# Quantum graph

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{{Short description|Type of graph in mathematics and physics}}
{{About|metric graphs equipped with a differential operator|quantum states represented by graphs|Graph state}}

{{More citations needed|date=August 2025}}
In [mathematics](/source/mathematics) and  [physics](/source/physics), a '''quantum graph''' is a linear, network-shaped structure  of vertices connected on edges (i.e., a [graph](/source/graph_(discrete_mathematics))) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge. An example would be a power network consisting of power lines (edges) connected at transformer stations (vertices); the differential equations would then describe the voltage along each of the lines, with boundary conditions for each edge provided at the adjacent vertices ensuring that the current added over all edges adds to zero at each vertex.

Quantum graphs were first studied by [Linus Pauling](/source/Linus_Pauling) as models of free electrons in organic molecules in the 1930s. They also arise in a variety of mathematical contexts,<ref name="BerkolaikoCarlson2005">{{cite book|last1=Berkolaiko|first1=Gregory|last2=Carlson|first2=Robert|last3=Kuchment|first3=Peter|last4=Fulling|first4=Stephen|title=Quantum Graphs and Their Applications (Contemporary Mathematics): Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their Applications|volume=415|year=2006|isbn=978-0821837658|publisher=American Mathematical Society}}</ref> e.g. as model systems in [quantum chaos](/source/quantum_chaos), in the study of [waveguides](/source/waveguides), in [photonic crystals](/source/photonic_crystals) and in [Anderson localization](/source/Anderson_localization), or as a limit on shrinking thin wires.  Quantum graphs have become prominent models in [mesoscopic physics](/source/mesoscopic_physics) used to obtain a theoretical understanding of [nanotechnology](/source/nanotechnology). Another, more simple notion of quantum graphs was introduced by Freedman et al.<ref name="FreedmanLovász2007">{{cite journal|last1=Freedman|first1=Michael|last2=Lovász|first2=László|last3=Schrijver|first3=Alexander|title=Reflection positivity, rank connectivity, and homomorphism of graphs|journal=Journal of the American Mathematical Society|volume=20|issue=1|year=2007|pages=37–52|issn=0894-0347|doi=10.1090/S0894-0347-06-00529-7|mr=2257396|arxiv=math/0404468|bibcode=2007JAMS...20...37F |s2cid=8208923 }}</ref>

Aside from actually solving the differential equations posed on a quantum graph for purposes of concrete applications, typical questions that arise are those of [controllability](/source/controllability) (what inputs have to be provided to bring the system into a desired state, for example providing sufficient power to all houses on a power network) and [identifiability](/source/identifiability) (how and where one has to measure something to obtain a complete picture of the state of the system, for example measuring the pressure of a water pipe network to determine whether or not there is a leaking pipe).

== Metric graphs ==

400px|right|thumb|A metric graph embedded in the plane 
with three open edges. The dashed line denotes the metric 
distance between two 
points <math>x</math> and <math>y</math>.
A '''metric graph'''
is a [graph](/source/Graph_(discrete_mathematics)) consisting of a set <math>V</math> of vertices and 
a set <math>E</math> of edges where each edge <math>e=(v_1,v_2)\in E</math> has been associated 
with an interval <math>[0,L_e]</math> so that <math>x_e</math> is the coordinate on the 
interval, the vertex <math>v_1</math> corresponds to <math>x_e=0</math> and  
<math>v_2</math> to <math>x_e=L_e</math> 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 <math>x,y</math> on the graph, <math>\rho(x,y)</math> 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 
<math>[0,\infty)</math> attached to a single vertex at <math>x_e=0</math>.  
A graph with one or more 
such open edges is referred to as an open graph.

== Quantum graphs ==

Quantum graphs are metric graphs equipped with a differential 
(or [pseudo-differential](/source/pseudo-differential_operator)) operator acting on functions on the graph.  
A function <math>f</math> on a metric graph is defined as the <math>|E|</math>-tuple of functions
<math>f_e(x_e)</math> on the intervals.   
The [Hilbert space](/source/Hilbert_space) of the graph is <math>\bigoplus_{e\in E} L^2([0,L_e])</math>
where the inner product of two functions is

:<math>\langle f,g \rangle = \sum_{e\in E} \int_{0}^{L_e} f_e^{*}(x_e)g_e(x_e) \, dx_e,</math>

