In mathematics, the Redheffer star product is a binary operation on linear operators that arises in connection to solving coupled systems of linear equations. It was introduced by Raymond Redheffer in 1959,[1] and has subsequently been widely adopted in computational methods for scattering matrices. Given two scattering matrices from different linear scatterers, the Redheffer star product yields the combined scattering matrix produced when some or all of the output channels of one scatterer are connected to inputs of another scatterer.
Suppose are the block matrices
and
,
whose blocks have the same shape when
.
The Redheffer star product is then defined by:
[1]
,
assuming that are invertible,
where is an identity matrix conformable
to or , respectively.
This can be rewritten several ways making use of the so-called
push-through identity.
Redheffer's definition extends beyond matrices to
linear operators on a Hilbert space.
[2]
.
By definition, are linear endomorphisms of ,
making linear endomorphisms of ,
where is the direct sum.
However, the star product still makes sense as long as the transformations are compatible,
which is possible when
and
so that .
If is the left matrix inverse of such that
, has a right inverse, and
exists, then .
[2]
Similarly, if is the left matrix inverse of such
that , has a right inverse, and
exists, then .
The star product arises from solving multiple linear systems of equations that share
variables in common.
Often, each linear system models the behavior of one subsystem in a physical process
and by connecting the multiple subsystems into a whole, one can eliminate variables
shared across subsystems in order to obtain the overall linear system.
For instance, let be elements of a Hilbert space such that
[4]
and
giving the following equations in variables:
.
By substituting the first equation into the last we find:
.
By substituting the last equation into the first we find:
.
Eliminating by substituting the two preceding equations
into those for results in the Redheffer star product
being the matrix such that:
[1]
Many scattering processes take on a form that motivates a different
convention for the block structure of the linear system of a scattering matrix.
Typically a physical device that performs a linear transformation on inputs, such as
linear dielectric media on electromagnetic waves or in quantum mechanical scattering,
can be encapsulated as a system which interacts with the environment through various
ports, each of which accepts inputs and returns outputs. It is conventional to use a different notation for the Hilbert space, , whose subscript
labels a port on the device.
Additionally, any element, , has an additional superscript labeling the direction of travel (where + indicates moving from port i to i+1 and - indicates the reverse).
The equivalent notation for a Redheffer transformation,
,
used in the previous section is
.
The action of the S-matrix,
,
is defined with an additional flip compared to Redheffer's definition:[5]
,
so
.
Note that for in order for the off-diagonal identity matrices to be defined,
we require be the same underlying Hilbert space.
(The subscript does not imply any difference, but is just a label for bookkeeping.)
The star product, ,
for two S-matrices, , is given by
[5]
These are analogues of the properties of for
Most of them follow from the correspondence
.
, the exchange operator, is also the S-matrix star identity defined below.
For the rest of this section, are S-matrices.
In physics, the Redheffer star product appears when constructing a total
scattering matrix from two or more subsystems.
If system has a scattering matrix and system
has scattering matrix , then the combined system
has scattering matrix .
[5]
Many physical processes, including radiative transfer, neutron diffusion, circuit theory, and others are described by scattering processes whose formulation depends on the dimension of the process and the representation of the operators.[6] For probabilistic problems, the scattering equation may appear in a Kolmogorov-type equation.
The Redheffer star product can be used to solve for the propagation of electromagnetic fields in stratified, multilayered media.[9] Each layer in the structure has its own scattering matrix and the total structure's scattering matrix can be described as the star product between all of the layers.[10] A free software program that simulates electromagnetism in layered media is the
Stanford Stratified Structure Solver.
Kinetic models of consecutive semiconductor interfaces can use a scattering matrix formulation to model the motion of electrons between the semiconductors.
[11]
In the analysis of Schrödinger operators on graphs, the scattering matrix of a graph can be obtained as a generalized star product of the scattering matrices corresponding to its subgraphs.[12]