Floquet's theorem

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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 .


Every fundamental matrix of the homogeneous system of linear differential equations

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


  • 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:


  • 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 ).