Computing Spectra -- On the Solvability Complexity Index Hierarchy and Towers of Algorithms

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 $\mathrm{sp}(A)$ of classes of bounded operators $A = \{a_{ij}\}_{i,j \in \mathbb{N}} \in \mathcal{B}(l^2(\mathbb{N}))$, given the matrix elements $\{a_{ij}\}_{i,j \in \mathbb{N}}$, that are sharp in the sense that they achieve the boundary of what a digital computer can achieve. Similarly, for a Schr\"odinger operator $H = -\Delta+V$, determine the existence of algorithms that can compute the spectrum $\mathrm{sp}(H)$ given point samples of the potential function $V$. 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 reach. The SCI is the smallest number of limits needed in the computation, yielding a classification hierarchy for all types of problems in computational mathematics that determines the boundaries of what computers can achieve in scientific computing. In addition, the SCI hierarchy provides classifications of computational problems that can be used in computer-assisted proofs. The SCI hierarchy captures many key computational issues in the history of mathematics including the insolvability of the quintic, Smale's problem on the existence of iterative generally convergent algorithm for polynomial root finding, the computational spectral problem, inverse problems, optimisation etc.