Analytic eigenvalue structure of a coupled oscillator system beyond the ground state

TL;DR

This paper analytically explores the eigenvalue structure of coupled harmonic oscillators beyond the ground state, revealing an eight-fold spectral connection structure that extends previous four-fold findings.

Contribution

It extends the spectral connection analysis from ground states to higher excitations, uncovering an eight-fold structure in the eigenvalue continuation of coupled oscillators.

Findings

Identified an eight-fold spectral connection structure for higher excited states.

Extended the known four-fold Riemann surface structure to higher energy levels.

Showed the structure's validity for equal oscillator frequencies.

Abstract

By analytically continuing the eigenvalue problem of a system of two coupled harmonic oscillators in the complex coupling constant $g$, we have found a continuation structure through which the conventional ground state of the decoupled system is connected to three other lower {\it unconventional} ground states that describe the different combinations of the two constituent oscillators, taking all possible spectral phases of these oscillators into account. In this work we calculate the connecting structures for the higher excitation states of the system and argue that - in contrast to the four-fold Riemann surface identified for the ground state - the general structure is eight-fold instead. Furthermore we show that this structure in principle remains valid for equal oscillator frequencies as well and comment on the similarity of the connection structure to that of the single complex harmonic oscillator.