Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential

Michael Bishop; Jan Wehr

doi:10.1007/s10955-012-0480-3

Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential

Michael Bishop
, Jan Wehr

Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

In this paper we show the that the ground state energy of the one-dimensional discrete random Schrödinger 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 π²/(ℓ_N+1)² in the sense that the ratio of the quantities goes to one.

Original language	English (US)
Pages (from-to)	529-541
Number of pages	13
Journal	Journal of Statistical Physics
Volume	147
Issue number	3
DOIs	https://doi.org/10.1007/s10955-012-0480-3
State	Published - May 2012

Keywords

Bernoulli
Discrete
Ground state energy
Longest run
Schrödinger operator

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s10955-012-0480-3

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@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 ℓN, 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 π2/(ℓN+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",

number = "3",

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TY - JOUR

T1 - Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential

AU - Bishop, Michael

AU - Wehr, Jan

N1 - 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 Schrödinger operator theory. We would like to thank M. Lewenstein, A. Sanpera, P. Massignan, and J. Stasińska 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.

PY - 2012/5

Y1 - 2012/5

N2 - In this paper we show the that the ground state energy of the one-dimensional discrete random Schrödinger operator with Bernoulli potential is controlled asymptotically as the system size N goes to infinity by the random variable ℓN, 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 π2/(ℓN+1)2 in the sense that the ratio of the quantities goes to one.

AB - In this paper we show the that the ground state energy of the one-dimensional discrete random Schrödinger operator with Bernoulli potential is controlled asymptotically as the system size N goes to infinity by the random variable ℓN, 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 π2/(ℓN+1)2 in the sense that the ratio of the quantities goes to one.

KW - Bernoulli

KW - Discrete

KW - Ground state energy

KW - Longest run

KW - Schrödinger operator

UR - https://www.scopus.com/pages/publications/84861190908

UR - https://www.scopus.com/pages/publications/84861190908#tab=citedBy

U2 - 10.1007/s10955-012-0480-3

DO - 10.1007/s10955-012-0480-3

M3 - Article

AN - SCOPUS:84861190908

SN - 0022-4715

VL - 147

SP - 529

EP - 541

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3

ER -

Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential

Abstract

Keywords

ASJC Scopus subject areas

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