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