quantum graph

[[Image:quantumgraph.png|400px|right|thumb|A metric graph embedded in the plane

with three open edges. The dashed line denotes the metric

distance between two

points $x$ and $y$.]]

A metric graph

is a graph consisting of a set $V$ of vertices and

a set $E$ of edges where each edge $e=(v\_1,v\_2)\backslash in\; E$ has been associated

with an interval $[0,L\_e]$ so that $x\_e$ is the coordinate on the

interval, the vertex $v\_1$ corresponds to $x\_e=0$ and

$v\_2$ to $x\_e=L\_e$ or vice versa. The choice of which vertex lies at zero is

arbitrary with the alternative corresponding to a change of coordinate on the

edge.

The graph has a natural metric: for two

points $x,y$ on the graph, $\backslash rho(x,y)$ is

the shortest distance between them

where distance is measured along the edges of the graph.

Open graphs: in the combinatorial graph model

edges always join pairs of vertices however in a quantum graph one may also

consider semi-infinite edges. These are edges associated with the interval

$[0,\backslash infty)$ attached to a single vertex at $x\_e=0$.

A graph with one or more

such open edges is referred to as an open graph.

Quantum graphs are metric graphs equipped with a differential

(or pseudo-differential) operator acting on functions on the graph.

A function $f$ on a metric graph is defined as the $|E|$-tuple of functions

$f\_e(x\_e)$ on the intervals.

The Hilbert space of the graph is $\backslash bigoplus\_\{e\backslash in\; E\}\; L^2([0,L\_e])$

where the inner product of two functions is

:$\backslash langle\; f,g\; \backslash rangle\; =\; \backslash sum\_\{e\backslash in\; E\}\; \backslash int\_\{0\}^\{L\_e\}\; f\_e^\{*\}(x\_e)g\_e(x\_e)\; \backslash ,\; dx\_e,$

$L\_e$ 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 $-\backslash frac\{\backslash textrm\{d\}^2\}\{\backslash textrm\{d\}\; x\_e^2\}$ where $x\_e$ 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 $H^2$ 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, $f\_e(0)=f\_e(L\_e)=0$ for every edge. An eigenfunction on a finite edge may be written as

:$f\_e(x\_e)\; =\; \backslash sin\; \backslash left(\; \backslash frac\{n\; \backslash pi\; x\_e\}\{L\_e\}\; \backslash right)$

for integer $n$. 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 $\backslash frac\{n^2\backslash pi^2\}\{L\_e^2\}$. 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 $f$ in the domain of the operator is continuous everywhere on the graph and the sum of the outgoing derivatives at a vertex is zero,

:$\backslash sum\_\{e\backslash sim\; v\}\; f\text{'}(v)\; =\; 0\; \backslash \; ,$

where $f\text{'}(v)=f\text{'}(0)$ if the vertex $v$ is at $x=0$ and $f\text{'}(v)=-f\text{'}(L\_e)$ if $v$ is at $x=L\_e$.

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

• These include the more general class of Schrödinger operators,

:$\backslash left(\; i\; \backslash frac\{\backslash textrm\{d\}\}\{\backslash textrm\{d\}\; x\_e\}\; +\; A\_e(x\_e)\; \backslash right)^2\; +\; V\_e(x\_e)\; \backslash \; ,$

where $A\_e$ is a "magnetic vector potential" on the edge and $V\_e$ 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.

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,{{cite journal|last1=Kuchment|first1=Peter|title=Quantum graphs: I. Some basic structures|journal=Waves in Random Media|volume=14|issue=1|year=2004|pages=S107–S128|issn=0959-7174|doi=10.1088/0959-7174/14/1/014|bibcode=2004WRM....14S.107K |s2cid=16874849 }} which automatically yields an operator in variational form.

