Basic properties of the current-current correlation measure for random Schrödinger operators

Peter D. Hislop; Olivier Lenoble

doi:10.1063/1.2378618

Basic properties of the current-current correlation measure for random Schrödinger operators

Peter D. Hislop, Olivier Lenoble

Mathematics

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

1 Scopus citations

Abstract

The current-current correlation measure plays a crucial role in the theory of conductivity for disordered systems. We prove a Pastur-Shubin-type formula for the current-current correlation measure expressing it as a thermodynamic limit for random Schrödinger operators on the lattice and the continuum. We prove that the limit is independent of the self-adjoint boundary conditions and independent of a large family of expanding regions. We relate this finite-volume definition to the definition obtained by using the infinite-volume operators and the trace-per-unit volume.

Original language	English
Article number	112106
Journal	Journal of Mathematical Physics
Volume	47
Issue number	11
DOIs	https://doi.org/10.1063/1.2378618
State	Published - 2006

Bibliographical note

Funding Information:
P.D.H. thanks R. M. Brown, M. Shubin, and especially H. Leschke, for many helpful discussions. P.D.H. is partially supposed by NSF Grant No. DMS-0503784.

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1063/1.2378618

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title = "Basic properties of the current-current correlation measure for random Schr{\"o}dinger operators",

abstract = "The current-current correlation measure plays a crucial role in the theory of conductivity for disordered systems. We prove a Pastur-Shubin-type formula for the current-current correlation measure expressing it as a thermodynamic limit for random Schr{\"o}dinger operators on the lattice and the continuum. We prove that the limit is independent of the self-adjoint boundary conditions and independent of a large family of expanding regions. We relate this finite-volume definition to the definition obtained by using the infinite-volume operators and the trace-per-unit volume.",

author = "Hislop, {Peter D.} and Olivier Lenoble",

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N2 - The current-current correlation measure plays a crucial role in the theory of conductivity for disordered systems. We prove a Pastur-Shubin-type formula for the current-current correlation measure expressing it as a thermodynamic limit for random Schrödinger operators on the lattice and the continuum. We prove that the limit is independent of the self-adjoint boundary conditions and independent of a large family of expanding regions. We relate this finite-volume definition to the definition obtained by using the infinite-volume operators and the trace-per-unit volume.

AB - The current-current correlation measure plays a crucial role in the theory of conductivity for disordered systems. We prove a Pastur-Shubin-type formula for the current-current correlation measure expressing it as a thermodynamic limit for random Schrödinger operators on the lattice and the continuum. We prove that the limit is independent of the self-adjoint boundary conditions and independent of a large family of expanding regions. We relate this finite-volume definition to the definition obtained by using the infinite-volume operators and the trace-per-unit volume.

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Basic properties of the current-current correlation measure for random Schrödinger operators

Abstract

Bibliographical note

ASJC Scopus subject areas

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