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 ${\ displaystyle \ Phi}$

{\ displaystyle \ y '(x) = A (x) y (x)}

with a continuous -periodic coefficient matrix can be written in the form ${\ displaystyle \ omega}$ ${\ displaystyle A: \ mathbb {R} \ rightarrow \ mathbb {R} ^ {m \ times m}}$

{\ displaystyle \ \ Phi (x) = P (x) \, \ exp (xR)}

wherein

${\ displaystyle P: \ mathbb {R} \ rightarrow GL (m; \ mathbb {C})}$ continuously differentiable and periodic ${\ displaystyle \ omega}$
${\ displaystyle R \ in \ mathbb {C} ^ {m \ times m}}$ is a constant matrix.
${\ displaystyle \ exp}$ the matrix exponential function .

If one is satisfied with the fact that only is periodic, real-valued can be chosen. ${\ displaystyle P}$ ${\ displaystyle 2 \ omega}$ ${\ displaystyle P, R}$

The transformation

{\ displaystyle \ z (x) = P ^ {- 1} (x) \, y (x)}

converts the differential equation system into one with constant coefficients:

{\ displaystyle \ z '(x) = R \, z (x)}

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 ).