Abstract
We study the multiparticle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multiparticle lower spectral edges are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multiparticle Anderson model at low energy. The proof based on the multiparticle multiscale analysis bounds needs the values of the external random potential to be independent and identically distributed, whose common probability distribution is at least Log-Hölder continuous.
Highlights
In their work [10], Klein and Nguyen developed the continuum multiparticle bootstrap multiscale analysis of the Anderson model with alloy type external potential. e method of Klein and Nguyen is very close in the spirit to that of our work [2]. e results of [2] were the first rigorous mathematical proof of localization for many body interacting Hamiltonians near the bottom of the spectrum on the lattice
We prove the exponential localization in the max-norm and the strong dynamical localization near the bottom of the spectrum
For 푛 = 1 the property holds true for all 푘 ≥ 0 by the single-particle localization theory [12]
Summary
Δ is the Laplacian on R , U represents the inter-particle interaction potential which acts as a multiplication operator in 2 R푁푑. V is the multiparticle random external potential, acting as multiplication operator on. E potential of inter-particle interaction U is bounded, nonnegative and of the form. E common probability distribution function, , of the i.i.d. random variables 푉(푥, ⋅), ∈ Z associated to the measure is defined by:. E random potential field 푉(푥, 휔); 푥 ∈ Z is i.i.d., of nonnegative values and the corresponding probability distribution function is log-. Some parts of the rest of the text overlap with the paper [14] but for the reader convenience we give all the details of the arguments
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