Paths of unitary access to exceptional points

Abstract

With an innovative ideas of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson's papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato's exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states L into a triplet (viz., in our notation, spaces K and H besides the conventional L). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

Identification of the metric for diagonalizable (anti-)pseudo-Hermitian Hamilton operators represented by two-dimensional matrices

“…Most recently [62] (see also [63]) there has been addressed the following Hamilton operator by M. Znojil in the context of the Wheeler-deWitt equation:…”

“…suggesting the following two identities for the particular choice of the normalizations N 1 and N 2 in [62]:…”