Applications of methods of random differential geometry to quantum statistical systems

TL;DR

This paper explores the application of random differential geometry to quantum statistical systems, analyzing eigenvalue problems of random operators within the framework of random matrix ensembles and their geometric structures.

Contribution

It introduces a novel approach linking random differential geometry with quantum statistical systems using random matrix ensembles and Liouville space structures.

Findings

Analysis of eigenvalue distributions for random quantum Hamiltonians.

Application of Gaussian and Ginibre ensembles to quantum operators.

Insights into the geometric structure of Liouville space in quantum systems.

Abstract

We apply concepts of random differential geometry connected to the random matrix ensembles of the random linear operators acting on finite dimensional Hilbert spaces. The values taken by random linear operators belong to the Liouville space. This Liouville space is endowed with topological and geometrical random structure. The considered random eigenproblems for the operators are applied to the quantum statistical systems. In the case of random quantum Hamiltonians we study both hermitean (self-adjoint) and non-hermitean (non-self-adjoint) operators leading to Gaussian and Ginibre ensembles Refs. [1], [2], [3]. [1] M. M. Duras, ``Finite-difference distributions for the Ginibre ensemble,'' {\em J. Opt. B: Quantum Semiclass. Opt.} {\bf 2}, 287-291, 2000. [2] M. M. Duras, K. Sokalski, ``Finite Element Distributions in Statistical Theory of Energy Levels in Quantum Systems,'' {\em Physica} {\bf D125}, 260-274, 1999. [3] M. M. Duras, K. Sokalski, ``Higher-order finite-element distributions in statistical theory of nuclear spectra,'' {\em Phys. Rev. E} {\bf 54}, 3142-3148, 1996.