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

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