  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.
