Quantum graph

<h2 id="metric-graphs">Metric graphs</h2>

A metric graph
is a <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graph</a> consisting of a set 
 
 
 
 V
 
 
 {\displaystyle V}
 
 of vertices and 
a set 
 
 
 
 E
 
 
 {\displaystyle E}
 
 of edges where each edge 
 
 
 
 e
 =
 (
 
 v
 
 1
 
 
 ,
 
 v
 
 2
 
 
 )
 ∈
 E
 
 
 {\displaystyle e=(v_{1},v_{2})\in E}
 
 has been associated 
with an interval 
 
 
 
 [
 0
 ,
 
 L
 
 e
 
 
 ]
 
 
 {\displaystyle [0,L_{e}]}
 
 so that 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle x_{e}}
 
 is the coordinate on the 
interval, the vertex 
 
 
 
 
 v
 
 1
 
 
 
 
 {\displaystyle v_{1}}
 
 corresponds to 
 
 
 
 
 x
 
 e
 
 
 =
 0
 
 
 {\displaystyle x_{e}=0}
 
 and

v
 
 2
 
 
 
 
 {\displaystyle v_{2}}
 
 to 
 
 
 
 
 x
 
 e
 
 
 =
 
 L
 
 e
 
 
 
 
 {\displaystyle 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
 
 
 {\displaystyle x,y}
 
 on the graph, 
 
 
 
 ρ
 (
 x
 ,
 y
 )
 
 
 {\displaystyle \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
 ,
 ∞
 )
 
 
 {\displaystyle [0,\infty )}
 
 attached to a single vertex at 
 
 
 
 
 x
 
 e
 
 
 =
 0
 
 
 {\displaystyle x_{e}=0}
 
. 
A graph with one or more 
such open edges is referred to as an open graph.

<h2 id="quantum-graphs">Quantum graphs</h2>
Quantum graphs are metric graphs equipped with a differential 
(or <a href="/facts/Pseudo-differential_operator/wEPLVm5c">pseudo-differential</a>) operator acting on functions on the graph. 
A function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on a metric graph is defined as the 
 
 
 
 
 |
 
 E
 
 |
 
 
 
 {\displaystyle |E|}
 
-tuple of functions

f
 
 e
 
 
 (
 
 x
 
 e
 
 
 )
 
 
 {\displaystyle f_{e}(x_{e})}
 
 on the intervals. 
The <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> of the graph is 
 
 
 
 
 ⨁
 
 e
 ∈
 E
 
 
 
 L
 
 2
 
 
 (
 [
 0
 ,
 
 L
 
 e
 
 
 ]
 )
 
 
 {\displaystyle \bigoplus _{e\in E}L^{2}([0,L_{e}])}

where the inner product of two functions is

⟨
        f
        ,
        g
        ⟩
        =
        
          ∑
          
            e
            ∈
            E
          
        
        
          ∫
          
            0
          
          
            
              L
              
                e
              
            
          
        
        
          f
          
            e
          
          
            ∗
          
        
        (
        
          x
          
            e
          
        
        )
        
          g
          
            e
          
        
        (
        
          x
          
            e
          
        
        )
        
        d
        
          x
          
            e
          
        
        ,
      
    
    {\displaystyle \langle f,g\rangle =\sum _{e\in E}\int _{0}^{L_{e}}f_{e}^{*}(x_{e})g_{e}(x_{e})\,dx_{e},}

L
 
 e
 
 
 
 
 {\displaystyle L_{e}}
 
 may be infinite in the case of an open edge. The simplest example of an operator on a metric graph is the <a href="/facts/Laplace_operator/kov9u3PT">Laplace operator</a>. The operator on an edge is 
 
 
 
 −
 
 
 
 
 
 d
 
 
 
 2
 
 
 
 
 
 d
 
 
 
 x
 
 e
 
 
 2
 
 
 
 
 
 
 
 {\displaystyle -{\frac {{\textrm {d}}^{2}}{{\textrm {d}}x_{e}^{2}}}}
 
 where 
 
 
 
 
 x
 
 e
 
 
 
