Kirchhoff's Rule for Quantum Wires

Abstract

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with $n$ open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with $n$ channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy $E>0$ is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms.

40 pages

References in corpus (8)

• Contact interactions on graph superlattices
• Weakly coupled states on branching graphs
• Four-Parameter Point-Interaction in 1-D Quantum Systems
• Dielectric resonances of binary random networks
• Scattering Theory Approach to Random Schroedinger Operators in One Dimension
• Ballistic transport: A view from the quantum theory of motion
• Evanescent modes in a multiple scattering factorization
• Quantum Mechanics of Multi-Prong Potentials

Cited by in corpus (215)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Can One Hear the Shape of a Graph?
• Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs
• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Convergence of spectra of graph-like thin manifolds
• On the spectra of carbon nano-structures
• Scattering solutions in a network of thin fibers: small diameter asymptotics
• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• Variational properties and orbital stability of standing waves for NLS equation on a star graph
• Scattering on compact manifolds with infinitely thin horns
• Nonlinear Schrödinger equation on graphs: recent results and open problems
• Spectra of Schroedinger operators on equilateral quantum graphs
• Vacuum Energy and Repulsive Casimir Forces in Quantum Star Graphs
• Spectral Statistics for the Dirac Operator on Graphs
• Repulsive Casimir Pistons
• Fast solitons on star graphs
• Geometrically induced spectrum in curved leaky wires
• The Generalized Star Product and the Factorization of Scattering Matrices on Graphs
• On the structure of critical energy levels for the cubic focusing NLS on star graphs
• Approximation of a general singular vertex coupling in quantum graphs
• The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums
• Bosonization and Scale Invariance on Quantum Wires
• Stationary scattering from a nonlinear network
• Quantum Fields on Star Graphs
• Functionals of the Brownian motion, localization and metric graphs
• Quantum spins on star graphs and the Kondo model
• Ground state and orbital stability for the NLS equation on a general starlike graph with potentials
• Coupling in the singular limit of thin quantum waveguides
• Leaky quantum graphs: approximations by point interaction Hamiltonians
• Scattering theory on graphs
• Spectral gap for quantum graphs and their connectivity
• Brownian Motions on Metric Graphs
• Resolvents of self-adjoint extensions with mixed boundary conditions
• Limits of Quantum Graph Operators With Shrinking Edges
• Topological Resonances in Scattering on Networks (Graphs)
• On an inverse problem for tree-like networks of elastic strings
• Convergence of resonances on thin branched quantum wave guides
• Dirac and magnetic Schrödinger operators on fractals
• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• Quantum graphs as holonomic constraints
• Eigenfunction Statistics on Quantum Graphs
• Large gaps in point-coupled periodic systems of manifolds
• Index theorems for quantum graphs
• An approximation to ${\delta }^{\prime }$ couplings on graphs
• Branched Quantization
• A single-mode quantum transport in serial-structure geometric scatterers
• Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory
• Stationary Nonlinear Schrödinger Equation on Simplest Graphs: Boundary conditions and exact solutions
• Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
• Approximations of singular vertex couplings in quantum graphs
• On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators
• A remark on Krein's resolvent formula and boundary conditions
• Linear Representations and Isospectrality with Boundary Conditions
• Irreversible quantum graphs
• Sine-Gordon solitons in networks: Scattering and transmission at vertices
• Quantum statistics on graphs
• On the ground state of quantum graphs with attractive $\delta$-coupling
• Green's function approach for quantum graphs: an overview
• A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds
• Transparent Quantum Graphs
• Finite pseudo orbit expansions for spectral quantities of quantum graphs
• Bi-Laplacians on graphs and networks
• Non-self-adjoint graphs
• Quantum graphs where back-scattering is prohibited
• Scattering theory on graphs (2): the Friedel sum rule
• Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit
• Time-Energy coherent states and adiabatic scattering
• Non-Weyl asymptotics for quantum graphs with general coupling conditions
• Quantum graphs with singular two-particle interactions
• Quantum graphs with vertices of a preferred orientation
• The Berry-Keating operator on $L^2(\rz_>,\ud x)$ and on compact quantum graphs with general self-adjoint realizations
