# The $J$-method for the Gross-Pitaevskii eigenvalue problem

## TL;DR

This paper analyzes the $J$-method for solving the Gross-Pitaevskii eigenvalue problem, demonstrating its global and local convergence properties, and showcasing its effectiveness in approximating excited states and localized phenomena.

## Contribution

It provides a rigorous convergence analysis of the $J$-method in a Hilbert space framework, including a damping modification and applications to complex physical states.

## Key findings

- Proves global convergence for a damped $J$-method.
- Establishes local linear convergence near eigenfunctions.
- Demonstrates effectiveness in numerical experiments with localized states.

## Abstract

This paper studies the $J$-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross-Pitaevskii eigenvalue problem, we prove global convergence towards an eigenfunction for a damped version of the $J$-method. More importantly, when the iterations are sufficiently close to an eigenfunction, the damping can be switched off and we recover a local linear convergence rate previously known from the discrete setting. This quantitative convergence analysis is closely connected to the~$J$-method's unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows, this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps.