Full Description
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.
Contents
Chapter 1 Introduction.- Chapter 2 The pseudo-Hermitian condition.- Chapter 3 Pseudo-Hermitian b-Hermite ensemble with real eigenvalues1.- Chapter 4 Pseudo-Hermitian 휷-Hermite ensemble with an unbound metric2.- Chapter 5 Pseudo-Hermitian 휷-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 휷-Laguerre ensemble with non-positive metric.- Chapter 9 The pseudo-Hermitian 휷-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.
