## UC Berkeley / Lawrence Berkeley Laboratory

#### The Foundations of Infinite-Dimensional Spectral Computations

**Matthew Colbrook, Cambridge University**

Spectral computations in infinite dimensions are ubiquitous in the
sciences and computing spectra is one of the most studied areas of
computational mathematics over the last half-century. However, the
infinite-dimensional problem is notoriously difficult, since standard
approaches do not, in general, produce correct solutions. We introduce
new classes of algorithms that solve longstanding computational spectral
problems for the first time, such as: computing spectra with error
control; spectral measures, decompositions and the functional calculus;
fractal dimensions, capacity and transport properties of spectra;
discrete and essential spectra; and the spectral gap problem.
Furthermore, these algorithms are optimal, encompassing both Turing's
classical work and Smale's program on the foundations of computational
mathematics, with ramifications beyond spectral theory (*e.g.* Fefferman's
work on the Dirac-Schwinger conjecture).

We show how to rigorously compute spectra of infinite-dimensional
operators with guaranteed error control (*e.g.* for general classes of
partial differential operators on L^2(R^d) via point sampling
coefficients). One surprising consequence is that the problem of
computing spectra of compact operators, whose resolution has been known
for decades, is strictly harder than the problem of computing spectra of
SchrÃ¶dinger operators on L^2(R^d), which was open for more than half a
century.

We show how to diagonalise general self-adjoint operators - such as
differential, integral and lattice operators - via computing their
spectral measures (previous methods are case specific, relying on
analytical formulae). The algorithm can achieve arbitrarily high-orders
of convergence and explicit pointwise and L^p -error bounds are derived
in terms of the local regularity of the measure. Moreover, this allows
the computation of the functional calculus and spectral type
decompositions.

Finally, these algorithms are embarrassingly parallelisable. We give
numerical examples, demonstrating efficiency and tackling difficult
problems in the sciences. The discussed computational problems are
samples of what is likely to be a very rich classification theory, with
applications beyond spectral theory.