How Dixmier Trace is used in Machine Learning part3
- A Dixmier trace formula for the density of states(arXiv)
Author : Nurulla Azamov, Edward McDonald, Fedor Sukochev, Dmitriy Zanin
Abstract : A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrödinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators — — the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances. △
2.Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples (arXiv)
Author : Leandro Cioletti, L. Y. Hataishi, Artur O. Lopes, M. Stadlbauer
Abstract : In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalize the Perron-Frobenius-Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined ζ-function at its critical parameter. We then generalize Sharp’s construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the ζ-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold
