Quantum Graph - Quantum Graphs

Quantum Graphs

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 Schrŏdinger 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.