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Register a new userMethod of measurements with random perturbation: Application in photoemission experiments
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We report an application of a simultaneous perturbation stochastic approximation (SPSA) algorithm to filtering systematic noise (SN) with non-zero mean value in photoemission data. In our analysis we have used a series of 50 single-scan photoemission spectra of W(110) surface where randomly chosen SN was added. It was found that the SPSA-evaluated spectrum is in good agreement with the spectrum measured without SN. On the basis of our results a wide application of SPSA for evaluation of experimental data is anticipated.
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We introduce a simple yet powerful calculational tool useful in calculating averages of ratios and products of characteristic polynomials. The method is based on Dyson Brownian motion and Grassmann integration formula for determinants. It is intended as an alternative to other RMT techniques applicable to general gaussian measures. Resulting formulas are exact for finite matrix size N and form integral representations convenient for large N asymptotics. Quantities obtained by the method can be interpreted as averages over matrix models with an external source. We provide several explicit and novel calculations showing a range of applications.
The quantum theory of indirect measurements in physical systems is studied. The example of an indirect measurement of an observable represented by a self-adjoint operator $mathcal{N}$ with finite spectrum is analysed in detail. The Hamiltonian generating the time evolution of the system in the absence of direct measurements is assumed to be given by the sum of a term commuting with $mathcal{N}$ and a small perturbation not commuting with $mathcal{N}$. The system is subject to repeated direct (projective) measurements using a single instrument whose action on the state of the system commutes with $mathcal{N}$. If the Hamiltonian commutes with the observable $mathcal{N}$ (i.e., if the perturbation vanishes) the state of the system approaches an eigenstate of $mathcal{N}$, as the number of direct measurements tends to $infty$. If the perturbation term in the Hamiltonian does textit{not} commute with $mathcal{N}$ the system exhibits jumps between different eigenstates of $mathcal{N}$. We determine the rate of these jumps to leading order in the strength of the perturbation and show that if time is re-scaled appropriately a maximum likelihood estimate of $mathcal{N}$ approaches a Markovian jump process on the spectrum of $mathcal{N}$, as the strength of the perturbation tends to $0$.
We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. First we calculate directly the moments of the density. Then, by studying some deformation of the moments, we get a family of differential equations of first order which the densities satisfy (see Theorem 1.2), and give the densities by solving them. Further, we prove that the density is invariant after the polynomial perturbation of the weight function (see Theorem 1.5).
Using operator methods, we generally present the level densities for kinds of random matrix unitary ensembles in weak sense. As a corollary, the limit spectral distributions of random matrices from Gaussian, Laguerre and Jacobi unitary ensembles are recovered. At the same time, we study the perturbation invariability of the level densities of random matrix unitary ensembles. After the weight function associated with the 1-level correlation function is appended a polynomial multiplicative factor, the level density is invariant in the weak sense.
In the last few years, the supersymmetry method was generalized to real-symmetric, Hermitean, and Hermitean self-dual random matrices drawn from ensembles invariant under the orthogonal, unitary, and unitary symplectic group, respectively. We extend this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way. Thereby we generalize a projection formula providing a direct link and, hence, a `short cut between the probability density in ordinary space and the one in superspace. We emphasize that this point was one of the main problems and critiques of the supersymmetry method since only implicit dualities between ordinary and superspace were known before. As examples we apply this approach to the calculation of the supersymmetric analogue of a Lorentzian (Cauchy) ensemble and an ensemble with a quartic potential. Moreover we consider the partially quenched partition function of the three chiral Gaussian ensembles corresponding to four-dimensional continuum QCD. We identify a natural splitting of the chiral Lagrangian in its lowest order into a part of the physical mesons and a part associated to source terms generating the observables, e.g. the level density of the Dirac operator.
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