# Spectral properties of nonlinear Schr\"odinger equation on a ring

## TL;DR

This paper investigates the spectral properties of the nonlinear Schrödinger equation on a ring with a defect, revealing unique phenomena like energy level crossings, shifts, and foam-like structures due to nonlinearity.

## Contribution

It introduces unconventional connection conditions and explores their effects on spectral properties, highlighting differences from the linear case and discovering new phenomena.

## Key findings

- Energy level crossings and shifts observed
- Existence of foam-like spectral structures confirmed
- Degeneracy occurs on a line in nonlinearity parameter space

## Abstract

The stationary states of nonlinear Schr{\"o}dinger equation on a ring with a defect is numerically analyzed. Unconventional connection conditions are imposed on the point defect, and it is shown that the system displays energy level crossings and level shifts and associated quantum holonomies in the space of system parameters, just as in the corresponding linear system. In the space of nonlinearity parameter, on the other hand, the degeneracy occurs on a line, excluding the possibility of any anholonomies. In contrast to the linear case, existence of exotic phenomena such as disappearance of energy level and foam-like structure are confirmed.