hamiltonian by performing the same scaling limit. Our asymptotic generator is a non linear diffusion generator with a killing rate governed by the GP order parameter. The associated stochastic process properly describes the single particle motion in the Bose-Einstein condensate including the self-interaction which acts here as a probability density-dependent killing rate. Finally, in a one-dimensional setting, we discuss under which conditions the process conditioned to survival converges to a quasi-stationary distribution having density with respect to the speed measure given by the principal eigenfunction of the diffusion generator with killing.
|Titolo:||A Doob h-Transform of the Gross-Pitaevskii Hamiltonian|
UGOLINI, STEFANIA (Corresponding)
|Parole Chiave:||Bose-Einstein Condensation; Gross-Pitaevskii scaling limit; Stochastic Mechanics; interacting Nelson diffusions; diffusions with killing; quasi-stationary distribution|
|Settore Scientifico Disciplinare:||Settore MAT/06 - Probabilita' e Statistica Matematica|
|Data di pubblicazione:||ott-2015|
|Digital Object Identifier (DOI):||10.1007/s10955-015-1337-3|
|Appare nelle tipologie:||01 - Articolo su periodico|