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12
Notes on infinite determinants of Hilbert space operators
 Adv. Math
, 1977
"... We present a novel approach to obtaining the basic facts (including Lidskii's theorem on the equality of the matrix and spectral traces) about determinants and traces of trace class operators on a separable Hilbert space. We also discuss Fredholm theory, "regularized " determinants an ..."
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Cited by 60 (2 self)
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We present a novel approach to obtaining the basic facts (including Lidskii's theorem on the equality of the matrix and spectral traces) about determinants and traces of trace class operators on a separable Hilbert space. We also discuss Fredholm theory, "regularized " determinants and Fredholm theory on the trace ideals, c#~(p < oo). 1.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... 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 ..."
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Cited by 43 (6 self)
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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 Painlevé 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 Nyström 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 and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
Quantum Mechanics and Semiclassics of Hyperbolic nDisk Scattering Systems
 Physics Reports 309
, 1999
"... The scattering problems of a scalar point particle from an assembly of 1 < n < ∞ nonoverlapping and disconnected hard disks, fixed in the twodimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Their simplicity allows for a detailed study of the ..."
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Cited by 17 (1 self)
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The scattering problems of a scalar point particle from an assembly of 1 < n < ∞ nonoverlapping and disconnected hard disks, fixed in the twodimensional plane, belong to the simplest realizations of classically hyperbolic scattering systems. Their simplicity allows for a detailed study of the quantum mechanics, semiclassics and classics of the scattering. Here, we investigate the connection between the spectral properties of the quantummechanical scattering matrix and its semiclassical equivalent based on the semiclassical zetafunction of Gutzwiller and Voros. We construct the scattering matrix and its determinant for any nonoverlapping ndisk system (with n < ∞) and rewrite the determinant in such a way that it separates into the product over n determinants of 1disk scattering matrices – representing the incoherent part of the scattering from the ndisk system – and the ratio of two mutually complex conjugate determinants of the genuine multiscattering matrix M which is of KorringaKohnRostokertype and which represents the coherent multidisk aspect of the ndisk scattering. Our quantummechanical calculation is welldefined at every step, as the onshell T–matrix and the multiscattering kernel M−1 are shown to be traceclass. The multiscattering determinant can be organized in terms of
A Priori Error Analysis of Stochastic Galerkin Mixed Approximations of Elliptic PDEs with Random Data
, 2011
"... Abstract. We construct stochastic Galerkin approximations to the solution of a firstorder system of PDEs with random coefficients. Under the standard finitedimensional noise assumption, we transform the variational saddle point problem to a parametric deterministic one. Approximations are construc ..."
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Cited by 4 (1 self)
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Abstract. We construct stochastic Galerkin approximations to the solution of a firstorder system of PDEs with random coefficients. Under the standard finitedimensional noise assumption, we transform the variational saddle point problem to a parametric deterministic one. Approximations are constructed by combining mixed finite elements on the computational domain with Mvariate tensor product polynomials. We study the infsup stability and wellposedness of the continuous and finitedimensional problems, the regularity of solutions with respect to the M parameters describing the random coefficients, and establish a priori error estimates for stochastic Galerkin finite element approximations.
Computing with functions in two dimensions
, 2014
"... New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs nearoptimal low rank approxim ..."
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Cited by 1 (0 self)
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New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs nearoptimal low rank approximations. Operations such as integration, differentiation, and function evaluation are particularly efficient. Explicit convergence rates are shown for the singular values of differentiable and separately analytic functions, and examples are given to demonstrate some paradoxical features of low rank approximation theory. Analogues of QR, LU, and Cholesky factorizations are introduced for matrices that are continuous in one or both directions, deriving a continuous linear algebra. New notions of triangular structures are proposed and the convergence of the infinite series associated with these factorizations is proved under certain smoothness assumptions.
Restoration of optical objects using regularization
, 1977
"... Using the regularization theory for improperly posed problems, we discuss object restoration beyond the diffraction limit in the presence of noise. Only the case of onedimensional coherent objects is considered. We focus attention on the estimation of the error on the restored objects, and we show ..."
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Cited by 1 (1 self)
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Using the regularization theory for improperly posed problems, we discuss object restoration beyond the diffraction limit in the presence of noise. Only the case of onedimensional coherent objects is considered. We focus attention on the estimation of the error on the restored objects, and we show that, in most realistic cases, it is at best proportional to an inverse power of I In t l, where E is the error on the data (logarithmic continuity). Finally we suggest the extension of this result to other inverse problems. Let us consider an ideal, diffractionlimited, spaceinvariant imaging system and a onedimensional, coherent object f, identically zero outside the interval (1,1). Its noiseless image is given by Af, where A is the linear integral operator, (Af)(x) = sin[c (x x')] f(x')dx'. (1) The quantity d = 7r/c is the Rayleighresolution distance. To restore the object, one has to invert the operator A. As is known, this problem is improperly posed. When the image is known only approximately,
METRIC AND PROBABILISTIC INFORMATION ASSOCIATED WITH FREDHOLM INTEGRAL EQUATIONS OF THE FIRST
, 2005
"... The problem of evaluating the information associated with Fredholm integral equations of the first kind, when the integral operator is self– adjoint and compact, is considered here. The data function is assumed to be perturbed gently by an additive noise so that it still belongs to the range of the ..."
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Cited by 1 (1 self)
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The problem of evaluating the information associated with Fredholm integral equations of the first kind, when the integral operator is self– adjoint and compact, is considered here. The data function is assumed to be perturbed gently by an additive noise so that it still belongs to the range of the operator. First we estimate upper and lower bounds for the ε–capacity (and then for the metric information), and explicit computations in some specific cases are given; then the problem is reformulated from a probabilistic viewpoint and use is made of the probabilistic information theory. The results obtained by these two approaches are then compared.
The Missing Link Between the Quantum Mechanical and SemiClassical Determination of Scattering Resonance Poles
, 1996
"... We investigate the twodimensional scattering of a pointparticle from n nonoverlapping fixed disks and study  using these systems  the connection between the spectral properties of the quantum mechanical scattering matrix and its semiclassical equivalent based on the GutzwillerVoros zetafunc ..."
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We investigate the twodimensional scattering of a pointparticle from n nonoverlapping fixed disks and study  using these systems  the connection between the spectral properties of the quantum mechanical scattering matrix and its semiclassical equivalent based on the GutzwillerVoros zetafunction. We rewrite the determinant of the scattering matrix in such a way that it separates into the product over n determinants of 1disk scattering matrices  representing the incoherent part of the scattering from the n disk system  and the ratio of two mutually complex conjugate determinants of the genuinely multiscattering kernel, M, of KKRtype which represents the coherent part of the ndisk scattering. Our result is welldefined at every step of the calculation, as the onshell Tmatrix and the kernel A = M  1 are shown to be traceclass. We stress that the cumulant expansion (which defines the determinant over an infinite, but trace class matrix) imposes the curvature regula...