Dual oscillators and Quantum Pendulums: spectrum and correlators

TL;DR

This paper explores a duality transformation in oscillator systems that links Hermitian and non-Hermitian Hamiltonians, enabling exact spectral calculations and dual representations of correlators in quantum rotators.

Contribution

It introduces a nonlinear path integral transformation revealing a duality between oscillator spectra and non-Hermitian Hamiltonians, providing exact solutions and correlator representations.

Findings

Exact spectra of quantum rotators calculated.

Duality transformation links Hermitian and non-Hermitian systems.

Dual representations of n-point correlators derived.

Abstract

We outline the nonlinear transformation in the path integral representation for partition function of O(N) symmetric oscillator systems bringing their duality to certain one-dimensional oscillators with unstable potential shapes. This duality transformation realizes the equivalence between spectra of a Hermitian and a non-Hermitian Hamiltonians. It is exploited to calculate exactly the spectra of quantum rotators . The zero-temperature limit is considered and the dual representation of n-point correlators of coordinate operators is obtained.