• A Random Necklace Model
• Small-Energy Analysis for the Selfadjoint Matrix Schroedinger Operator on the Half Line
• Quantum Fields on Star Graphs with Bound States at the Vertex
• The very unusual properties of the resolvent, heat kernel, and zeta function for the operator $-d^2/d{r}^2-1/(4r^2)$
• Distributions of the Wigner reaction matrix for microwave networks with symplectic symmetry in the presence of absorption
• Experimental investigation of distributions of the off-diagonal elements of the scattering and the Wigner's $\hat{K}$ matrices for networks with broken time reversal invariance
• Irregular Dynamics in a Solvable One-Dimensional Quantum Graph
• Local Friedel sum rule on graphs
• Localization effects in a periodic quantum graph with magnetic field and spin-orbit interaction
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• High-Energy analysis and Levinson's theorem for the selfadjoint matrix Schroedinger operator on the half line
• Quantum graphs with two-particle contact interactions
• Duality and Anholonomy in Quantum Mechanics of 1D Contact Interactions
• Dynamics of nodal points and the nodal count on a family of quantum graphs
• Quantum ergodicity for graphs related to interval maps
• Finite propagation speed and causal free quantum fields on networks
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions
• Eigenvalue estimates for Schroedinger operators on metric trees
• Zeta functions of quantum graphs
• Nodal Statistics On Quantum Graphs
• Potential-controlled filtering in quantum star graphs
• Continuity of the integrated density of states on random length metric graphs
• Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta trees
• Scattering Theory for the matrix Schrödinger operator on the half line with general boundary conditions
• The first eigenvalue of the $p-$Laplacian on quantum graphs
• Direct computation of scattering matrices for general quantum graphs
• On the inverse problem of the two-velocity tree-like graph
• Simple quantum graphs proposal for quantum devices
• Spectral enclosures for non-self-adjoint extensions of symmetric operators
• Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone
• Dirac-Krein systems on star graphs
• Charged solitons in branched conducting polymers
• Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs
• Fulop-Tsutsui interactions on quantum graphs
• PT-symmetric quantum graphs
• Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over ${\mathbb{Z}}^d$
• Linear Hyperbolic Systems on Networks
• Dirac Particles in Transparent Quantum Graphs: Tunable transport of relativistic quasiparticles in branched structures
• Dressed quantum graphs with optical nonlinearities approaching the fundamental limit
• Boundary conditions: The path integral approach
• Approximations of quantum-graph vertex couplings by singularly scaled rank-one operators
• Exciton dynamics in branched conducting polymers: Quantum graphs based approach
• Small-energy analysis for the selfadjoint matrix Schroedinger operator on the half line. II
• Quantum field theory on quantum graphs and application to their conductance
• Unbounded quantum graphs with unbounded boundary conditions
• Spectral filtering in quantum Y-junction
• Schrödinger operators on star graphs with singularly scaled potentials supported near the vertices
• The Analogue of Bohm-Bell Processes on a Graph
• Network Models: Action formulation
• Quantum Graphs: $\mathcal{P}\mathcal{T}$-symmetry and reflection symmetry of the spectrum
• On ${\delta }^{\prime }$-like potential scattering on star graphs
• Well-Posedness and Symmetries of Strongly Coupled Network Equations
• Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
• Quantum Wire Network with Magnetic Flux
• Exactly solvable interacting two-particle quantum graphs
• Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover
• On the absence of absolutely continuous spectra for Schrödinger operators on radial tree graphs
• Trace Formulae for quantum graphs with edge potentials
• Integrable Models From Twisted Half Loop Algebras
• Dynamics of an electron confined to a "hybrid plane" and interacting with a magnetic field
• High-energy asymptotics of the spectrum of a periodic square-lattice quantum graph
• Tripartite connection condition for quantum graph vertex
• Well-posedness of networks for 1-D hyperbolic partial differential equations
• Spectral determinants and zeta functions of Schrödinger operators on metric graphs
• Bohmian Trajectories for Hamiltonians with Interior-Boundary Conditions
• Approximations by graphs and emergence of global structures
• Algebraic approach to multiple defects on the line and application to Casimir force
• Bound states in point-interaction star-graphs
• Experimental study of closed and open microwave waveguide graphs with preserved and partially violated time-reversal invariance
