# Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency

## TL;DR

This paper introduces a new Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem, demonstrating exponential convergence to ground states and an efficient numerical method that outperforms existing approaches.

## Contribution

It proposes a novel normalized Sobolev gradient flow with a time-dependent inner product, providing global convergence and an efficient discretization for the nonlinear eigenvalue problem.

## Key findings

- Flow converges exponentially fast to eigenfunctions.
- Numerical method reduces energy at each step and converges globally.
- Method is competitive with established discretizations.

## Abstract

We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as exponentially fast where the rate depends on spectral gaps of a linearized operator. The forward Euler time discretization of the flow yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally in $H^1$ to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem.