This book is a comprehensive guide to pseudo-Hermitian random matrices, their properties, and their role in many models that are relevant to physical processes. The book starts by showing how the concept of pseudo-Hermiticity emerged from studies of PT-symmetric systems which aroused the interest of the random matrix theory community. The chapters that follow discuss the consequences of the pseudo-Hermitian condition to the eigen-decomposition of non-Hermitian matrices, and an investigation of pseudo-Hermitian random matrices in tridiagonal form, discussing the scenario with real eigenvalues, and the appearance of complex eigenvalues generated by unbound and non-positive metrics. Subsequently, the author introduces pseudo-Hermitian Gaussian matrices and their properties including characteristic polynomials, and statistical properties of their eigenvalues. Finally, in the last chapter, the time invariance of the metric is upended and a pseudo-Hermitian model with a time dependent metricis constructed to discuss the time evolution of entangled states.

About the Author

Mauricio Porto Pato is a Senior Professor at the University of Sao Paulo with a large experience in the field of random matrices theory and applications. In the early 90's, in a collaboration with the nuclear physicist M. S. Hussein, he began a study of random matrices that resulted in the construction of an ensemble to be applied to a situation of partial conservation of a quantum number. The model was then, successfully, applied to the description of isospin data. In a collaboration with O. Bohigas, another important contribution of him to be highlighted, was the formalism to deal with missing levels in correlated spectra, a study that evolved from his work with the experimentalist G. E. Mitchell. About ten years ago, his interest moved from Hermitian to non-Hermitian operators and this led to his involvement with the studies of the class of pseudo-Hermitian matrices associated to PT-symmetric systems, that is, systems invariant under parity and time-reversal transformations. This investigation started with the introduction of the pseudo-Hermiticity condition in the sparse tridiagonal matrices of the so-called beta-ensembles of the random matrix theory. Next, the pseudo-Hermiticity condition was extended to the standard Gaussian matrices with the creation of the pseudo-Hermitian Gaussian ensembles. All this effort, along a decade, comprises about one dozen of works among articles and thesis.

Book Information
ISBN 9783031602931
Author Mauricio Porto Pato
Format Hardback
Page Count 178
Imprint Springer International Publishing AG
Publisher Springer International Publishing AG