Computing Spectra -- On the Solvability Complexity Index Hierarchy and Towers of Algorithms
Jonathan Ben-Artzi, Matthew J. Colbrook, Anders C. Hansen, Olavi, Nevanlinna, Markus Seidel

TL;DR
This paper introduces the Solvability Complexity Index hierarchy to classify the computational difficulty of spectral problems, establishing fundamental limits and new algorithms for computing spectra of operators and Schrödinger operators.
Contribution
It develops the SCI hierarchy to determine the minimal number of limits needed for algorithms, providing a framework for understanding computational boundaries in spectral theory and related fields.
Findings
Established the SCI hierarchy as a classification tool.
Proposed new algorithms for spectral computation.
Solved long-standing spectral computability problems.
Abstract
This paper establishes some of the fundamental barriers in the theory of computations and finally settles the long-standing computational spectral problem. That is to determine the existence of algorithms that can compute spectra of classes of bounded operators , given the matrix elements , that are sharp in the sense that they achieve the boundary of what a digital computer can achieve. Similarly, for a Schr\"odinger operator , determine the existence of algorithms that can compute the spectrum given point samples of the potential function . In order to solve these problems, we establish the Solvability Complexity Index (SCI) hierarchy and provide a collection of new algorithms that allow for problems that were previously out of…
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Taxonomy
TopicsDigital Image Processing Techniques · Topological and Geometric Data Analysis · Mathematical Analysis and Transform Methods
