DIFFERENTIAL GEOMETRY/PDE SEMINAR

Monday, March 5 (Note: Special Day) Padelford C-36 3:45 PM

Numerical linear algebra and solvability of PDEs

Maciej Zworski (UC Berkeley)

One of the goals of Numerical Linear Algebra is the accurate computation of eigenvalues of matrices. This is quite successful in the symmetric case but the situation is different for non-self-adjoint matrices. These arise in the discretisation of engineering or physical problems involving friction or dissipation, and it is an important problem to compute their eigenvalues.

It has been observed long time ago that the obstruction to accurately computing eigenvalues of non-self-adjoint matrices is inherent in the problem, and cannot be circumvented by using more powerful computers. The basic idea is that any algorithm for locating the eigenvalues will also find some ‘false eigenvalues’.

These false eigenvalues also explain one of the most surprising phenomena in linear PDEs, namely the fact (discovered by Hans Lewy in 1957, in Berkeley) that one cannot always locally solve the PDE P u = f. Local solvability is always possible if P is self-adjoint or has constant coefficients, but non-self-adjointness can destroy that property: Lewy’s example was a simple vector-field with complex, non-constant coefficients arising in the study of several complex variables.

Almost immediately after that discovery, Hörmander provided an explanation of Lewy’s example showing that almost all non-self-adjoint operators are not locally solvable. That was done by considering the essentially dual problem of existence of non-propagating singularities. I will give an elementary quantum mechanical interpretation of these issues of local solvability and non-propagation of singularities in terms of creation and annihilation operators.

Finally, I will explain how the non-propagating singularities are the source of (at least some of) the computational problems in finding eigenvalues of non-self-adjoint matrices arising in discretisation of high energy or semi-classical operators.