T he goal of this talk is to introduce classes of re solvent based algorithms that compute spectral pro perties of operators on separable Hilbert spaces. As well as solving computational problems for the first time\, these algorithms are proven to be opt imal\, and the computational problems themselves c an be classed in a hierarchy (the SCI hierarchy) w ith ramifications beyond spectral theory.

For concreteness\, I shall focus on two problems f or a very general class of operators on $l^2(\\mat hbb{N})$\, where algorithms access the matrix elem ents of the operator:

1) Computing spectra of closed operators in the Attouch-Wets topology (local uniform convergence of closed sets). This a lgorithm uses estimates of the norm of the resolve nt operator and a local minimisation scheme. As we ll as solving the long-standing computational spec tral problem\, this algorithm computes spectra wit h error control. It can also be extended to partia l differential operators with coefficients of loca lly bounded total variation with algorithms point sampling the coefficients.

2) Computing (p rojection-valued) spectral measures of self-adjoin t operators as given by the spectral theorem. This algorithm uses computation of the full resolvent operator (with asymptotic error control) to comput e convolutions of rational kernels with the measur e before taking a limit. I shall discuss local con vergence properties and extensions to computing sp ectral decompositions (pure point\, absolutely con tinuous and singular continuous parts).

Fi nally\, these algorithms are embarrassingly parall elisable. Numerical examples will be given\, demon strating efficiency\, and tackling difficult probl ems taken from mathematics and other fields such a s chemistry and physics. LOCATION:Seminar Room 1\, Newton Institute CONTACT:info@newton.ac.uk END:VEVENT END:VCALENDAR