Quantum toboggans: models exhibiting a multisheeted PT symmetry

TL;DR

This paper introduces a generalized framework for PT-symmetric quantum models using complex potentials and multi-sheeted Riemann surfaces, expanding the scope of non-Hermitian quantum mechanics.

Contribution

It extends PT-symmetric Hamiltonians to include analytic potentials with singularities and multi-sheeted paths, defining nontrivial toboggans with topologically complex coordinate trajectories.

Findings

Defined the concept of toboggans in PT-symmetric models.

Extended the class of PT-symmetric Hamiltonians to include multi-sheeted Riemann surfaces.

Provided a mathematical framework for analyzing complex quantum trajectories.

Abstract

A generalization of the concept of PT-symmetric Hamiltonians H=p^2+V(x) is described. It uses analytic potentials V(x) (with singularities) and a generalized concept of PT-symmetric asymptotic boundary conditions. Nontrivial toboggans are defined as integrated along topologically nontrivial paths of coordinates running over several Riemann sheets of wave functions.