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Sektion Plenum
Dienstag, 19.09.2000, 11.30 Uhr, Großer Mathematik-Hörsaal, Trefftz-Bau

Adiabatic Perturbations of Linear Periodic Problems

Vladimir S. Buslaev, St. Petersburg State University

If the coefficients of a linear differential equation on Rd are slow functions of the variables,

L(ex, i @x)y  = 0,
the equation can possess wide classes of formal semiclassical solutions that vary faster compared with the coefficients. It is well known that these solutions admit very natural geometric interpretations and their theory is a source of significant analytical, geometrical and even physical ideas and constructions.

We shall show that all general classes of such solutions and the general constructions related to them can be naturally generalized to the wider class of the equations:

        1 @
L(x, ex,-----)y  = 0,
        i @x
the symbols L(x, y, p) of which are periodic functions of x. The combination of such two dependencies (periodic and slow) generates many new geometrical questions and changes essentially the spectral properties of the corresponding operators.

These generalized semiclassical equations and operators have also many important physical and technical applications. They were objects of intensive study during the last decades.