Leaky Quantum Graphs: A Review

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.

42 pages, to appear in the proceedings of the Isaac Newton Institute programme "Analysis on Graphs and Applications"; the final version with minor improvements

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Cited by in corpus (23)

• Little Magnetic Book
• Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
• Two-dimensional Dirac operators with singular interactions supported on closed curves
• Klein's Paradox and the Relativistic $\delta$-shell Interaction in ${\mathbb{R}}^3$
• Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc
• Strong coupling asymptotics for Schrödinger operators with an interaction supported by an open arc in three dimensions
• General $\delta$-shell interactions for the two-dimensional Dirac operator: self-adjointness and approximation
• Spectral estimates for resolvent differences of self-adjoint elliptic operators
• Spectral asymptotics for ${\delta }^{\prime }$ interaction supported by a infinite curve
• Discrete spectrum of interactions concentrated near conical surfaces
• Spectral enclosures for non-self-adjoint extensions of symmetric operators
• Direct computation of scattering matrices for general quantum graphs
• Gap asymptotics in a weakly bent leaky quantum wire
• The relativistic spherical $\delta$-shell interaction in ${\mathbb{R}}^3$: spectrum and approximation
• Spectral asymptotics of a strong ${\delta }^{\prime }$ interaction supported by a surface
• Singular interactions supported by embedded curves
• Attractive conical surfaces create infinitely many bound states
• Spectral asymptotics for $\delta$-interactions on sharp cones
• Strong coupling asymptotics for $\delta$-interactions supported by curves with cusps
• Lower bound on the spectrum of the Schrödinger operator in the plane with delta-potential supported by a curve
• Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions
• On the bound states of magnetic Laplacians on wedges
• Trace formulae for Schrödinger operators with singular interactions