1999/01/31 by Michael J. Gruber
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.OA #math.QA #math.SP #msc:35Q40 #msc:46L89 #msc:46N50 #msc:47F05 #msc:47N50 #msc:58C40 #msc:58G25 #msc:58Gxx #msc:58Z05 #quant-ph
paper · pdf · doi:10.1063/1.1369122
published as J.Math.Phys. 42 (2001) 2438-2465 · 8 pages; final version, to appear in Rep. Math. Phys. (conference proceedings "XVII-th Workshop on Geometric Methods in Physics")
arxiv created 1999/06/29 · openalex publication_date 2001/06/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
For differential operators which are invariant under the action of an abelian group Bloch theory is the tool of choice to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a non-commutative Bloch theory for elliptic operators on Hilbert C*-modules. It relates properties of C*-algebras to spectral properties of module operators such as band structure, weak genericity of cantor spectra, and absence of discrete spectrum. It applies e.g. to differential operators invariant under a projective group action, such as Schroedinger operators with periodic magnetic field.