Let $v$ be a vertex with $d$ edges emanating from it. For simplicity we choose the coordinates on the edges so that $v$ lies at $x\_e=0$ for each edge meeting at $v$. For a function $f$ on the graph let

:$\backslash mathbf\{f\}=(f\_\{e\_1\}(0),f\_\{e\_2\}(0),\backslash dots,f\_\{e\_\{d\}\}(0))^T\; ,\; \backslash qquad\; \backslash mathbf\{f\}\text{'}=(f\text{'}\_\{e\_1\}(0),f\text{'}\_\{e\_2\}(0),\backslash dots,f\text{'}\_\{e\_\{d\}\}(0))^T.$

Matching conditions at $v$ can be specified by a pair of matrices

$A$ and $B$ through the linear equation,

:$A\; \backslash mathbf\{f\}\; +B\; \backslash mathbf\{f\}\text{'}=\backslash mathbf\{0\}.$

The matching conditions define a self-adjoint operator if

$(A,\; B)$ has the maximal rank $d$ and $AB^\{*\}=BA^\{*\}.$

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

using a scattering matrix approach introduced by Kottos and Smilansky

.{{cite journal|last1=Kottos|first1=Tsampikos|last2=Smilansky|first2=Uzy|title=Periodic Orbit Theory and Spectral Statistics for Quantum Graphs|journal=Annals of Physics|volume=274|issue=1|year=1999|pages=76–124|issn=0003-4916|doi=10.1006/aphy.1999.5904 |arxiv=chao-dyn/9812005|bibcode=1999AnPhy.274...76K |s2cid=17510999 }}{{cite journal|last1=Gnutzmann∥|first1=Sven|last2=Smilansky|first2=Uzy|title=Quantum graphs: Applications to quantum chaos and universal spectral statistics|journal=Advances in Physics|volume=55|issue=5–6|year=2006|pages=527–625|issn=0001-8732|doi=10.1080/00018730600908042|arxiv=nlin/0605028|bibcode=2006AdPhy..55..527G |s2cid=119424306 }} The eigenvalue problem on an edge is,

:$-\backslash frac\{d^2\}\{dx\_e^2\}\; f\_e(x\_e)=k^2\; f\_e(x\_e).\backslash ,$

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

:$f\_e(x\_e)\; =\; c\_e\; \backslash textrm\{e\}^\{i\; k\; x\_e\}\; +\; \backslash hat\{c\}\_e\; \backslash textrm\{e\}^\{-i\; k\; x\_e\}.\backslash ,$

where in a time-dependent Schrödinger equation $c$ is the coefficient

of the outgoing plane wave at $0$ and $\backslash hat\{c\}$ coefficient of the incoming

plane wave at $0$.

The matching conditions at $v$ define a scattering matrix

:$S(k)=-(A+i\; kB)^\{-1\}(A-ikB).\backslash ,$

The scattering matrix relates the vectors of incoming and outgoing plane-wave

coefficients at $v$, $\backslash mathbf\{c\}=S(k)\backslash hat\{\backslash mathbf\{c\}\}$.

For self-adjoint matching conditions $S$ is unitary. An element of

$\backslash sigma\_\{(uv)(vw)\}$ of $S$ is a complex transition amplitude

from a directed edge $(uv)$

to the edge $(vw)$ which in general depends on $k$.

However, for a large class of matching conditions

the S-matrix is independent of $k$.

With Neumann matching conditions for example

:$$

A=\left( \begin{array}{ccccc}

1& -1 & 0 & 0 & \dots \\

0 & 1 & -1 & 0 & \dots \\

& & \ddots & \ddots & \\

0& \dots & 0 & 1 & -1 \\

0 &\dots & 0 & 0& 0 \\

\end{array} \right) , \quad B=\left( \begin{array}{cccc}

0& 0 & \dots & 0 \\

\vdots & \vdots & & \vdots \\

0& 0 & \dots & 0 \\

1 &1 & \dots & 1 \\

\end{array} \right).

