Lie systems and Schr\"odinger equations

TL;DR

This paper demonstrates that certain Schr"odinger equations can be described using Lie systems with K"ahler vector fields, enabling nonlinear superposition rules and applications to quantum systems like n-qubits.

Contribution

It extends Lie system methods to Schr"odinger equations, revealing new geometric structures and nonlinear superposition rules, especially in quantum information contexts.

Findings

Schr"odinger equations can be modeled as Lie systems with K"ahler vector fields.

Derived nonlinear superposition rules depending on fewer solutions.

Applications demonstrated in n-qubit quantum systems.

Abstract

We prove that $t$-dependent Schr\"odinger equations on finite-dimensional Hilbert spaces determined by $t$-dependent Hermitian Hamiltonian operators can be described through Lie systems admitting a Vessiot--Guldberg Lie algebra of K\"ahler vector fields. This result is extended to other related Schr\"odinger equations, e.g. projective ones, and their properties are studied through Poisson, presymplectic and K\"ahler structures. This leads to derive nonlinear superposition rules for them depending in a lower (or equal) number of solutions than standard linear ones. Special attention is paid to applications in $n$-qubit systems.