# Floquet's theorem

The **Floquet theory** (after Gaston Floquet ) makes a statement about the structure of the Fundamentalmatrizen a homogeneous linear ordinary differential equation system with periodic coefficient matrix .

This theorem is used in quantum mechanics : the defined eigenstates of an undisturbed system are periodically changed in their energy by the application of a *temporally * periodic field or potential ; they then correspond exactly to the periodic part of the fundamental solution and are called *Floquet states* . For example, a Fourier expansion of these states can simplify work with them considerably.

Applied to *spatially* periodic potentials, Floquet's theorem in quantum theory is better known as *Bloch's theorem* . The eigen-states are called Bloch functions here .

## formulation

Every fundamental matrix of the homogeneous system of linear differential equations

with a continuous -periodic coefficient matrix can be written in the form

wherein

- continuously differentiable and periodic
- is a constant matrix.
- the matrix exponential function .

If one is satisfied with the fact that only is periodic, real-valued can be chosen.

The transformation

converts the differential equation system into one with constant coefficients:

## literature

- Carmen Chicone:
*Ordinary Differential Equations with Applications*. 2nd Edition. Texts in Applied Mathematics 34. Springer-Verlag, 2006, ISBN 0-387-30769-9 . - Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (= Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).