---
abstract: |
    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 Schrödinger 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.
address: 'Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068 Řež near Prague, and Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Břehová 7, 11519 Prague, Czechia'
author:
- Pavel Exner
date: 'September ??, 2002 and, in revised form, ????, 2002.'
title: 'Leaky Quantum Graphs: A Review'
---

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Introduction
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In this paper we are going to review results concerning a class of 