# Coalescing versus merging of energy levels in one-dimensional potentials

## TL;DR

This paper explores how energy levels in a symmetric double well potential behave differently under Hermitian and PT-symmetric perturbations, revealing a transition from merging to coalescing at an exceptional point.

## Contribution

It demonstrates that mild PT-symmetric perturbations convert level merging into coalescence at an exceptional point, linking phenomena in Hermitian and non-Hermitian quantum systems.

## Key findings

- Levels merge into one in Hermitian case as distance increases.
- PT-symmetric perturbation causes levels to coalesce at an exceptional point.
- Beyond the exceptional point, eigenvalues become complex conjugates with real parts matching the original ground state energy.

## Abstract

The sub-barrier pairs of energy levels of a Hermitian one-dimensional symmetric double well potential are known to merge into one, if the inter-well distance ($a$) is increased slowly. The energy at which the doublets merge are the ground state eigenvalues of independent wells ($\epsilon_0$). We show that if the double well is perturbed mildly by a complex PT-symmetric potential the merging of levels turns into the coalescing of two levels at an exceptional point $a=a_*$. For $a>a_*$, the real part of complex-conjugate eigenvalues coincides with $\epsilon_0$ again. This is an interesting and rare connection between the two phenomena in two domains: Hermiticity and complex PT-symmetry.