### 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 π ^{2}/(ℓ _{N}+1) ^{2} in the sense that the ratio of the quantities goes to one.

Original language | English (US) |
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Pages (from-to) | 529-541 |

Number of pages | 13 |

Journal | Journal of Statistical Physics |

Volume | 147 |

Issue number | 3 |

DOIs | |

State | Published - May 2012 |

### Fingerprint

### Keywords

- Bernoulli
- Discrete
- Ground state energy
- Longest run
- Schrödinger operator

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Ground State Energy of the One-Dimensional Discrete Random Schrödinger Operator with Bernoulli Potential.** / Bishop, Michael; Wehr, Jan.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 147, no. 3, pp. 529-541. https://doi.org/10.1007/s10955-012-0480-3

}

TY - JOUR

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

AU - Bishop, Michael

AU - Wehr, Jan

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 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.

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 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.

KW - Bernoulli

KW - Discrete

KW - Ground state energy

KW - Longest run

KW - Schrödinger operator

UR - http://www.scopus.com/inward/record.url?scp=84861190908&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861190908&partnerID=8YFLogxK

U2 - 10.1007/s10955-012-0480-3

DO - 10.1007/s10955-012-0480-3

M3 - Article

VL - 147

SP - 529

EP - 541

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -