Ad hoc physical Hilbert spaces in Quantum Mechanics

TL;DR

This paper explores PT-symmetric quantum mechanics, demonstrating how to construct physical Hilbert spaces where non-Hermitian Hamiltonians yield real energies and unitary evolution, with explicit models and solutions.

Contribution

It provides a detailed formalism for PT-symmetric models, including explicit solutions for bound states and a method to ensure unitarity in a modified Hilbert space.

Findings

Real low-lying energies can be obtained in closed form.

Non-Hermitian Hamiltonians can produce unitary evolution in a modified Hilbert space.

Models effectively regularize singular potentials.

Abstract

The overall principles of what is now widely known as PT-symmetric quantum mechanics are listed, explained and illustrated via a few examples. In particular, models based on an elementary local interaction V(x) are discussed as motivated by the naturally emergent possibility of an efficient regularization of an otherwise unacceptable presence of a strongly singular repulsive core in the origin. The emphasis is put on the constructive aspects of the models. Besides the overall outline of the formalism we show how the low-lying energies of bound states may be found in closed form in certain dynamical regimes. Finally, once these energies are found real we explain that in spite of a manifest non-Hermiticity of the Hamiltonian the time-evolution of the system becomes unitary in a properly amended physical Hilbert space.