Operator (computer programming)

About: Operator (computer programming) is a research topic. Over the lifetime, 40896 publications have been published within this topic receiving 671452 citations. The topic is also known as: operator symbol & operator name.

Resolvent Positive Operators

Abstract: Resolvent positive operators on an ordered Banach space (with generating and normal positive cone) are by definition linear (possibly unbounded) operators whose resolvent exists and is positive on a right half-line. Even though these operators are defined by a simple (purely algebraic) condition, analogues of the basic results of the theory of Q-semigroups can be proved for them. In fact, if A is resolvent positive and has a dense domain, then the Cauchy problem associated with A has a unique solution for every initial value in the domain of A, and the solution is positive if the initial value is positive. Also the converse is true (if we assume that A has a non-empty resolvent set and D(A) n E+ is dense in £+). Moreover, every positive resolvent is a Laplace-Stieltjes transform of a so-called integrated semigroup; and conversely every such (increasing, non-degenerate) integrated semigroup defines a unique resolvent positive operator.

On the numerical evaluation of Fredholm determinants

Abstract: Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painleve transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nystrom method for the solution of Fredholm equations of the second kind. Using Gauss—Legendre or Clenshaw—Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk scaling limit and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the two-point correlation functions of the more recently studied Airy and Airy 1 processes.

Gaussian-Wigner distributions in quantum mechanics and optics

Abstract: Gaussian kernels representing operators on the Hilbert space scrH=L2(openRn) are studied. Necessary and sufficient conditions on such a kernel in order that the corresponding operator be positive semidefinite, corresponding to a density matrix (cross-spectral density) in quantum mechanics (optics), are derived. The Wigner distribution method is shown to be a convenient framework for characterizing Gaussian kernels and their unitary evolution under Sp(2n,openR) action. The nontrivial role played by a phase term in the kernel is brought out. The entire analysis is presented in a form which is directly applicable to n-dimensional oscillator systems in quantum mechanics and to Gaussian Schell-model partially coherent fields in optics.