Quantum Graph - Theorems

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, see, 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 . 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.