2007/10/31 by Pavel Exner, Exner, Pavel · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #35J10 #35P15 #58J05 (Secondary) #81Q05 (Primary) #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mesoscale and Nanoscale Physics (cond-mat.mes-hall) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #Quantum and electron transport phenomena #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.0710.5903
openalex publication_date 2007/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝν, 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.