The problem of computing spectra of operators is arguably one of the most
investigated areas of computational mathematics. Recent progress and the
current paper reveal that, unlike the finite-dimensional case,
infinite-dimensional problems yield a highly intricate infinite classification
theory determining which spectral problems can be solved and with which type of
algorithms. Classifying spectral problems and providing optimal algorithms is
uncharted territory in the foundations of computational mathematics. This paper
is the first of a two-part series establishing the foundations of computational
spectral theory through the Solvability Complexity Index (SCI) hierarchy and
has three purposes. First, we establish answers to many longstanding open
questions on the existence of algorithms. We show that for large classes of
partial differential operators on unbounded domains, spectra can be computed
with error control from point sampling operator coefficients. Further results
include computing spectra of operators on graphs with error control, the
spectral gap problem, spectral classifications, and discrete spectra,
multiplicities and eigenspaces. Second, these classifications determine which
types of problems can be used in computer-assisted proofs. The theory for this
is virtually non-existent, and we provide some of the first results in this
infinite classification theory. Third, our proofs are constructive, yielding a
library of new algorithms and techniques that handle problems that before were
out of reach. We show several examples on contemporary problems in the physical
sciences. Our approach is closely related to Smale's program on the foundations
of computational mathematics initiated in the 1980s, as many spectral problems
can only be computed via several limits, a phenomenon shared with the
foundations of polynomial root finding with rational maps, as proved by
McMullen