Professor Man Kam Kwong

Lucent Technologies Inc., Naperville, USA

Floquet Theory and the Oscillation of a Forced Hill's Equation

Abstract

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A nontrivial solution of a second order linear equation of the form

y"(t) + q(t) y(t) = f (t) t ³ 0

is said to be oscillatory if it has arbitrarily large zeros. It is wellknown that in the case f (t) is identically zero, if one nontrivial solution is oscillatory, then all nontrivial solutions are oscillatory (Sturm's Theorem). Oscillation of homogeneous equations have been extensively studied.

However, relatively little is known about the forced equation (when f (t) is not identically zero).

In this talk, the following recent result is reported:

All solutions of the differential equation

y"(t) + c sin(t) y(t) = cos(t) t ³ 0

are oscillatory, where c is any non-zero constant.

The proof of this seemingly easy-to-state result is surprisingly complicated. It involves some new results in the Floquet theory of Hill's equations (equations whose coefficients are periodic functions), Lyapunov-type inequalities and a series solution algorithm.

This is joint work with James S.W. Wong.