<math>L_e</math> may be infinite in the case of an open edge.  The simplest example of an operator on a metric graph is the [Laplace operator](/source/Laplace_operator).  The operator on an edge is <math>-\frac{\textrm{d}^2}{\textrm{d} x_e^2}</math> where <math>x_e</math> 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](/source/Sobolev_space) <math>H^2</math> 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](/source/Dirichlet_boundary_conditions), <math>f_e(0)=f_e(L_e)=0</math> for every edge.  An eigenfunction on a finite edge may be written as

:<math>f_e(x_e) = \sin \left( \frac{n \pi x_e}{L_e} \right)</math>

for integer <math>n</math>.  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 <math>\frac{n^2\pi^2}{L_e^2}</math>.  The Dirichlet conditions 
don't allow interaction between the intervals so the [spectrum](/source/Spectrum_(functional_analysis)) 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](/source/Neumann_boundary_condition) or natural matching conditions.  A function <math>f</math> in the domain of the operator is continuous everywhere on the graph and the sum of the outgoing derivatives at a vertex is zero,

:<math>\sum_{e\sim v} f'(v) = 0 \ ,</math>

where <math>f'(v)=f'(0)</math> if the vertex <math>v</math> is at <math>x=0</math> and <math>f'(v)=-f'(L_e)</math> if <math>v</math> is at <math>x=L_e</math>.

The properties of other operators on metric graphs have also been studied.  
* These include the more general class of Schrödinger operators, <math display=block>\left( i \frac{\textrm{d}}{\textrm{d} x_e} + A_e(x_e) \right)^2 + V_e(x_e) \ ,</math> where <math>A_e</math> is a "[magnetic vector potential](/source/magnetic_vector_potential)" on the edge and <math>V_e</math> is a [scalar potential](/source/scalar_potential).
* Another example is the [Dirac operator](/source/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](/source/angular_momentum) of one half such as the [electron](/source/electron).
* The Dirichlet-to-Neumann operator on a graph is a pseudo-differential operator that arises in the  study of [photonic crystals](/source/photonic_crystals).

== Theorems ==

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,<ref name="Kuchment2004">{{cite journal|last1=Kuchment|first1=Peter|title=Quantum graphs: I. Some basic structures|journal=[Waves in Random Media](/source/Waves_in_Random_and_Complex_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 }}</ref> which automatically yields an operator in variational form.

Let <math>v</math> be a vertex with <math>d</math> edges emanating from it.  For simplicity we choose the coordinates on the edges so that <math>v</math> lies at <math>x_e=0</math> for each edge meeting at <math>v</math>.  For a function <math>f</math> on the graph let

:<math>\mathbf{f}=(f_{e_1}(0),f_{e_2}(0),\dots,f_{e_{d}}(0))^T , \qquad \mathbf{f}'=(f'_{e_1}(0),f'_{e_2}(0),\dots,f'_{e_{d}}(0))^T.</math>
 
Matching conditions at <math>v</math> can be specified by a pair of matrices 
<math>A</math> and <math>B</math> through the [linear equation](/source/linear_equation),

:<math>A \mathbf{f} +B \mathbf{f}'=\mathbf{0}. </math>

The matching conditions define a [self-adjoint operator](/source/self-adjoint_operator) if 
<math>(A, B)</math> has the maximal rank <math>d</math> and <math>AB^{*}=BA^{*}.</math>

The spectrum of the Laplace operator on a finite graph can be conveniently described 
using a '''scattering matrix''' approach introduced by Kottos and Smilansky.<ref name="KottosSmilansky1999">{{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 }}</ref><ref name="Gnutzmann∥Smilansky2006">{{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 }}</ref>  The eigenvalue problem on an edge is,

:<math>-\frac{d^2}{dx_e^2} f_e(x_e)=k^2 f_e(x_e).\,</math>

So a solution on the edge can be written as a [linear combination](/source/linear_combination) of [plane waves](/source/plane_waves).

:<math>f_e(x_e) = c_e \textrm{e}^{i k x_e} + \hat{c}_e \textrm{e}^{-i k x_e}.\,</math>

where in a time-dependent Schrödinger equation <math>c</math> is the coefficient 
of the outgoing plane wave at <math>0</math> and <math>\hat{c}</math> coefficient of the incoming 
plane wave at <math>0</math>.
The matching conditions at <math>v</math> define a scattering matrix

:<math>S(k)=-(A+i kB)^{-1}(A-ikB).\,</math>

The scattering matrix relates the vectors of incoming and outgoing plane-wave 
coefficients  at <math>v</math>, <math>\mathbf{c}=S(k)\hat{\mathbf{c}}</math>.
For self-adjoint matching conditions <math>S</math> is unitary.  An element of 
<math>\sigma_{(uv)(vw)}</math> of <math>S</math> is a complex transition amplitude 
from a directed edge <math>(uv)</math>
to the edge <math>(vw)</math> which in general depends on <math>k</math>.  
However, for a large class of matching conditions 
the S-matrix is independent of <math>k</math>.  
With Neumann matching conditions for example

