# How to Compute Spectra with Error Control.

@article{Colbrook2019HowTC, title={How to Compute Spectra with Error Control.}, author={Matthew J. Colbrook and Bogdan Roman and Anders C. Hansen}, journal={Physical review letters}, year={2019}, volume={122 25}, pages={ 250201 } }

Computing the spectra of operators is a fundamental problem in the sciences, with wide-ranging applications in condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that in certain cases converge to the spectrum, no general procedure is known that (a) always converges, (b) provides bounds on the errors of approximation, and (c) provides approximate eigenvectors. This may lead to incorrect simulations. It has been an open problem since…

## 23 Citations

ON THE COMPUTATION OF GEOMETRIC FEATURES OF SPECTRA OF LINEAR OPERATORS ON HILBERT SPACES

- Mathematics, Computer Science
- 2021

The computation of geometric features of spectra in the SCI hierarchy is classified, allowing us to precisely determine the boundaries of what computers can achieve and prove that the authors' algorithms are optimal.

Computing Spectral Measures and Spectral Types: New Algorithms and Classifications

- MathematicsArXiv
- 2019

It is shown that if each matrix column decays at infinity at a known asymptotic rate, then it is possible to compute spectral measures of self-adjoint and unitary linear operators on separable Hilbert spaces and is classified in the SCI hierarchy for such operators.

Pseudoergodic operators and periodic boundary conditions

- MathematicsMath. Comput.
- 2020

A simple proof that the finite section method with periodic boundary conditions converges to the pseudospectrum of the full operator is given, and it is shown that the result carries over to pseudoergodic operators acting on lp spaces for p ∈ [1,∞].

Computing solutions of Schrödinger equations on unbounded domains- On the brink of numerical algorithms

- Computer Science, MathematicsArXiv
- 2020

The results provide classifications of which mathematical problems may be solved by computer assisted proofs and are a part of the Solvability Complexity Index (SCI) hierarchy and Smale's program on the foundations of computational mathematics.

Universal algorithms for computing spectra of periodic operators

- Mathematics, Computer ScienceNumerische Mathematik
- 2022

It is shown that for periodic banded matrices this can be done, as well as for Schrödinger operators with periodic potentials that are sufficiently smooth, and implementable algorithms are provided, along with examples.

Computing semigroups with error control

- MathematicsSIAM Journal on Numerical Analysis
- 2022

It is shown that it is possible, even when only allowing pointwise evaluation of coefficients, to compute semigroups on the unbounded domain L2(Rd) that are generated by partial differential operators with polynomially bounded coefficients of locally bounded total variation.

Computing spectral properties of topological insulators without artificial truncation or supercell approximation

- PhysicsArXiv
- 2021

These tools completely avoid such artificial restrictions and allow one to probe the spectral properties of the infinitedimensional operator directly, even in the presence of material defects and disorder.

On Numerical Approximations of the Koopman Operator

- MathematicsMathematics
- 2022

The error in the Krylov subspace version of the finite section method is studied and it is indicated that Krylov sequence-based approximations can have low error without an exponential-in-dimension increase in the number of functions needed for approximation.

Spectra of Jacobi Operators via Connection Coefficient Matrices

- Mathematics
- 2017

We address the computational spectral theory of Jacobi operators that are compact perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In…

Computing eigenvalues of the Laplacian on rough domains

- Mathematics, Computer ScienceArXiv
- 2021

A general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses is proved, which implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence.

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