Substituting in the equation for $S$

produces $k$-independent transition amplitudes

:$\backslash sigma\_\{(uv)(vw)\}=\backslash frac\{2\}\{d\}-\backslash delta\_\{uw\}.\backslash ,$

where $\backslash delta\_\{uw\}$ is the Kronecker delta function that is one if $u=w$ and

zero otherwise. From the transition amplitudes we may define a

$2|E|\backslash times\; 2|E|$ matrix

:$U\_\{(uv)(lm)\}(k)=\; \backslash delta\_\{vl\}\; \backslash sigma\_\{(uv)(vm)\}(k)\; \backslash textrm\{e\}^\{i\; kL\_\{(uv)\}\}.\backslash ,$

$U$ 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 $2|E|$ plane-wave coefficients for the

graph where $c\_\{(uv)\}$ is the coefficient of

the plane wave traveling from $u$ to $v$.

The phase $\backslash textrm\{e\}^\{i\; kL\_\{(uv)\}\}$ is the phase acquired by the plane wave

when propagating from vertex $u$ to vertex $v$.

Quantization condition: An eigenfunction on the graph

can be defined through its associated $2|E|$ 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.

:$|U(k)-I|=0.\backslash ,$

Eigenvalues $k\_j$ occur at values of $k$ where the matrix $U(k)$ has an

eigenvalue one. We will order the spectrum with

$0\backslash leqslant\; k\_0\; \backslash leqslant\; k\_1\; \backslash leqslant\; \backslash dots$.

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 $k$.

The trace formula links the spectrum with periodic orbits on the graph.

:$d(k):=\backslash sum\_\{j=0\}^\{\backslash infty\}\; \backslash delta(k-k\_j)=\backslash frac\{L\}\{\backslash pi\}+\backslash frac\{1\}\{\backslash pi\}$

\sum_p \frac{L_p}{r_p} A_p \cos(kL_p).

$d(k)$ is called the density of states. The right hand side of the trace

formula is made up of two terms, the Weyl

term $\backslash frac\{L\}\{\backslash pi\}$

is the mean separation of eigenvalues and the oscillating part is a sum

over all periodic orbits $p=(e\_1,e\_2,\backslash dots,e\_n)$ on the graph.

$L\_p=\backslash sum\_\{e\backslash in\; p\}\; L\_e$ is the length of the orbit and

$L=\backslash sum\_\{e\backslash in\; E\}L\_e$ is

the total length of the graph. For an orbit generated by repeating a

shorter primitive orbit, $r\_p$ counts the number of repartitions.

$A\_p=\backslash sigma\_\{e\_1\; e\_2\}\; \backslash sigma\_\{e\_2\; e\_3\}\; \backslash dots\; \backslash sigma\_\{e\_n\; e\_1\}$ is

the product of the transition amplitudes at the vertices of the graph around

the orbit.

Image:naphthalene2.png

Quantum graphs were first employed in the 1930s

to model the spectrum of free electrons in organic molecules like

Naphthalene, see figure. As a first approximation the

atoms are taken to be vertices while the

σ-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 waveguide 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{{cite journal|last1=Kottos|first1=Tsampikos|last2=Smilansky|first2=Uzy|title=Quantum Chaos on Graphs|journal=Physical Review Letters|volume=79|issue=24|year=1997|pages=4794–4797|issn=0031-9007|doi=10.1103/PhysRevLett.79.4794|bibcode=1997PhRvL..79.4794K }} 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 $e$ to edge $f$ is given by the absolute value of the

quantum transition amplitude squared, $|\backslash sigma\_\{ef\}|^2$. 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.{{cite journal|last1=Kuchment|first1=Peter|last2=Kunyansky|first2=Leonid|title=Differential Operators on Graphs and Photonic Crystals|journal=Advances in Computational Mathematics|volume=16|issue=24|year=2002|pages=263–290|doi=10.1023/A:1014481629504|s2cid=17506556 }} 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 $\{\backslash mathbb\; R\}^2$ 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 edge of spectral bands in the presence of disorder.

• Schild's Ladder, a novel dealing with a fictional quantum graph theory
• Feynman diagram

Category:Quantum mechanics

Category:Extensions and generalizations of graphs