Floquet multiplier

Second-order periodic equations

The values of ${\displaystyle E}$ for which the Floquet multiplier satisfies ${\displaystyle \xi =\pm 1}$ are called the periodic and antiperiodic eigenvalues. They form the endpoints of the spectral bands of the periodic Schrödinger operator. For real values of ${\displaystyle E}$, the behavior of solutions is determined by the magnitude of the Floquet multipliers. A nontrivial solution remains bounded on the real line if and only if the corresponding multipliers satisfy ${\displaystyle |\xi |=1}$. The set of such energies consists of a union of intervals $${\displaystyle [E_{1},E_{2}],\;[E_{3},E_{4}],\;\ldots ,}$$ called the spectral bands, separated by intervals (band gaps) where solutions are not bounded. As ${\displaystyle E}$ varies across a spectral band, the Floquet multipliers move along the unit circle in the complex plane, and the endpoints of each band occur precisely when ${\displaystyle \xi =\pm 1}$.