# Approximation theorems for the Schr\"odinger equation and quantum vortex reconnection

## TL;DR

This paper proves the existence of smooth solutions to the Gross-Pitaevskii equation with complex vortex reconnections, enabling detailed tracking of vortex evolution and verifying properties observed in experiments and simulations.

## Contribution

It introduces novel global approximation theorems for the Schrödinger equation, applicable to complex vortex dynamics in quantum fluids.

## Key findings

- Verified vortex reconnection scenarios exhibit $t^{1/2}$ and parity change laws.
- Developed new approximation theorems for Schrödinger solutions on general spacetime sets.
- Established frequency-dependent estimates for the Helmholtz-Yukawa equation.

## Abstract

We prove the existence of smooth solutions to the Gross-Pitaevskii equation on $\mathbf{R}^3$ that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits to describe the reconnection events in detail and verify that this scenario exhibits the properties observed in experiments and numerics, such as the $t^{1/2}$ and change of parity laws. We are mostly interested in solutions tending to1 at infinity, which have finite Ginzburg-Landau energy and physically correspond to the presence of a background chemical potential, but we also consider the cases of Schwartz initial data and of the Gross-Pitaevskii equation on the torus. An essential ingredient in the proofs is the development of novel global approximation theorems for the Schr\"odinger equation on $\mathbf{R}^n$. Specifically, we prove a qualitative approximation result that applies for solutions defined on very general spacetime sets and also a quantitative result for solutions on product sets in spacetime $D \times \mathbf{R}$. This hinges on frequency-dependent estimates for the Helmholtz-Yukawa equation that are of independent interest.