# Analytic structure of eigenvalues of coupled quantum systems

## TL;DR

This paper explores the analytic continuation of coupling constants in quantum systems to understand eigenvalue structures, revealing unconventional states with energies below the ground state, and discusses the physical implications of such continuations.

## Contribution

It introduces a method to analyze eigenvalues of coupled quantum systems through complex coupling continuation, highlighting new states and their energetic properties.

## Key findings

- Analytic continuation can lead to states with energies below the original ground state.
- Exceptional points in the complex plane influence eigenvalue behavior.
- The process requires work, questioning the physical realization of these states.

## Abstract

By analytically continuing the coupling constant $g$ of a coupled quantum theory, one can, at least in principle, arrive at a state whose energy is lower than the ground state of the theory. The idea is to begin with the uncoupled $g=0$ theory in its ground state, to analytically continue around an exceptional point (square-root singularity) in the complex-coupling-constant plane, and finally to return to the point $g=0$. In the course of this analytic continuation, the uncoupled theory ends up in an unconventional state whose energy is lower than the original ground state energy. However, it is unclear whether one can use this analytic continuation to extract energy from the conventional vacuum state; this process appears to be exothermic but one must do work to vary the coupling constant $g$.