Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userABCD Matrices as Similarity Transformations of Wigner Matrices and Periodic Systems in Optics
234
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The beam transfer matrix, often called the $ABCD$ matrix, is a two-by-two matrix with unit determinant, and with three independent parameters. It is noted that this matrix cannot always be diagonalized. It can however be brought by rotation to a matrix with equal diagonal elements. This equi-diagonal matrix can then be squeeze-transformed to a rotation, to a squeeze, or to one of the two shear matrices. It is noted that these one-parameter matrices constitute the basic elements of the Wigners little group for space-time symmetries of elementary particles. Thus every $ABCD$ matrix can be written as a similarity transformation of one of the Wigner matrices, while the transformation matrix is a rotation preceded by a squeeze. This mathematical property enables us to compute scattering processes in periodic systems. Laser cavities and multilayer optics are discussed in detail. For both cases, it is shown possible to write the one-cycle transfer matrix as a similarity transformation of one of the Wigner matrices. It is thus possible to calculate the $ABCD$ matrix for an arbitrary number of cycles.
rate research
Read More
We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.
We consider the single eigenvalue fluctuations of random matrices of general Wigner-type, under a one-cut assumption on the density of states. For eigenvalues in the bulk, we prove that the asymptotic fluctuations of a single eigenvalue around its classical location are Gaussian with a universal variance which agrees with the GOE and GUE cases. Our method is based on a dynamical approach to mesoscopic linear spectral statistics which reduces their behavior on short scales to that on larger scales. We prove a central limit theorem for linear spectral statistics on larger scales via resolvent techniques and show that for certain classes of test functions, the leading order contribution to the variance is universal, agreeing with the GOE/GUE cases.
We consider a general Langevin dynamics for the one-dimensional N-particle Coulomb gas with confining potential $V$ at temperature $beta$. These dynamics describe for $beta=2$ the time evolution of the eigenvalues of $Ntimes N$ random Hermitian matrices. The equilibrium partition function -- equal to the normalization constant of the Laughlin wave function in fractional quantum Hall effect -- is known to satisfy an infinite number of constraints called Virasoro or loop constraints. We introduce here a dynamical generating function on the space of random trajectories which satisfies a large class of constraints of geometric origin. We focus in this article on a subclass induced by the invariance under the Schrodinger-Virasoro algebra.
We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the infinite matrix which encodes all the moment data of a Borel measure on R^d or C. To encode the salient features of a given IFS into precise moment data, we establish an interdependence between IFS equilibrium measures, the encoding of the sequence of moments of these measures into operators, and a new correspondence between the IFS moments and this family of operators in Hilbert space. For a given IFS, our aim is to establish a functorial correspondence in such a way that the geometric transformations of the IFS turn into transformations of moment matrices, or rather transformations of the operators that are associated with them. We first examine the classical existence problem for moments, culminating in a new proof of the existence of a Borel measure on R or C with a specified list of moments. Next, we consider moment problems associated with affine and non-affine IFSs. Our main goal is to determine conditions under which an intertwining relation is satisfied by the moment matrix of an equilibrium measure of an IFS. Finally, using the famous Hilbert matrix as our prototypical example, we study boundedness and spectral properties of moment matrices viewed as Kato-Friedrichs operators on weighted l^2 spaces.
We show that Wigner semi-circle law holds for Hermitian matrices with dependent entries, provided the deviation of the cumulants from the normalised Gaussian case obeys a simple power law bound in the size of the matrix. To establish this result, we use replicas interpreted as a zero-dimensional quantum field theoretical model whose effective potential obey a renormalisation group equation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
