Leaky quantum graphs: A review

Abstract

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 −∆ − αδ(x − Γ) with a singular attractive interaction supported by a graph-like set in 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 leakygraph 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.

Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

“…Since T is a restriction of the orthogonal sum S * i ⊕ S * e and f i | C = f e | C , g i | C = g e | C for f, g ∈ dom T we conclude from (18) and (19) that…”

“…Schrödinger operators with δ-interactions are frequently used in mathematical physics to model interactions of quantum particles; we refer to the monographs [1] and [23], to the review article [18] and to [2, 3, 8, 16, 17, 19-22, 24, 25, 32, 33, 36, 37, 42] for a small selection of related papers on spectral analysis of such operators. Let A free be the usual self-adjoint realization of −∆ in L 2 (R n ) and let A δ,α be the self-adjoint operator with δ-interaction on C in (1).…”

New isotope technologies in environmental physics

“…Another generalization of quantum graphs are so-called "leaky" graphs where tunneling between nearby edges is allowed (see review [Exn07] and references therein). In this model the spectral problem −∆u = α(x)δ(x − Γ)u + λu is considered, where δ(x − Γ) is a delta distribution supported on the metric graph Γ, α(x) is the coefficient giving strength of the interaction, λ is the spectral parameter and u is the wavefunction in two or three dimensions.…”

Asymptotic spectral analysis in colliding leaky quantum layers

“…Theoretical studies have lead to interesting mathematical problems which involve an interaction of differential geometry, spectral analysis and theory of partial differential equations. In this paper, we rely on the mathematical concept of leaky quantum graphs or waveguides introduced by Exner and Ichinose in 2001 [15] (see [14] for a survey), where the quantum Hamiltonian is modelled by the Schrödinger operator with a Dirac-measure potential supported on a hypersurface in R d . The situations d = 1, 2, 3 are of particular interest in the context of mesoscopic physics of nanostructures, where they are sometimes referred to as quantum dots, wires or layers, respectively.…”