Universal algorithms for computing spectra of periodic operators

TL;DR

This paper investigates the possibility of universal algorithms for computing spectra of various periodic operators, providing solutions for some cases and proving impossibility for others, thus advancing spectral computation methods.

Contribution

It demonstrates that universal algorithms exist for periodic banded matrices and smooth Schr"odinger operators, and proves their non-existence for certain divergent potentials.

Findings

Algorithms are provided for periodic banded matrices.

Algorithms are provided for Schr"odinger operators with smooth periodic potentials.

No universal algorithm exists for Schr"odinger operators with potentials diverging at a point.

Abstract

Schr\"odinger operators with periodic (possibly complex-valued) potentials and discrete periodic operators (possibly with complex-valued entries) are considered, and in both cases the computational spectral problem is investigated: namely, under what conditions can a "one-size-fits-all" algorithm for computing their spectra be devised? It is shown that for periodic banded matrices this can be done, as well as for Schr\"odinger operators with periodic potentials that are sufficiently smooth. In both cases implementable algorithms are provided, along with examples. For certain Schr\"odinger operators whose potentials may diverge at a single point (but are otherwise well-behaved) it is shown that there does not exist such an algorithm, though it is shown that the computation is possible if one allows for two successive limits.