quantum graph - wikipedia, the free encyclopedia

7/30/2019 Quantum Graph - Wikipedia, The Free Encyclopedia

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intervals. The Hilbert space of the graph is where the inner product of two functions is

may be infinite in the case of an open edge. The simplest example of an operator on a metric graph is the

Laplace operator. The operator on an edge is where is the coordinate on the edge. To make the

operator self-adjoint a suitable domain must be specified. This is typically achieved by taking the Sobolev space

of functions on the edges of the graph and specifying matching conditions at the vertices.

The trivial example of matching conditions that make the operator self-adjoint are the Dirichlet boundary

conditions, for every edge. An eigenfunction on a finite edge may be written as

for integer . If the graph is closed with no infinite edges and the lengths of the edges of the graph are rationally

independent then an eigenfunction is supported on a single graph edge and the eigenvalues are . The

Dirichlet conditions don't allow interaction between the intervals so the spectrum is the same as that of the set of

disconnected edges.

More interesting self-adjoint matching conditions that allow interaction between edges are the Neumann or

natural matching conditions. A function in the domain of the operator is continuous everywhere on the graph

and the sum of the outgoing derivatives at a vertex is zero,

where if the vertex is at and if is at .

The properties of other operators on metric graphs have also been studied.

These include the more general class of Schrdinger operators,

where is a "magnetic vector potential" on the edge and is a scalar potential.

Another example is the Dirac operator on a graph which is a matrix valued operator acting on vector

valued functions that describe the quantum mechanics of particles with an intrinsic angular momentum of

one half such as the electron.

The Dirichlet-to-Neumann operator on a graph is a pseudo-differential operator that arises in the study of

photonic crystals.


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All self-adjoint matching conditions of the Laplace operator on a graph can be classified according to a

scheme of Kostrykin and Schrader. In practice, it is often more convenient to adopt a formalism introduced by

Kuchment, see [1], which automatically yields an operator in variational form.

Let be a vertex with edges emanating from it. For simplicity we choose the coordinates on the edges so that

lies at for each edge meeting at . For a function on the graph let

Matching conditions at can be specified by a pair of matrices and through the linear equation,

The matching conditions define a self-adjoint operator if has the maximal rank and

The spectrum of the Laplace operator on a finite graph can be conveniently described using a scattering matrix

approach introduced by Kottos and Smilansky [2]. The eigenvalue problem on an edge is,

So a solution on the edge can be written as a linear combination of plane waves.

where in a time-dependent Schrdinger equation is the coefficient of the outgoing plane wave at and

coefficient of the incoming plane wave at . The matching conditions at define a scattering matrix

The scattering matrix relates the vectors of incoming and outgoing plane-wave coefficients at , .

For self-adjoint matching conditions is unitary. An element of of is a complex transition

amplitude from a directed edge to the edge which in general depends on . However, for a large

class of matching conditions the S-matrix is independent of . With Neumann matching conditions for example

Substituting in the equation for produces -independent transition amplitudes


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where is the Kronecker delta function that is one if and zero otherwise. From the transition

amplitudes we may define a matrix

is called the bond scattering matrix and can be thought of as a quantum evolution operator on the graph. It is

unitary and acts on the vector of plane-wave coefficients for the graph where is the coefficient of

the plane wave traveling from to . The phase is the phase acquired by the plane wave whenpropagating from vertex to vertex .

Quantization condition: An eigenfunction on the graph can be defined through its associated plane-wave

coefficients. As the eigenfunction is stationary under the quantum evolution a quantization condition for the

graph can be written using the evolution operator.

Eigenvalues occur at values of where the matrix has an eigenvalue one. We will order the spectrum

with .

The first trace formula for a graph was derived by Roth (1983). In 1997 Kottos and Smilansky used the

quantization condition above to obtain the following trace formula for the Laplace operator on a graph when the

transition amplitudes are independent of . The trace formula links the spectrum with periodic orbits on the

graph.

is called the density of states. The right hand side of the trace formula is made up of two terms, the Weyl

term is the mean separation of eigenvalues and the oscillating part is a sum over all periodic orbits

on the graph. is the length of the orbit and is the total

length of the graph. For an orbit generated by repeating a shorter primitive orbit, counts the number of

repartitions. is the product of the transition amplitudes at the vertices of the graph

around the orbit.

Quantum graphs were first employed in the 1930s to modelthe spectrum of free electrons in organic molecules like

Naphthalene, see figure. As a first approximation the atoms

are taken to be vertices while the -electrons form bonds

that fix a frame in the shape of the molecule on which the

free electrons are confined.

A similar problem appears when considering quantum

waveguides. These are mesoscopic systems - systems built

with a width on the scale of nanometers. A quantum


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Naphthalene moleculewaveguide can be thought of as a fattened graph where the

edges are thin tubes. The spectrum of the Laplace operator

on this domain converges to the spectrum of the Laplace

operator on the graph under certain conditions. Understanding mesoscopic systems plays an important role in

the field of nanotechnology.

In 1997 Kottos and Smilansky proposed quantum graphs as a model to study quantum chaos, the quantum

mechanics of systems that are classically chaotic. Classical motion on the graph can be defined as a probabilistic

Markov chain where the probability of scattering from edge to edge is given by the absolute value of the

quantum transition amplitude squared, . For almost all finite connected quantum graphs the probabilistic

dynamics is ergodic and mixing, in other words chaotic.

Quantum graphs embedded in two or three dimensions appear in the study of photonic crystals. In two

dimensions a simple model of a photonic crystal consists of polygonal cells of a dense dielectric with narrow

interfaces between the cells filled with air. Studying dielectric modes that stay mostly in the dielectric gives rise

to a pseudo-differential operator on the graph that follows the narrow interfaces.

Periodic quantum graphs like the lattice in are common models of periodic systems and quantum graphs

have been applied to the study the phenomena of Anderson localization where localized states occur at the edgeof spectral bands in the presence of disorder.

Event symmetry

Schild's Ladder, for fictional quantum graph theory

Feynman diagram

^ P. Kuchment, Quantum graphs I. Some basic structures, Waves in Random Media14, S107-S128 (2004)1.

^ S. Gnutzman & U. Smilansky, Quantum graphs: applications to quantum chaos and universal spectral statistics,

Adv. Phys.55 527-625 (2006)

2.

Quantum graphs on arxiv.org (http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-339567)

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