2020/03/13 by Pavel Exner, Exner, Pavel
Mathematics · Physics and Astronomy · #34L40 #35J10 #81Q35 #Advanced Mathematical Theories and Applications #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Spectral Theory (math.SP) #advanced mathematical theories #math-ph #math.MP #math.SP #msc:34L40 #msc:35J10 #msc:81Q35 #quant-ph
paper · pdf · doi:10.48550/arxiv.2003.06189
to appear in the Proceedings 8th International Congress of Chinese Mathematicians
arxiv created 2020/03/13 · openalex publication_date 2020/03/13 · arxiv updated 2020/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This review paper summarizes the contents of the talk given by the author at the 8th International Congress of Chinese Mathematicians. Using examples of Schrödinger operators on metric graphs, it is shown that a nontrivial topology of the configuration space can give rise to a rich variety of spectral types. In particular, it is shown that the spectrum may be of a pure point type or to have a Cantor structure. We also address the question about the number of open spectral gaps and show that it could be nonzero and finite. Finally, inspired by a recent attempt to model the anomalous Hall effect we analyze a vertex coupling which exhibits high-energy behavior determined by the vertex degree parity.