 
 {\displaystyle 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 <a href="/facts/Sobolev_space/34qxWCyr">Sobolev space</a> 
 
 
 
 
 H
 
 2
 
 
 
 
 {\displaystyle 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 <a href="/facts/Dirichlet_boundary_conditions/m9wsgG89">Dirichlet boundary conditions</a>, 
 
 
 
 
 f
 
 e
 
 
 (
 0
 )
 =
 
 f
 
 e
 
 
 (
 
 L
 
 e
 
 
 )
 =
 0
 
 
 {\displaystyle 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
          
        
        )
        =
        sin
        ⁡
        
          (
          
            
              
                n
                π
                
                  x
                  
                    e
                  
                
              
              
                L
                
                  e
                
              
            
          
          )
        
      
    
    {\displaystyle f_{e}(x_{e})=\sin \left({\frac {n\pi x_{e}}{L_{e}}}\right)}

for integer 
 
 
 
 n
 
 
 {\displaystyle 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 
 
 
 
 
 
 
 
 n
 
 2
 
 
 
 π
 
 2
 
 
 
 
 L
 
 e
 
 
 2
 
 
 
 
 
 
 {\displaystyle {\frac {n^{2}\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 <a href="/facts/Neumann_boundary_condition/0e0YBW4G">Neumann</a> or natural matching conditions. A function 
 
 
 
 f
 
 
 {\displaystyle 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,

∑
          
            e
            ∼
            v
          
        
        
          f
          ′
        
        (
        v
        )
        =
        0
         
        ,
      
    
    {\displaystyle \sum _{e\sim v}f'(v)=0\ ,}

where 
 
 
 
 
 f
 ′
 
 (
 v
 )
 =
 
 f
 ′
 
 (
 0
 )
 
 
 {\displaystyle f'(v)=f'(0)}
 
 if the vertex 
 
 
 
 v
 
 
 {\displaystyle v}
 
 is at 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
 and 
 
 
 
 
 f
 ′
 
 (
 v
 )
 =
 −
 
 f
 ′
 
 (
 
 L
 
 e
 
 
 )
 
 
 {\displaystyle f'(v)=-f'(L_{e})}
 
 if 
 
 
 
 v
 
 
 {\displaystyle v}
 
 is at 
 
 
 
 x
 =
 
 L
 
 e
 
 
 
 
 {\displaystyle x=L_{e}}
 
.
The properties of other operators on metric graphs have also been studied. 

<ul><li>These include the more general class of Schrödinger operators,</li></ul>

(
            
              i
              
                
                  
                    d
                  
                  
                    
                      
                        d
                      
                    
                    
                      x
                      
                        e
                      
                    
                  
                
              
              +
              
                A
                
                  e
                
              
              (
              
                x
                
                  e
                
              
              )
            
            )
          
          
            2
          
        
        +
        
          V
          
            e
          
        
        (
        
          x
          
            e
          
        
        )
         
        ,
      
    
    {\displaystyle \left(i{\frac {\textrm {d}}{{\textrm {d}}x_{e}}}+A_{e}(x_{e})\right)^{2}+V_{e}(x_{e})\ ,}

where 
 
 
 
 
 A
 
 e
 
 
 
 
 {\displaystyle A_{e}}
 
 is a "magnetic vector potential" on the edge and 
 
 
 
 
 V
 
 e
 
 
 
 
 {\displaystyle V_{e}}
 
 is a scalar potential.

