@article{72f42d72b0bc47bf9c98f9611720278a,

title = "Ground State Energy of the One-Dimensional Discrete Random Schr{\"o}dinger Operator with Bernoulli Potential",

abstract = "In this paper we show the that the ground state energy of the one-dimensional discrete random Schr{\"o}dinger operator with Bernoulli potential is controlled asymptotically as the system size N goes to infinity by the random variable ℓ N, the length the longest consecutive sequence of sites on the lattice with potential equal to zero. Specifically, we will show that for almost every realization of the potential the ground state energy behaves asymptotically as π 2/(ℓ N+1) 2 in the sense that the ratio of the quantities goes to one.",

keywords = "Bernoulli, Discrete, Ground state energy, Longest run, Schr{\"o}dinger operator",

author = "Michael Bishop and Jan Wehr",

note = "Funding Information: Acknowledgements We would like to thank W. Faris for the suggestion to look at potentials with Bernoulli distributions which turned out to be a great starting point. We would like to thank R. Sims for useful discussions, in particular of the broader context of random Schr{\"o}dinger operator theory. We would like to thank M. Lewenstein, A. Sanpera, P. Massignan, and J. Stasi{\'n}ska for collaborating with us on related projects as well as providing supporting numerical results. The first idea of the present paper grew out of discussions with J. Xin. Both authors were supported in part by NSF grant DMS-1009508 and M. Bishop was in addition funded by NSF VIGRE grant DMS-0602173 at the University of Arizona and by the NSF under Grant No DGE-0841234.",

year = "2012",

month = may,

doi = "10.1007/s10955-012-0480-3",

language = "English (US)",

volume = "147",

pages = "529--541",

journal = "Journal of Statistical Physics",

issn = "0022-4715",

publisher = "Springer New York",

number = "3",

}