Physics – Mathematical Physics
Scientific paper
2007-10-31
Physics
Mathematical Physics
42 pages, to appear in the proceedings of the Isaac Newton Institute programme "Analysis on Graphs and Applications"; the fina
Scientific paper
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 $-\Delta -\alpha \delta (x-\Gamma)$ with a singular attractive interaction supported by a graph-like set in $\mathbb{R}^\nu,\: \nu=2,3$. We will explain how such singular Schr\"odinger operators can be properly defined for different codimensions of $\Gamma$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $\Gamma$ 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.
No associations
LandOfFree
Leaky Quantum Graphs: A Review does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Leaky Quantum Graphs: A Review, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Leaky Quantum Graphs: A Review will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-14950