Recent progress in the spectral theory of first order elliptic systems

Dmitri Vassiliev (University College London)

The talk deals with the distribution of eigenvalues of a linear self-adjoint elliptic operator. The eigenvalue problem is considered in the deterministic setting, i.e. the coefficients of the operator are prescribed smooth functions. The objective is to derive a two-term asymptotic formula for the counting function (number of eigenvalues between zero and a positive lambda) as lambda tends to plus infinity.

There is an extensive literature on the subject (see, for example, [1]), mostly dealing with scalar operators. It has always been taken for granted that all results extend in a straightforward manner to systems. However, the author has recently discovered [2,3] that all previous publications on first order systems give formulae for the second asymptotic coefficient that are either incorrect or incomplete (i.e. an algorithm for the calculation of the second asymptotic coefficient rather than an explicit formula). The aim of the talk is to explain the spectral theoretic difference between scalar operators and systems and to present the correct formula for the second asymptotic coefficient.

[1] Yu.Safarov and D.Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, American Mathematical Society, 1997 (hardcover), 1998 (softcover).

[2] Preprint arXiv:1204.6567.

[3] Preprint arXiv:1208.6015.