<ul><li>Another example is the <a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a> 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 <a href="/facts/Electron/L0KBo5cW">electron</a>.</li>
<li>The Dirichlet-to-Neumann operator on a graph is a pseudo-differential operator that arises in the study of <a href="/facts/Photonic_crystals/R763CtuW">photonic crystals</a>.</li></ul>
<h2 id="theorems">Theorems</h2>
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,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> which automatically yields an operator in variational form.
Let 
 
 
 
 v
 
 
 {\displaystyle v}
 
 be a vertex with 
 
 
 
 d
 
 
 {\displaystyle d}
 
 edges emanating from it. For simplicity we choose the coordinates on the edges so that 
 
 
 
 v
 
 
 {\displaystyle v}
 
 lies at 
 
 
 
 
 x
 
 e
 
 
 =
 0
 
 
 {\displaystyle x_{e}=0}
 
 for each edge meeting at 
 
 
 
 v
 
 
 {\displaystyle v}
 
. For a function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 on the graph let

f
        
        =
        (
        
          f
          
            
              e
              
                1
              
            
          
        
        (
        0
        )
        ,
        
          f
          
            
              e
              
                2
              
            
          
        
        (
        0
        )
        ,
        …
        ,
        
          f
          
            
              e
              
                d
              
            
          
        
        (
        0
        )
        
          )
          
            T
          
        
        ,
        
        
          
            f
          
          ′
        
        =
        (
        
          f
          
            
              e
              
                1
              
            
          
          ′
        
        (
        0
        )
        ,
        
          f
          
            
              e
              
                2
              
            
          
          ′
        
        (
        0
        )
        ,
        …
        ,
        
          f
          
            
              e
              
                d
              
            
          
          ′
        
        (
        0
        )
        
          )
          
            T
          
        
        .
      
    
    {\displaystyle \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}.}

Matching conditions at 
 
 
 
 v
 
 
 {\displaystyle v}
 
 can be specified by a pair of matrices

A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 through the linear equation,

A
        
          f
        
        +
        B
        
          
            f
          
          ′
        
        =
        
          0
        
        .
      
    
    {\displaystyle A\mathbf {f} +B\mathbf {f} '=\mathbf {0} .}

The matching conditions define a self-adjoint operator if

(
        A
        ,
        B
        )
      
    
    {\displaystyle (A,B)}
  
 has the maximal rank 
  
    
      
        d
      
    
    {\displaystyle d}
  
 and 
  
    
      
        A
        
          B
          
            ∗
          
        
        =
        B
        
          A
          
            ∗
          
        
        .
      
    
    {\displaystyle 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
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> The eigenvalue problem on an edge is,

−
        
          
            
              d
              
                2
              
            
            
              d
              
                x
                
                  e
                
                
                  2
                
              
            
          
        
        
          f
          
            e
          
        
        (
        
          x
          
            e
          
        
        )
        =
        
          k
          
            2
          
        
        
          f
          
            e
          
        
        (
        
          x
          
            e
          
        
        )
        .
        
      
    
    {\displaystyle -{\frac {d^{2}}{dx_{e}^{2}}}f_{e}(x_{e})=k^{2}f_{e}(x_{e}).\,}

So a solution on the edge can be written as a linear combination of <a href="/facts/Plane_waves/0aDUbNhQ">plane waves</a>.

f
          
            e
          
        
        (
        
          x
          
            e
          
        
        )
        =
        
          c
          
            e
          
        
        
          
            
              e
            
          
          
            i
            k
            
              x
              
                e
              
            
          
        
        +
        
          
            
              
                c
                ^
              
            
          
          
            e
          
        
        
          
            
              e
            
          
          
            −
            i
            k
            
              x
              
                e
              
            
          
        
        .
        
      
    
    {\displaystyle f_{e}(x_{e})=c_{e}{\textrm {e}}^{ikx_{e}}+{\hat {c}}_{e}{\textrm {e}}^{-ikx_{e}}.\,}

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

S
        (
        k
        )
        =
        −
        (
        A
        +
        i
        k
        B
        
          )
          
            −
            1
          
        
        (
        A
        −
        i
        k
        B
        )
        .
        
