The Foundations of Infinite-Dimensional Spectral Computations

Matthew Colbrook, Cambridge University
4/15, 2020 at 10:10AM-11AM in

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 Schrödinger 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.