Stepan S. Manko | Searcharxiv

arXiv2014

Spectra of magnetic chain graphs: coupling constant perturbations

We analyze spectral properties of a quantum graph in the form of a ring chain with a $δ$ coupling in the vertices exposed to a homogeneous magnetic field perpendicular to the graph plane. We find the band spectrum in the case when the chain exhibits a translational symmetry and study the discrete spectrum in the gaps resulting from changing a finite number of vertex coupling constants. In particular, we discuss in details some examples such as perturbations of one or two vertices, weak perturbation asymptotics, and a pair of distant perturbations.

math-ph↗

arXiv2013

Approximations of quantum-graph vertex couplings by singularly scaled potentials

We investigate the limit properties of a family of Schrödinger operators of the form $H_\varepsilon= -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+ \frac{λ(\varepsilon)}{\varepsilon^2}Q \big(\frac{x}{\varepsilon}\big)$ acting on $n$-edge star graphs with Kirchhoff conditions imposed at the vertex. The real-valued potential $Q$ is supposed to have compact support and $λ(\cdot)$ to be analytic around $\varepsilon=0$ with $λ(0)=1$. We show that if the operator has a zero-energy resonance of order $m$ for $\varepsilon=1$ and $λ(1)=1$, in the limit $\varepsilon\to 0$ one obtains the Laplacian with a vertex coupling depending on $1+\frac12 m(2n-m+1)$ parameters. We prove the norm-resolvent convergence as well as the convergence of the corresponding on-shell scattering matrices. The obtained vertex couplings are of scale-invariant type provided $λ'(0)=0$; otherwise the scattering matrix depends on energy and the scaled potential becomes asymptotically opaque in the low-energy limit.

math-ph↗