Linear differential equations with periodic coefficients constitute
a well developed part of the theory of ordinary differential
equations [17, 94, 156, 177, 178, 272, 389]. They arise in many
physical and technical applications [177, 178, 272]. A new wave of
interest in this subject has been stimulated during the last two
decades by the development of the inverse scattering method for
integration of nonlinear differential equations. This has led to
significant progress in this traditional area [27, 71, 72, 111 119,
250, 276, 277, 284, 286, 287, 312, 313, 337, 349, 354, 392, 393,
403, 404]. At the same time, many theoretical and applied problems
lead to periodic partial differential equations. We can mention,
for instance, quantum mechanics [14, 18, 40, 54, 60, 91, 92, 107,
123, 157-160, 192, 193, 204, 315, 367, 412, 414, 415, 417],
hydrodynamics [179, 180], elasticity theory [395], the theory of
guided waves [87-89, 208, 300], homogenization theory [29, 41,
348], direct and inverse scattering [175, 206, 216, 314, 388,
406-408], parametric resonance theory [122, 178], and spectral
theory and spectral geometry [103 105, 381, 382, 389]. There is a
sjgnificant distinction between the cases of ordinary and partial
differential periodic equations. The main tool of the theory of
periodic ordinary differential equations is the so-called Floquet
theory [17, 94, 120, 156, 177, 267, 272, 389]. Its central result
is the following theorem (sometimes called Floquet-Lyapunov
theorem) [120, 267].
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