:<math>
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).
</math>

Substituting in the equation for <math>S</math> 
produces <math>k</math>-independent transition amplitudes

:<math>\sigma_{(uv)(vw)}=\frac{2}{d}-\delta_{uw}.\,</math>

where <math>\delta_{uw}</math> is the [Kronecker delta](/source/Kronecker_delta) function that is one if <math>u=w</math> and 
zero otherwise.  From the transition amplitudes we may define a 
<math>2|E|\times 2|E|</math> matrix

:<math>U_{(uv)(lm)}(k)= \delta_{vl} \sigma_{(uv)(vm)}(k) \textrm{e}^{i kL_{(uv)}}.\,</math>

<math>U</math> 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 <math>2|E|</math> plane-wave coefficients for the 
graph where <math>c_{(uv)}</math> is the coefficient of 
the plane wave traveling from <math>u</math> to <math>v</math>.  
The phase <math>\textrm{e}^{i kL_{(uv)}}</math> is the phase acquired by the plane wave 
when propagating from vertex <math>u</math> to vertex <math>v</math>.

'''Quantization condition:''' An eigenfunction on the graph 
can be defined through its associated <math>2|E|</math> 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.

:<math>|U(k)-I|=0.\,</math>

Eigenvalues <math>k_j</math> occur at values of <math>k</math> where the matrix <math>U(k)</math> has an 
eigenvalue one.  We will order the spectrum with 
<math>0\leqslant k_0 \leqslant k_1 \leqslant \dots</math>.

The first '''trace formula''' for a graph was derived by Roth (1983).{{full citation needed|date=August 2025}}
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 <math>k</math>.
The trace formula links the spectrum with periodic orbits on the graph.

:<math>d(k):=\sum_{j=0}^{\infty} \delta(k-k_j)=\frac{L}{\pi}+\frac{1}{\pi} 
\sum_p \frac{L_p}{r_p} A_p \cos(kL_p).</math>

<math>d(k)</math> is called the density of states.  The right hand side of the trace 
formula is made up of two terms, the Weyl 
term <math>\frac{L}{\pi}</math> 
is the mean separation of eigenvalues and the oscillating part is a sum 
over all periodic orbits <math>p=(e_1,e_2,\dots,e_n)</math> on the graph.  
<math>L_p=\sum_{e\in p} L_e</math> is the length of the orbit and 
<math>L=\sum_{e\in E}L_e</math> is
the total length of the graph.  For an orbit generated by repeating a 
shorter primitive orbit, <math>r_p</math> counts the number of repartitions.  
<math>A_p=\sigma_{e_1 e_2} \sigma_{e_2 e_3} \dots \sigma_{e_n e_1}</math> is 
the product of the transition amplitudes at the vertices of the graph around 
the orbit.

== Applications ==

300px|right|thumb|Naphthalene molecule
Quantum graphs were first employed in the 1930s 
to model the spectrum of free electrons in organic molecules like 
[Naphthalene](/source/Naphthalene).  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](/source/nanotechnology).

In 1997<ref name="KottosSmilansky1997">{{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 }}</ref> Kottos and Smilansky proposed quantum graphs as a model to study  
[quantum chaos](/source/quantum_chaos), the quantum mechanics of systems that 
are classically chaotic.  Classical motion on the graph can be defined as 
a probabilistic [Markov chain](/source/Markov_chain) where the probability of scattering 
from edge <math>e</math> to edge <math>f</math> is given by the absolute value of the 
quantum transition amplitude squared, <math>|\sigma_{ef}|^2</math>.  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](/source/photonic_crystals).<ref name="KunyanskyKuchment2002">{{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 }}</ref>  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 <math>{\mathbb R}^2</math> are common models of
periodic systems and quantum graphs have been applied 
to the study the phenomena of [Anderson localization](/source/Anderson_localization) where localized 
states occur at the edge of spectral bands in the presence of disorder.

== See also ==

*''[Schild's Ladder](/source/Schild's_Ladder)'', a novel dealing with a fictional quantum graph theory
*[Feynman diagram](/source/Feynman_diagram)

== References ==
<references/>

Category:Quantum mechanics
Category:Extensions and generalizations of graphs

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Adapted from the Wikipedia article [Quantum graph](https://en.wikipedia.org/wiki/Quantum_graph) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Quantum_graph?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.