To quantify the difference of $\eta$-inner products in $\mathcal{P}\mathcal{T}$-symmetric theory

Abstract

In this paper, we consider a typical continuous two dimensional $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian and propose two different approaches to quantitatively show the difference between the $\eta$-inner products. Despite the continuity of Hamiltonian, the $\eta$-inner product is not continuous in some sense. It is shown that the difference between the $\eta$-inner products of broken and unbroken $\mathcal{P}\mathcal{T}$-symmetry is lower bounded. Moreover, such a property can lead to an uncertainty relation.

Minor corrections

References in corpus (3)

• Making Sense of Non-Hermitian Hamiltonians
• Observation of parity-time symmetry breaking in a single spin system
• The Naimark dilated PT-symmetric brachistochrone