Lie systems and Schrödinger equations

Abstract

We prove that $t$-dependent Schrödinger 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 Kähler vector fields. This result is extended to other related Schrödinger equations, e.g. projective ones, and their properties are studied through Poisson, presymplectic and Kähler 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.

arXiv comment36 pages

References in corpus (8)

• Geometric Quantum Mechanics
• Superposition rules, Lie theorem and partial differential equations
• Geometrization of Quantum Mechanics
• Recent Applications of the Theory of Lie Systems in Ermakov Systems
• From constants of motion to superposition rules for Lie-Hamilton systems
• Dirac--Lie systems and Schwarzian equations
• Superposition rules for higher-order systems and their applications
• k-symplectic Lie systems: theory and applications

Cited by in corpus (1)

• Lie-Hamilton systems on curved spaces: A geometrical approach