## Details

Original language | English |
---|---|

Pages (from-to) | 515-533 |

Number of pages | 19 |

Journal | Journal of functional analysis |

Volume | 253 |

Issue number | 2 |

Publication status | Published - 15 Dec 2007 |

Externally published | Yes |

## Abstract

We consider ergodic random Schrödinger operators on the metric graph Z^{d} with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin-Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.

## Keywords

- Integrated density of states, Metric graph, Quantum graph, Random Schrödinger operator

## ASJC Scopus subject areas

- Mathematics(all)
**Analysis**

## Cite this

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**Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over Z**/ Gruber, Michael J.; Lenz, Daniel H.; Veselić, Ivan.

^{d}.In: Journal of functional analysis, Vol. 253, No. 2, 15.12.2007, p. 515-533.

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

^{d}',

*Journal of functional analysis*, vol. 253, no. 2, pp. 515-533. https://doi.org/10.1016/j.jfa.2007.09.003

^{d}.

*Journal of functional analysis*,

*253*(2), 515-533. https://doi.org/10.1016/j.jfa.2007.09.003

^{d}. Journal of functional analysis. 2007 Dec 15;253(2):515-533. doi: 10.1016/j.jfa.2007.09.003

}

TY - JOUR

T1 - Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over Zd

AU - Gruber, Michael J.

AU - Lenz, Daniel H.

AU - Veselić, Ivan

N1 - Funding information: It is a pleasure to thank Mario Helm for interesting discussions. The authors were financially supported by the DFG, two of them (M.G. and I.V.) under grant Ve 253/2-2 within the Emmy-Noether-Programme.

PY - 2007/12/15

Y1 - 2007/12/15

N2 - We consider ergodic random Schrödinger operators on the metric graph Zd with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin-Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.

AB - We consider ergodic random Schrödinger operators on the metric graph Zd with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin-Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.

KW - Integrated density of states

KW - Metric graph

KW - Quantum graph

KW - Random Schrödinger operator

UR - http://www.scopus.com/inward/record.url?scp=36048954578&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2007.09.003

DO - 10.1016/j.jfa.2007.09.003

M3 - Article

AN - SCOPUS:36048954578

VL - 253

SP - 515

EP - 533

JO - Journal of functional analysis

JF - Journal of functional analysis

SN - 0022-1236

IS - 2

ER -