Contact interactions on graph superlattices

Abstract

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or delta type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the delta' interaction and its symmetrized version for particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig-Penney analogues: the delta' lattices have bands whose widths are asymptotically bounded and do not approach zero, while the delta lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the delta coupling constant is small enough, the delta lattice has no ggaps above the threshold of the spectrum. On the other hand, infinitely many gaps emerge above a critical value of the coupling constant; for almost all ratios this value is zero.

LaTeX, 20 pages, 2 figures available upon request

References in corpus (1)

• Four-Parameter Point-Interaction in 1-D Quantum Systems

Cited by in corpus (44)

• Kirchhoff's Rule for Quantum Wires
• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Realizing Discontinuous Wave Functions with Renormalized Short-Range Potentials
• Fast solitons on star graphs
• Weakly coupled states on branching graphs
• The Generalized Star Product and the Factorization of Scattering Matrices on Graphs
• Functionals of the Brownian motion, localization and metric graphs
• Point interaction in dimension two and three as models of small scatterers
• Double Spiral Energy Surface in One-dimensional Quantum Mechanics of Generalized Pointlike Potentials
• Spectral Theory of Infinite Quantum Graphs
• Magnetoresonances on a lasso graph
• An approximation to ${\delta }^{\prime }$ couplings on graphs
• A single-mode quantum transport in serial-structure geometric scatterers
• Approximations of singular vertex couplings in quantum graphs
• Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs
• On the ground state of quantum graphs with attractive $\delta$-coupling
• Green's function approach for quantum graphs: an overview
• Distribution theory for Schrödinger's integral equation
• Distributional approach to point interactions in one-dimensional quantum mechanics
• Statistical properties of resonance widths for open Quantum Graphs
• Duality and Anholonomy in Quantum Mechanics of 1D Contact Interactions
• Irregular Dynamics in a Solvable One-Dimensional Quantum Graph
• Single point potentials with total resonant tunneling
• Approximations of quantum-graph vertex couplings by singularly scaled rank-one operators
• Periodic quantum graphs from the Bethe-Sommerfeld perspective
• Spectral filtering in quantum Y-junction
• Schrödinger operators on star graphs with singularly scaled potentials supported near the vertices
• Quantum Abacus
• Ring chains with vertex coupling of a preferred orientation
• On ${\delta }^{\prime }$-like potential scattering on star graphs
• Equivalence of Local and Separable Realizations of the Discontinuity-Inducing Contact Interaction and Its Perturbative Renormalizability
• High-energy asymptotics of the spectrum of a periodic square-lattice quantum graph
• A quasi-free position-dependent-mass jump and self-scattering correspondence
• One-sided Diophantine approximations
• Role of a Phase Factor in the Boundary Condition of a One-Dimensional Junction
• Approximations of quantum-graph vertex couplings by singularly scaled potentials
• Anomalous Hall conductivity and quantum friction
• Spectrum of a dilated honeycomb network
• Reducible boundary conditions in coupled channels
• Dynamical and variational properties of the NLS-${\delta }_s^{\prime }$ equation on the star graph
• Quantum graph vertices with permutation-symmetric scattering probabilities
• The heat equation under linear conditions on the moments in higher dimensions
• Quantum transport in randomized quantum graphs
• Gaps in the spectrum of a cuboidal periodic lattice graph