• An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics
• A linear Wegner estimate for alloy type Schroedinger operators on metric graphs
• Zeta Functions of the Dirac Operator on Quantum Graphs
• Instability of Bose-Einstein condensation on quantum graphs under repulsive perturbations
• Hermitian unitary matrices with modular permutation symmetry
• On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach
• Grassmann-Gaussian integrals and generalized star products
• Delay-time distribution in the scattering of time-narrow wave packets (II) - Quantum Graphs
• Semigroups for dynamical processes on metric graphs
• Optimization of eigenstates and spectra for quasi-linear nonlinear optical systems
• Mathematical Aspects of Vacuum Energy on Quantum Graphs
• Reducible boundary conditions in coupled channels
• Self-adjointness in Quantum Mechanics: a pedagogical path
• The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs
• Sign-indefinite second order differential operators on finite metric graphs
• Kagome network with vertex coupling of a preferred orientation
• A quantum-network approach to spin interferometry driven by Abelian and non-Abelian fields
• Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph
• Trace Identities for the matrix Schrödinger operator on the half line with general boundary conditions
• On the multilevel internal structure of the asymptotic distribution of resonances
• Entanglement entropy on finitely ramified graphs
• 5d Dirac fermion on quantum graph
• A trace formula for metric graphs with piecewise constant potentials and multi-mode graphs
• Quantum graphs: Coulomb-type potentials and exactly solvable models
• Closed and open superconducting microwave waveguide networks as a model for quantum graphs
• Approximation by point potentials in a magnetic field
• $\zeta$-regularised spectral determinants on metric graphs
• The ubiquitous $\zeta$-function and some of its "usual" and "unusual" meromorphic properties
• Path Integral Junctions
• Dirac cones for bi- and trilayer Bernal-stacked graphene in a quantum graph model
• Particle escapes in an open quantum network via multiple leads
• Coupled unidirectional chaotic microwave graphs
• Inverse scattering problem for quantum graph vertices
• Recovering quantum graph spectrum from vertex data
• Rényi and Tsallis entropies related to eigenfunctions of quantum graphs
• Bound states without potentials: localization at singularities
• Instantons and Berry's connections on quantum graph
• Zero modes of quantum graph Laplacians and an index theorem
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• Quantizing graphs, one way or two?
• The number of eigenvalues of the matrix Schrödinger operator on the half line with general boundary conditions
• Bounds on the negative eigenvalues of Laplacians on finite metric graphs
• Wave propagation in periodic networks of thin fibers
• Hidden symmetries in non-self-adjoint graphs
• Physical approach to quantum networks with massive particles
• Dynamical processes on metric networks
• Exact scattering matrix of graphs in magnetic field and quantum noise
• $L^p-L^{p^{\prime }}$ estimates for matrix Schrödinger equations
• Isospectrality for graph Laplacians under the change of coupling at graph vertices: necessary and sufficient conditions
• Exotic eigenvalues of shrinking metric graphs
• Boundary conditions and violations of bulk-edge correspondence in a hydrodynamic model
• Experimental study of the distributions of off-diagonal scattering-matrix elements of quantum graphs with symplectic symmetry
• The Ablowitz-Ladik system on a graph
• Magnetic square lattice with vertex coupling of a preferred orientation
• Integrable Defects at Junctions within a Network
• The semiclassical limit on a star-graph with Kirchhoff conditions
• Quantum graph as a quantum spectral filter
• Boundary Value Problems for Dirac Operators on Graphs
• Uniform existence of the integrated density of states on metric Cayley graphs
• Schroedinger Operators on Regular Metric Trees with Long Range Potentials: Weak Coupling Behavior
• Time evolution and the Schrödinger equation on time dependent quantum graphs
• Escape behaviour of a quantum particle in a loop coupled to a lead
• Exotic eigenvalues and analytic resolvent for a graph with a shrinking edge
• A Flux-Tunable Discrete Angular Filter
• Vacuum energy of Schrödinger operators on metric graphs
• Emerging entanglement on network histories
• Asymmetric quantum transport in a double-stranded Kronig-Penney model
• Quantum graph vertices with minimal number of passbands
• Non-equilibrium current via geometric scatterers
• Scattering off a junction
• Quantum Systems with jump-discontinuous mass. I
• Correspondence of topological classification between quantum graph extra dimension and topological matter
• Dirac particles on periodic quantum graphs
• Short-range interaction in three-dimensional quantum mechanics