Continuous spectra and numerical eigenvalues

Some spectral problems for differential operators are naturally posed on the whole real line, often leading to eigenvalues plus continuous spectrum. Then the numerical approximation typically involves three processes: (a) reduction to a finite interval; (b) discretization; (c) application of a numerical eigenvalue solver such as the QR-algorithm.Reduction to a finite interval and discretization typically eliminate the continuous spectrum. However, through round-off error, the continuous spectrum may show up again when the eigenvalue solver is applied. (In some sense, three wrongs make a right.) Interestingly, not all parts of the continuous spectrum show up in the same way, however. We illustrate this observation by numerical examples. A perturbation argument, though non-rigorous, explains the observation.