We assess the probability of resonances between sufficiently distant states of an -particle disordered quantum system in a combinatorial graph . In the -particle configuration space, there are arbitrarily distant pairs of configurations giving rise to pairs of local (random) Hamiltonians which are strongly coupled, so that the eigenvalue concentration (EVC) bounds are difficult to obtain. We extend to any number of particles the efficient EVC bounds, obtained earlier for the 2-particle systems. 1. Introduction We study quantum systems in a disordered environment, usually referred to as Anderson-type models, due to the seminal paper by Anderson [1]. For nearly fifty years following its publication, the localization phenomena have been studied in the single-particle approximation, that is, under the assumption that the interaction between particles subject to the common random external potential is sufficiently weak to be neglected in the analysis of the decay properties of eigenstates of the multiparticle system in question. A detailed discussion of recent developments in the physics of disordered media is most certainly beyond the scope of this paper; we simply refer to the papers by Basko et al. [2] and by Gornyi et al. [3] where it was shown, in the framework of physical models and methods, that the localization phenomena, firmly established for the noninteracting systems, persist in presence of nontrivial interactions. The mathematical Anderson localization theory has motivated a large number of studies of random differential and finite-difference operators during the last forty years, but only recently a significant progress has been made in the rigorous theory of multiparticle quantum systems in a disordered environment with a nontrivial interparticle interaction (cf. [4–6] and more recent works [7–10]), and there still remain many challenging open problems in this area of mathematical physics. Such problems are often related to the EVC bounds, which in the single-particle setting go back to the celebrated Wegner bound [11]. Building on our results from [12], we discuss in the present paper some important implications for the Hamiltonians of the multiparticle disordered quantum systems. Specifically, we consider a system of quantum particles in a finite or countable, locally finite connected graph , endowed with the canonical graph distance , with the Hamiltonian of the following form: where is a random field on the graph , relative to a probability space , is the graph Laplacian on , namely: and the interaction operator is the multiplication by a
References
[1]
P. W. Anderson, “Absence of diffusion in certain random lattices,” Physical Review, vol. 109, no. 5, pp. 1492–1505, 1958.
[2]
D. M. Basko, I. L. Aleiner, and B. L. Altshuler, “Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states,” Annals of Physics, vol. 321, no. 5, pp. 1126–1205, 2005.
[3]
I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, “Interacting electrons in disordered wires: anderson localization and low-T transport,” Physical Review Letters, vol. 95, Article ID 206603, 2005.
[4]
V. Chulaevsky and Y. Suhov, “Eigenfunctions in a two-particle Anderson tight binding model,” Communications in Mathematical Physics, vol. 289, no. 2, pp. 701–723, 2009.
[5]
V. Chulaevsky and Y. Suhov, “Multi-particle Anderson localisation: induction on the number of particles,” Mathematical Physics, Analysis and Geometry, vol. 12, no. 2, pp. 117–139, 2009.
[6]
M. Aizenman and S. Warzel, “Localization bounds for multiparticle systems,” Communications in Mathematical Physics, vol. 290, no. 3, pp. 903–934, 2009.
[7]
V. Chulaevsky, A. Boutet de Monvel, and Y. Suhov, “Dynamical localization for a multi-particle model with an alloy-type external random potential,” Nonlinearity, vol. 24, no. 5, pp. 1451–1472, 2011.
[8]
A. Klein and S. T. Nguyen, “The bootstrap multiscale analysis of the multi-particle Anderson model,” Journal of Statistical Physics, vol. 151, no. 5, pp. 938–973, 2013.
[9]
A. Klein and S. T. Nguyen, “Bootstrap multiscale analysis and localization for multi-particle continuous Anderson Hamiltonians,” 2013, http://arxiv.org/abs/1311.4220.
[10]
M. Fauser and S. Warzel, “Multiparticle localization for disordered systems on continuous space via the fractional moment method,” http://arxiv.org/abs/1402.5832.
[11]
F. Wegner, “Bounds on the density of states in disordered systems,” Zeitschrift für Physik B Condensed Matter and Quanta, vol. 44, no. 1-2, pp. 9–15, 1981.
[12]
V. Chulaevsky, “On the regularity of the conditional distribution of the sample mean,” http://arxiv.org/abs/1304.6913.
[13]
J. Fr？hlich and T. Spencer, “Absence of diffusion in the Anderson tight binding model for large disorder or low energy,” Communications in Mathematical Physics, vol. 88, no. 2, pp. 151–184, 1983.
[14]
M. Aizenman and S. Molchanov, “Localization at large disorder and at extreme energies: an elementary derivation,” Communications in Mathematical Physics, vol. 157, no. 2, pp. 245–278, 1993.
[15]
V. Chulaevsky and Y. Suhov, “Wegner bounds for a two-particle tight binding model,” Communications in Mathematical Physics, vol. 283, no. 2, pp. 479–489, 2008.
[16]
V. Chulaevsky and Y. Suhov, “Anderson localisation for an interacting two-particle quantum system on ,” http://arxiv.org/abs/0705.0657.
