Eva-Maria Graefe

PT-symmetric random matrix ensembles using split-quaternonic numbers

Random matrices play a crucial role in various fields of mathematics and physics. In particular in the field of quantum chaos Hermitian random matrix ensembles represent universality classes for spectral features of Hamiltonians with classically chaotic counterparts. In recent years the study of non-Hermitian but PT-symmetric quantum systems has attracted a lot of attention. These are non-Hermitian systems that have an anti-unitary symmetry, which is often interpreted as a balance of loss and gain in a system.

In this talk the question of whether and how the standard ensembles of Hermitian quantum mechanics can be modified to yield PT-symmetric counterparts is addressed. In particular it is argued that using split-complex and split-quaternionic numbers two new PT-symmetric random matrix ensembles can be constructed. These matrices have either real or complex conjugate eigenvalues, the statistical features of which are analysed for 2 × 2 matrices.