Remarks on PT-pseudo-norm in PT-symmetric quantum mechanics

TL;DR

This paper explores the mathematical relationship between the PT-pseudo-norm in PT-symmetric quantum mechanics and the Stokes multiplier, revealing how sign changes in the pseudo-norm are governed by the derivative of the Stokes multiplier.

Contribution

It establishes an analytical link between the PT-pseudo-norm and the Stokes multiplier, providing insight into the sign alternation phenomenon in PT-symmetric systems.

Findings

Sign alternation of PT-pseudo-norm is controlled by the derivative of the Stokes multiplier.

The paper reveals a fundamental analytical relationship between PT-pseudo-norm and Stokes multiplier.

Provides a new perspective on the mathematical structure of PT-symmetric quantum mechanics.

Abstract

This paper presents an underlying analytical relationship between the $\mathcal{PT}$-pseudo-norm associated with the $\mathcal{PT}$-symmetric Hamiltonian $H = p^2+ V(q)$ and the Stokes multiplier of the differential equation corresponding to this Hamiltonian. We show that the sign alternation of $\mathcal{PT}$-pseudo-norm, which has been observed as a generic feature of the $\mathcal{PT}$-inner product, is essentially controlled by the derivative of Stokes multiplier with respect to eigenparameter.