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Matt Colbrook: Spectral analysis and new resolvent based methods

Submitted by Jeremy Upsal on April 29, 2019 - 6:07pm

Speaker:  Matt Colbrook, University of Cambridge

Date: May 2, 2019

Title: Spectral analysis and new resolvent based methods

Abstract:

I will present new results on spectral computations for linear
operators on separable Hilbert spaces. This problem has a rich history,
leading to the Solvability Complexity Index (SCI) hierarchy - a measure
of difficulty of a computational problem. I will discuss classifications
of spectral problems in this hierarchy, which also link with potential
applications in computer assisted proofs (this may be useful in the
study of stability after linearisation for example). Some longstanding
problems are solved revealing potential surprises. For example, the
problem of computing spectra of compact operators, for which the method
has been known for decades, is strictly harder than the problem of
computing spectra of Schrodinger operators with bounded potentials,
which has been open for more than half a century. This latter problem
can be solved with a precise form of error control and without spectral
pollution using computational estimates of resolvent norms. These
techniques can also be carried over to the non-self-adjoint setting.
Another resolvent based example I will discuss is the first set of
general algorithms for computing spectral measures of self-adjoint and
unitary operators. Numerical examples are given throughout, showing the
new algorithms not only converge, but are also competitive with
state-of-the-art methods (which do not converge in general). Other
problems such as point spectra, and fractal dimension of spectra will
also make an appearance. These problems are samples of what is likely to
be a very rich classification theory with applications in spectral and
PDE theory.

[1] M.J. Colbrook, B. Roman and A.C. Hansen, "How to compute spectra
with error control", submitted 2019.

[2] M.J. Colbrook and A.C. Hansen, "On the infinite-dimensional QR
algorithm", Numerische Mathematik, 2019.

[3] M.J. Colbrook, "Computing spectral measures and spectral types: new
algorithms and classifications", submitted 2019.

[4] A.C. Hansen, "On the solvability complexity index, the
n-pseudospectrum and approximations of spectra of operators", J. Amer.
Math. Soc., 2011.

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