## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 44, Number 3 (2004), 615-653.

### On the integrated density of states of random Pauli Hamiltonians

#### Abstract

The difference of the integrated densities of states (IDS) of the two components of a random Pauli Hamiltonian is shown to equal a constant given in terms of the expectation of the magnetic field. This formula is a random version of the Aharonov and Casher theory or that of the Atiyah and Singer index theorem. By this formula, the IDS is shown to jump at 0 if the expectation of the magnetic field is nonzero. For simple cases where the expectation of the magnetic field is zero, a lower estimate of the asymptotics of the IDS at 0 is given. This lower estimate shows that the IDS decays slower than known results for random Schrödinger operators whose infimum of the spectrum is 0. Moreover the strong-magnetic-field limit of the IDS is identified in a general setting.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 44, Number 3 (2004), 615-653.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250283087

**Digital Object Identifier**

doi:10.1215/kjm/1250283087

**Mathematical Reviews number (MathSciNet)**

MR2103786

**Zentralblatt MATH identifier**

1087.82014

**Subjects**

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Secondary: 47B80: Random operators [See also 47H40, 60H25] 47N55 60G60: Random fields

#### Citation

Ueki, Naomasa. On the integrated density of states of random Pauli Hamiltonians. J. Math. Kyoto Univ. 44 (2004), no. 3, 615--653. doi:10.1215/kjm/1250283087. https://projecteuclid.org/euclid.kjm/1250283087