The Maslov index and the spectrum of differential operators

Speaker: Yuri Latushkin (Missouri)

Abstract:

We will review some recent results on connections between the Maslov and the Morse indices

for differential operators. The Morse index is a spectral quantity defined as the number of

negative eigenvalues counting multiplicities while the Maslov index is a geometric

characteristic defined as the signed number of intersections of a path in the space of

Lagrangian planes with the train of a given plane. The problem of relating these two quantities

is rooted in Sturm's Theory and has a long history going back to the classical work by Arnold,

Bott and Smale, and has attracted recent attention of several groups of mathematicians.

We will briefly mention how the relation between the two indices helps to prove the fact

that a pulse in a gradient system of reaction diffusion equations is unstable.

We will also discuss a fairly general theorem relating the indices for a broad class

of multidimensional elliptic selfadjoint operators. Connections of the Maslov index and

Hadamard-type formulas for the derivative of eigenvalues will be also discussed.

This talk is based on a joint work with M. Beck, G. Cox, C. Jones, P. Howard, R. Marangell,

K. McQuighan, A. Sukhtayev, and S. Sukhtaiev.