To Quantify the Difference of η-Inner Products in $\mathcal {P}\mathcal {T}$-Symmetric Theory

Abstract

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