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Floquet's Theorem


Let Q(x) be a real or complex piecewise-continuous function defined for all values of the real variable x and that is periodic with minimum period pi so that

 Q(x+pi)=Q(x).
(1)

Then the differential equation

 y^('')+Q(x)y=0
(2)

has two continuously differentiable solutions y_1(x) and y_2(x), and the characteristic equation is

 rho^2-[y_1(pi)+y_2^'(pi)]rho+1=0,
(3)

with eigenvalues rho_1=e^(ialphapi) and rho_2=e^(-ialphapi). Then Floquet's theorem states that if the roots rho_1 and rho_2 are different from each other, then (2) has two linearly independent solutions

f_1(x)=e^(ialphax)p_1(x)
(4)
f_2(x)=e^(-ialphax)p_2(x),
(5)

where p_1(x) and p_2(x) are periodic with period pi (Magnus and Winkler 1979, p. 4).


See also

Floquet Analysis, Hill's Differential Equation

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References

Magnus, W. and Winkler, S. "Floquet's Theorem." §1.2 in Hill's Equation. New York: Dover, pp. 3-8, 1979.

Referenced on Wolfram|Alpha

Floquet's Theorem

Cite this as:

Weisstein, Eric W. "Floquet's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FloquetsTheorem.html

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