Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction : General results and a case study with Landau Hamiltonian

TL;DR

This paper demonstrates that adding Lorentz interaction to Hamiltonian systems with balanced loss and gain stabilizes the kinetic energy, enabling a consistent quantum theory and revealing modified classical and quantum behaviors exemplified by the Landau Hamiltonian.

Contribution

It introduces a method to stabilize Hamiltonian systems with balanced loss and gain using Lorentz interaction, allowing for a well-defined quantum theory and analyzing the Landau Hamiltonian case.

Findings

Kinetic energy becomes semi-positive-definite with Lorentz interaction.

Classical particles move on elliptical orbits with reduced cyclotron frequency.

Quantum properties resemble standard Landau Hamiltonian but with modified cyclotron frequency.

Abstract

The kinetic energy term of Hamiltonian systems with balanced loss and gain is not semi-positive-definite, leading to instabilities at the classical as well quantum level. It is shown that an additional Lorentz interaction in the Hamiltonian allows the kinetic energy term to be semi-positive-definite and thereby, improving the stability properties of the system. Further, a consistent quantum theory admitting bound states may be obtained on the real line instead of Stoke wedges on the complex plane. The Landau Hamiltonian in presence of balanced loss and gain is considered for elucidating the general result. The kinetic energy term is semi-positive-definite provided the magnitude of the applied external magnetic field is greater than the magnitude of the `analogous magnetic field' due to the loss gain terms. It is shown that the classical particle moves on an elliptical orbit with a cyclotron frequency that is less than its value in absence of the loss-gain terms. The quantum system share the properties of the standard Landau Hamiltonian, but, with the modified cyclotron frequency. It is shown that the Hall current has non-vanishing components along the direction of the external uniform electric field and to its transverse direction. The Pauli equation in presence of balanced loss and gain is shown to be supersymmetric.