To quantify the difference of $\eta$-inner products in $\cal PT$-symmetric theory

TL;DR

This paper investigates the differences in $ ext{eta}$-inner products within a 2D $ ext{PT}$-symmetric quantum system, revealing lower bounds and implications for uncertainty relations, highlighting discontinuities despite Hamiltonian continuity.

Contribution

It introduces two approaches to quantify $ ext{eta}$-inner product differences in $ ext{PT}$-symmetric systems, emphasizing their non-continuity and lower bounds.

Findings

The $ ext{eta}$-inner product difference is lower bounded.

Discontinuity of $ ext{eta}$-inner products despite Hamiltonian continuity.

A potential link to an uncertainty relation is established.

Abstract

In this paper, we consider a typical continuous two dimensional $\cal PT$-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 $\cal PT$-symmetry is lower bounded. Moreover, such a property can lead to an uncertainty relation.