Computability in basic quantum mechanics

The basic notions of quantum mechanics are formulated in terms of separable infinite dimensional Hilbert space H. In terms of the Hilbert lattice L of closed linear subspaces of H the notions of state and observable can be formulated as kinds of measures as in [21]. The aim of this paper is to show that there is a good notion of computability for these data structures in the sense of Weihrauch’s Type Two Effectivity (TTE) [26]. Instead of explicitly exhibiting admissible representations for the data types under consideration we show that they do live within the category QCB0 which is equivalent to the category AdmRep of admissible representations and continuously realizable maps between them. For this purpose in case of observables we have to replace measures by valuations which allows us to prove an effective version of von Neumann’s Spectral Theorem

Towards computability of elliptic boundary value problems in variational formulation

AbstractWe present computable versions of the Fréchet–Riesz Representation Theorem and the Lax–Milgram Theorem. The classical versions of these theorems play important roles in various problems of mathematical analysis, including boundary value problems of elliptic equations. We demonstrate how their computable versions yield computable solutions of the Neumann and Dirichlet boundary value problems for a simple non-symmetric elliptic differential equation in the one-dimensional case. For the discussion of these elementary boundary value problems, we also provide a computable version of the Theorem of Schauder, which shows that the adjoint of a computably compact operator on Hilbert spaces is computably compact again

Computability of the Spectrum of Self-Adjoint Operators

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 self-adjoint 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

Computability of the Spectrum of Self-Adjoint Operators and the Computable Operational Calculus

AbstractSelf-adjoint operators and their spectra play a crucial rôle in analysis and physics. Therefore, it is a natural question whether the spectrum of a self-adjoint operator and its eigenvalues 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 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 eigenvalues of self-adjoint operators can be computed in the sense that we can compute a list of indices such that those elements of the already computed dense subset of the spectrum, whose indices are not enumerated in this list, form the set of eigenvalues. Beside these main results we prove some computability results about the operational calculus, which we need in our proofs