Leaky Quantum Graphs: A Review
| dc.creator | Exner, Pavel | |
| dc.date | 2007-10-31 | |
| dc.date | 2008-01-07 | |
| dc.date.accessioned | 2026-07-07T08:52:31Z | |
| dc.date.available | 2026-07-07T08:52:31Z | |
| dc.description | The aim of this review is to provide an overview of a recent work concerning ``leaky'' quantum graphs described by Hamiltonians given formally by the expression $-Δ-αδ(x-Γ)$ with a singular attractive interaction supported by a graph-like set in $\mathbb{R}^ν,\: ν=2,3$. We will explain how such singular Schrödinger operators can be properly defined for different codimensions of $Γ$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $Γ$ influences their spectra and the scattering, strong-coupling asymptotic behavior, and a discrete counterpart to leaky-graph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the reader will take some of them as a challenge. | |
| dc.description | 42 pages, to appear in the proceedings of the Isaac Newton Institute programme "Analysis on Graphs and Applications"; the final version with minor improvements | |
| dc.identifier | https://arxiv.org/abs/0710.5903 | |
| dc.identifier | http://arxiv.org/abs/0710.5903 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/145304 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Mesoscale and Nanoscale Physics | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Spectral Theory | |
| dc.subject | Quantum Physics | |
| dc.subject | 81Q05 (Primary); 35J10, 35P15, 58J05 (Secondary) | |
| dc.title | Leaky Quantum Graphs: A Review | |
| dc.type | text |