      
    
    {\displaystyle S(k)=-(A+ikB)^{-1}(A-ikB).\,}

The scattering matrix relates the vectors of incoming and outgoing plane-wave 
coefficients at 
 
 
 
 v
 
 
 {\displaystyle v}
 
, 
 
 
 
 
 c
 
 =
 S
 (
 k
 )
 
 
 
 
 c
 
 ^
 
 
 
 
 
 {\displaystyle \mathbf {c} =S(k){\hat {\mathbf {c} }}}
 
.
For self-adjoint matching conditions 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is unitary. An element of

σ
          
            (
            u
            v
            )
            (
            v
            w
            )
          
        
      
    
    {\displaystyle \sigma _{(uv)(vw)}}
  
 of 
  
    
      
        S
      
    
    {\displaystyle S}
  
 is a complex transition amplitude 
from a directed edge 
  
    
      
        (
        u
        v
        )
      
    
    {\displaystyle (uv)}

to the edge 
 
 
 
 (
 v
 w
 )
 
 
 {\displaystyle (vw)}
 
 which in general depends on 
 
 
 
 k
 
 
 {\displaystyle k}
 
. 
However, for a large class of matching conditions 
the S-matrix is independent of 
 
 
 
 k
 
 
 {\displaystyle k}
 
. 
With Neumann matching conditions for example

A
        =
        
          (
          
            
              
                
                  1
                
                
                  −
                  1
                
                
                  0
                
                
                  0
                
                
                  …
                
              
              
                
                  0
                
                
                  1
                
                
                  −
                  1
                
                
                  0
                
                
                  …
                
              
              
                
                
                
                  ⋱
                
                
                  ⋱
                
                
              
              
                
                  0
                
                
                  …
                
                
                  0
                
                
                  1
                
                
                  −
                  1
                
              
              
                
                  0
                
                
                  …
                
                
                  0
                
                
                  0
                
                
                  0
                
              
            
          
          )
        
        ,
        
        B
        =
        
          (
          
            
              
                
                  0
                
                
                  0
                
                
                  …
                
                
                  0
                
              
              
                
                  ⋮
                
                
                  ⋮
                
                
                
                  ⋮
                
              
              
                
                  0
                
                
                  0
                
                
                  …
                
                
                  0
                
              
              
                
                  1
                
                
                  1
                
                
                  …
                
                
                  1
                
              
            
          
          )
        
        .
      
    
    {\displaystyle 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
 
 
 {\displaystyle S}
 
 
produces 
 
 
 
 k
 
 
 {\displaystyle k}
 
-independent transition amplitudes

σ
          
            (
            u
            v
            )
            (
            v
            w
            )
          
        
        =
        
          
            2
            d
          
        
        −
        
          δ
          
            u
            w
          
        
        .
        
      
    
    {\displaystyle \sigma _{(uv)(vw)}={\frac {2}{d}}-\delta _{uw}.\,}

where 
 
 
 
 
 δ
 
 u
 w
 
 
 
 
 {\displaystyle \delta _{uw}}
 
 is the Kronecker delta function that is one if 
 
 
 
 u
 =
 w
 
 
 {\displaystyle u=w}
 
 and 
zero otherwise. From the transition amplitudes we may define a

2
 
 |
 
 E
 
 |
 
 ×
 2
 
 |
 
 E
 
 |
 
 
 
 {\displaystyle 2|E|\times 2|E|}
 
 matrix

U
          
            (
            u
            v
            )
            (
            l
            m
            )
          
        
        (
        k
        )
        =
        
          δ
          
            v
            l
          
        
        
          σ
          
            (
            u
            v
            )
            (
            v
            m
            )
          
        
        (
        k
        )
        
          
            
              e
            
          
          
            i
            k
            
              L
              
                (
                u
                v
                )
              
            
          
        
        .
        
      
    
    {\displaystyle U_{(uv)(lm)}(k)=\delta _{vl}\sigma _{(uv)(vm)}(k){\textrm {e}}^{ikL_{(uv)}}.\,}

U
 
 
 {\displaystyle 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
 
 |
 
 
 
 {\displaystyle 2|E|}
 
 plane-wave coefficients for the 
graph where 
 
 
 
 
 c
 
 (
 u
 v
 )
 
 
 
 
 {\displaystyle c_{(uv)}}
 
 is the coefficient of 
the plane wave traveling from 
 
 
 
 u
 
 
 {\displaystyle u}
 
 to 
 
 
 
 v
 
 
 {\displaystyle v}
 
. 
The phase 
 
 
 
