Cargando…

A new spectral invariant for quantum graphs

The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introdu...

Descripción completa

Detalles Bibliográficos
Autores principales: Ławniczak, Michał, Kurasov, Pavel, Bauch, Szymon, Białous, Małgorzata, Akhshani, Afshin, Sirko, Leszek
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Nature Publishing Group UK 2021
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8319202/
https://www.ncbi.nlm.nih.gov/pubmed/34321508
http://dx.doi.org/10.1038/s41598-021-94331-0
Descripción
Sumario:The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic [Formula: see text] , with [Formula: see text] denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic [Formula: see text] of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic [Formula: see text] can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic [Formula: see text] a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.