In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) 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 as models of free electrons in organic molecules in the 1930s. They also arise in a variety of mathematical contexts,[1] e.g. as model systems in quantum chaos, in the study of waveguides, in photonic crystals and in Anderson localization, or as limit on shrinking thin wires. Quantum graphs have become prominent models in mesoscopic physics used to obtain a theoretical understanding of nanotechnology. Another, more simple notion of quantum graphs was introduced by Freedman et al.[2]

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 (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 (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

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A metric graph embedded in the plane with three open edges. The dashed line denotes the metric distance between two points   and  .

A metric graph is a graph consisting of a set   of vertices and a set   of edges where each edge   has been associated with an interval   so that   is the coordinate on the interval, the vertex   corresponds to   and   to   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   on the graph,   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   attached to a single vertex at  . A graph with one or more such open edges is referred to as an open graph.

Quantum graphs

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Quantum graphs are metric graphs equipped with a differential (or pseudo-differential) operator acting on functions on the graph. A function   on a metric graph is defined as the  -tuple of functions   on the 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 Schrödinger 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.

Theorems

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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,[3] 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 .[4][5] 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 Schrödinger 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

 

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 when propagating 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.

Applications

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Naphthalene molecule

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[6] 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.[7] 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 edge of spectral bands in the presence of disorder.

See also

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References

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  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.
  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.
  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.
  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.
  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.
  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.
  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.