 
 
 
 e
 
 
 
 i
 k
 
 L
 
 (
 u
 v
 )
 
 
 
 
 
 
 {\displaystyle {\textrm {e}}^{ikL_{(uv)}}}
 
 is the phase acquired by the plane wave 
when propagating from vertex 
 
 
 
 u
 
 
 {\displaystyle u}
 
 to vertex 
 
 
 
 v
 
 
 {\displaystyle v}
 
.
Quantization condition: An eigenfunction on the graph 
can be defined through its associated 
 
 
 
 2
 
 |
 
 E
 
 |
 
 
 
 {\displaystyle 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.
        
      
    
    {\displaystyle |U(k)-I|=0.\,}

Eigenvalues 
 
 
 
 
 k
 
 j
 
 
 
 
 {\displaystyle k_{j}}
 
 occur at values of 
 
 
 
 k
 
 
 {\displaystyle k}
 
 where the matrix 
 
 
 
 U
 (
 k
 )
 
 
 {\displaystyle U(k)}
 
 has an 
eigenvalue one. We will order the spectrum with

0
 ⩽
 
 k
 
 0
 
 
 ⩽
 
 k
 
 1
 
 
 ⩽
 …
 
 
 {\displaystyle 0\leqslant k_{0}\leqslant k_{1}\leqslant \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
 
 
 {\displaystyle k}
 
.
The trace formula links the spectrum with periodic orbits on the graph.

d
        (
        k
        )
        :=
        
          ∑
          
            j
            =
            0
          
          
            ∞
          
        
        δ
        (
        k
        −
        
          k
          
            j
          
        
        )
        =
        
          
            L
            π
          
        
        +
        
          
            1
            π
          
        
        
          ∑
          
            p
          
        
        
          
            
              L
              
                p
              
            
            
              r
              
                p
              
            
          
        
        
          A
          
            p
          
        
        cos
        ⁡
        (
        k
        
          L
          
            p
          
        
        )
        .
      
    
    {\displaystyle 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}).}

d
 (
 k
 )
 
 
 {\displaystyle 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 
 
 
 
 
 
 L
 π
 
 
 
 
 {\displaystyle {\frac {L}{\pi }}}
 
 
is the mean separation of eigenvalues and the oscillating part is a sum 
over all periodic orbits 
 
 
 
 p
 =
 (
 
 e
 
 1
 
 
 ,
 
 e
 
 2
 
 
 ,
 …
 ,
 
 e
 
 n
 
 
 )
 
 
 {\displaystyle p=(e_{1},e_{2},\dots ,e_{n})}
 
 on the graph.

L
          
            p
          
        
        =
        
          ∑
          
            e
            ∈
            p
          
        
        
          L
          
            e
          
        
      
    
    {\displaystyle L_{p}=\sum _{e\in p}L_{e}}
  
 is the length of the orbit and

L
        =
        
          ∑
          
            e
            ∈
            E
          
        
        
          L
          
            e
          
        
      
    
    {\displaystyle L=\sum _{e\in E}L_{e}}
  
 is
the total length of the graph.  For an orbit generated by repeating a 
shorter primitive orbit, 
  
    
      
        
          r
          
            p
          
        
      
    
    {\displaystyle r_{p}}
  
 counts the number of repartitions.

A
 
 p
 
 
 =
 
 σ
 
 
 e
 
 1
 
 
 
 e
 
 2
 
 
 
 
 
 σ
 
 
 e
 
 2
 
 
 
 e
 
 3
 
 
 
 
 …
 
 σ
 
 
 e
 
 n
 
 
 
 e
 
 1
 
 
 
 
 
 
 {\displaystyle A_{p}=\sigma _{e_{1}e_{2}}\sigma _{e_{2}e_{3}}\dots \sigma _{e_{n}e_{1}}}
 
 is 
the product of the transition amplitudes at the vertices of the graph around 
the orbit.

