Open AccessJournal Article
Computability of the Spectrum of Self-Adjoint Operators.
Vasco Brattka,Ruth Dillhage +1 more
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It is proved that given a “program” of the operator one can obtain positive information on the spectrum as a compact set in the sense that a dense subset of the spectrum can be enumerated and a bound on the set can be computed.Abstract:
Self-adjoint operators and their spectra play a crucial role in analysis and physics. For instance, in quantum physics self-adjoint operators are used to describe measurements and the spectrum represents the set of possible measurement results. Therefore, it is a natural question whether the spectrum of a self-adjoint operator can be computed from a description of the operator. We prove that given a “program” of the operator one can obtain positive information on the spectrum as a compact set in the sense that a dense subset of the spectrum can be enumerated (or equivalently: its distance function can be computed from above) and a bound on the set can be computed. This generalizes some non-uniform results obtained by Pour-El and Richards, which imply that the spectrum of any computable self-adjoint operator is a recursively enumerable compact set. Additionally, we show that the spectrum of compact selfadjoint operators can even be computed in the sense that also negative information is available (i.e. the distance function can be fully computed). Finally, we also discuss computability properties of the resolvent map.read more
Citations
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Towards computability of elliptic boundary value problems in variational formulation
Vasco Brattka,Atsushi Yoshikawa +1 more
TL;DR: It is demonstrated how the computable versions of the Frechet-Riesz Representation Theorem and the Lax-Milgram Theorem yield computable solutions of the Neumann and Dirichlet boundary value problems for a simple non-symmetric elliptic differential equation in the one-dimensional case.
Journal Article
Computational Complexity on Computable Metric Spaces
TL;DR: In this paper, a new Turing machine based concept of time complexity for functions on computable metric spaces was introduced, which generalizes the ordinary complexity of word functions and the complexity of real functions studied by Ko [19] et al.
Journal ArticleDOI
Computability of compact operators on computable Banach spaces with bases
Vasco Brattka,Ruth Dillhage +1 more
TL;DR: It is proved that the space of compact operators on Banach spaces with monotone, computably shrinking, and computable bases is a computable Banach space itself and operations such as composition with bounded linear operators from left are computable.
Journal ArticleDOI
A Splitting Iterative Method for Solving the Neutron Transport Equation
Onana Awono,Jacques Tagoudjeu +1 more
TL;DR: In this article, an iterative method based on self-adjoint and maccretive splitting for the numerical treatment of the steady state neutron transport equation is presented, which converges unconditionally to the unique solution of the transport equation.
Journal Article
Computability in Basic Quantum Mechanics
TL;DR: This work discusses computability of basic notions of quantum mechanics like states and observables in the sense of Type Two Effectivity (TTE) via function realizability using topological domain theory as developed by Schroeder and Simpson.
References
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Computable Analysis : An Introduction
TL;DR: This book provides a solid fundament for studying various aspects of computability and complexity in analysis and is written in a style suitable for graduate-level and senior students in computer science and mathematics.
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Computability in analysis and physics
TL;DR: This book represents the first treatment of computable analysis at the graduate level within the tradition of classical mathematical reasoning and is sufficiently detailed to provide an introduction to research in this area.
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Computable Analysis
TL;DR: Computable analysis as discussed by the authors studies functions on the real numbers and related sets which can be computed by machines such as digital computers, and connects two traditionally disjoint fields, namely Analysis/Numerical Analysis on the one hand and Computability/Computational Complexity on the other hand, combining concepts of approximation and computation.
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Complexity theory of real functions
TL;DR: " polynomial complexity theory extends the notions and tools of the theory of computability to provide a solid theoretical foundation for the study of computational complexity of practical problems.
Journal ArticleDOI
Computability on subsets of metric spaces
Vasco Brattka,Gero Presser +1 more
TL;DR: This work investigates the special situation of compact subsets by studying the basic notions of effectivity in classical recursion theory and presents all results in the framework of "Type-2 Theory of Effectivity" which allows to express effectivity properties in a very uniform way.