Pseudo-Hermitian Random Matrices

Mauricio Porto Pato is a Senior Professor at the University of São 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.

Chapter 1 Introduction.- Chapter 2 The pseudo-Hermitian condition.- Chapter 3 Pseudo-Hermitian b-Hermite ensemble with real eigenvalues1.- Chapter 4 Pseudo-Hermitian -Hermite ensemble with an unbound metric2.- Chapter 5 Pseudo-Hermitian -Hermite ensemble with an unbound metric.- Chapter 6 Pseudo-Hermitian b-Laguerre ensemble with real eigenvalues3.- Chapter 7 Pseudo-Hermitian b-Laguerre ensemble with unbound metric.- Chapter 8 Pseudo-Hermitian -Laguerre ensemble with non-positive metric.- Chapter 9 The pseudo-Hermitian -Jacobi ensemble4.- Chapter 10 Pseudo-Hermitian Gaussian matrices5.- Chapter 11 Pseudo-Hermitian anti-Hermitian Gaussian matrices6.- Chapter 12 Average characteristic polynomials7.- Chapter 13 Spectral properties of pseudo-Hermitian matrices8.- Chapter 14 Eigenvalues as quasi-particles.- Chapter 15 Entanglement of pseudo-Hermitian random states9.