<h2 id="applications">Applications</h2>

Quantum graphs were first employed in the 1930s 
to model the spectrum of free electrons in organic molecules like 
<a href="/facts/Naphthalene/RU2sV8mN">Naphthalene</a>, 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 <a href="/facts/Nanotechnology/Co4FFNfY">nanotechnology</a>.
In 1997<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> Kottos and Smilansky proposed quantum graphs as a model to study 
<a href="/facts/Quantum_chaos/p1JVxSTD">quantum chaos</a>, the quantum mechanics of systems that 
are classically chaotic. Classical motion on the graph can be defined as 
a probabilistic <a href="/facts/Markov_chain/5oP5hntk">Markov chain</a> where the probability of scattering 
from edge 
 
 
 
 e
 
 
 {\displaystyle e}
 
 to edge 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is given by the absolute value of the 
quantum transition amplitude squared, 
 
 
 
 
 |
 
 
 σ
 
 e
 f
 
 
 
 
 |
 
 
 2
 
 
 
 
 {\displaystyle |\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 <a href="/facts/Photonic_crystals/R763CtuW">photonic crystals</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> 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 
 
 
 
 
 
 
 R
 
 
 
 2
 
 
 
 
 {\displaystyle {\mathbb {R} }^{2}}
 
 are common models of
periodic systems and quantum graphs have been applied 
to the study the phenomena of <a href="/facts/Anderson_localization/LB5Cf5Q5">Anderson localization</a> where localized 
states occur at the edge of spectral bands in the presence of disorder.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Schild%2527s_Ladder/QpRPJNpp">Schild's Ladder</a>, a novel dealing with a fictional quantum graph theory</li>
<li><a href="/facts/Feynman_diagram/BRFNvDdr">Feynman diagram</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Berkolaiko, Gregory; Carlson, Robert; Kuchment, Peter; Fulling, Stephen (2006). Quantum Graphs and Their Applications (Contemporary Mathematics): Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their Applications. Vol. 415. American Mathematical Society. ISBN 978-0821837658. <a href="978-0821837658" target="_blank">978-0821837658</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Freedman, Michael; Lovász, László; Schrijver, Alexander (2007). "Reflection positivity, rank connectivity, and homomorphism of graphs". Journal of the American Mathematical Society. 20 (1): 37–52. arXiv:math/0404468. Bibcode:2007JAMS...20...37F. doi:10.1090/S0894-0347-06-00529-7. ISSN 0894-0347. MR 2257396. S2CID 8208923. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Kuchment, Peter (2004). "Quantum graphs: I. Some basic structures". Waves in Random Media. 14 (1): S107 – S128. Bibcode:2004WRM....14S.107K. doi:10.1088/0959-7174/14/1/014. ISSN 0959-7174. S2CID 16874849. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Kottos, Tsampikos; Smilansky, Uzy (1999). "Periodic Orbit Theory and Spectral Statistics for Quantum Graphs". Annals of Physics. 274 (1): 76–124. arXiv:chao-dyn/9812005. Bibcode:1999AnPhy.274...76K. doi:10.1006/aphy.1999.5904. ISSN 0003-4916. S2CID 17510999. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Gnutzmann∥, Sven; Smilansky, Uzy (2006). "Quantum graphs: Applications to quantum chaos and universal spectral statistics". Advances in Physics. 55 (5–6): 527–625. arXiv:nlin/0605028. Bibcode:2006AdPhy..55..527G. doi:10.1080/00018730600908042. ISSN 0001-8732. S2CID 119424306. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Kottos, Tsampikos; Smilansky, Uzy (1997). "Quantum Chaos on Graphs". Physical Review Letters. 79 (24): 4794–4797. Bibcode:1997PhRvL..79.4794K. doi:10.1103/PhysRevLett.79.4794. ISSN 0031-9007. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Kuchment, Peter; Kunyansky, Leonid (2002). "Differential Operators on Graphs and Photonic Crystals". Advances in Computational Mathematics. 16 (24): 263–290. doi:10.1023/A:1014481629504. S2CID 17506556. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Quantum graph open-in-new

Quantum graph