Quantum mechanics in Riemannian spacetime. II. Operators of observables

The formulation of the generally covariant analog of standard (nonrelativistic) quantum mechanics in a general Riemannian spacetime begun in earlier studies of the author is continued with the introduction of asymptotic (with respect toc ^–2) operators of the spatial position of a spinless particle and of the projection of its momentum onto an arbitrary spacetime direction. The connection between the position operator and the generalization of theV _1,3 Newton—Wigner operator is established. It is shown that the projection of the momentum onto the 4-velocity of the frame of reference (the energy operator) is unitarily equivalent to the Hamiltonian in the Schrödinger equation.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 412–423, March, 1992.     

Spectral distributions and generalization of Stone's theorem

In this paper, we introduce the notion ofspectral distribution which is a generalization of the spectral measure. This notion is closely related to distribution semigroups and generalized scalar operators. The associated operator (called themomentum of the spectral distribution) has a functional calculus defined for infinitely differentiable functions on the real line. Our main result says thatA generating a smooth distribution group of orderk is equivalent to having ak-times integrated group that are O(¦t¦ ^k ) oriA being the momentum of a spectral distribution of degreek. We obtain the standard version of Stone's theorem as a special case of this result. The standard properties of a functional calculus together with spectral mapping theorem are derived. Finally, we show how the degree of a spectral distribution is related to the degree of the nilpotent operators which separate its momentum from its scalar part.     

The Vlasov-Maxwell-Boltzmann system near Maxwellians

Perhaps the most fundamental model for dynamics of dilute charged particles is described by the Vlasov-Maxwell-Boltzmann system, in which particles interact with themselves through collisions and with their self-consistent electromagnetic field. Despite its importance, no global in time solutions, weak or strong, have been constructed so far. It is shown in this article that any initially smooth, periodic small perturbation of a given global Maxwellian, which preserves the same mass, total momentum and reduced total energy (22), leads to a unique global in time classical solution for such a master system. The construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation: the linear collision operator L, not its time integration, is positive definite for any solutionf(t,x,v) with small amplitude to the Vlasov-Maxwell-Boltzmann system (8) and (12). As a by-product, such an estimate also yields an exponential decay for the simpler Vlasov-Poisson-Boltzmann system (24).     

Hardy Spaces of Harmonic Functions on Homogeneous Isotropic Trees

A variational method for operator equations of the form Pu + δβ(u) ? f has been given in Dinca [1]. Here P is a (generally) nonlinear operator in a Hilbert space, β: H → ? ∞ is a convex, proper (β ≠ + ∞) and lower-semicontinuous functional and δβ(u) stands for the subdifferential of β at the point u. The present paper has two parts. The first part contents the main results of Dinca [1] without proofs. The second part discusses particular cases and applications to mechanics among which “the climatisation problem for non-linear elliptic equations” and its applications.     

The Iterated Aluthge Transform of an Operator

The Aluthge transform (defined below) of an operator T on Hilbert space has been studied extensively, most often in connection with p-hyponormal operators. In [6] the present authors initiated a study of various relations between an arbitrary operator T and its associated , and this study was continued in [7], in which relations between the spectral pictures of T and were obtained. This article is a continuation of [6] and [7]. Here we pursue the study of the sequence of Aluthge iterates { ⁽ⁿ⁾} associated with an arbitrary operator T. In particular, we verify that in certain cases the sequence { ⁽ⁿ⁾} converges to a normal operator, which partially answers Conjecture 1.11 in [6] and its modified version below (Conjecture 5.6). Submitted: December 5, 2000? Revised: August 30, 2001.     

On the image of the limit q‐Bernstein operator

The limit q‐Bernstein operator B_q emerges naturally as an analogue to the Szász–Mirakyan operator related to the Euler distribution. Alternatively, B_q comes out as a limit for a sequence of q‐Bernstein polynomials in the case 0<q<1. Lately, different properties of the limit q‐Bernstein operator and its iterates have been studied by a number of authors. In particular, it has been shown that B_q is a positive shape‐preserving linear operator on C[0, 1] with ∥B_q∥=1, which possesses the following remarkable property: in general, it improves the analytic properties of a function. In this paper, new results on the properties of the image of B_q are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

The Matrix Multidisk Problem

The solutions of a class of matrix optimization problems (including the Nehari problem and its multidisk generalization) can be identified with the solutions of an abstract operator equation of the form T(., ., .) = 0. This equation can be solved numerically by Newton's method if the differential T' of T is invertible at the points of interest. This is typically too difficult to verify. However, it turns out that under reasonably broad conditions we can identify T' as the sum of a block Toeplitz operator and a compact block Hankel operator. Moreover, we can show that the block Toeplitz operator is a Fredholm operator and and in some cases can calculate its Fredholm index. Thus, T' will also be a Fredholm operator of the same index. In a number of cases that have been checked todate, numerical methods perform well when the Fredholm index is equal to zero and poorly otherwise. The main focus of this paper is on the multidisk problem alluded to above. However, a number of analogies with existing work on matrix optimization have been worked out and incorporated. Submitted: April 23, 2002.     

THE LEGENDRE TYPE OPERATOR IN A LEFT DEFINITE HILBERT SPACE

Abstract

The techniques for discussing linear differential operators in left definite spaces, developed earlier for regular fourth order and singular second order operators, are applied the Legendre type operator. It is shown that the operator, with its domain merely restricted to the new space, remains self-adjoint and has the same spectrum, inverse and spectral resolution (an eigenfunction expansion) as the original L ² operator.     

Epigraph of operator functions

It is known that a real function f is convex if and only if the set E(f) = {(x, y) ∈ ? × ?; f (x) ≤ y}, the epigraph of f is a convex set in ?². We state an extension of this result for operator convex functions and C^?-convex sets as well as operator log-convex functions and C^?-log-convex sets. Moreover, the C^?-convex hull of a Hermitian matrix has been represented in terms of its eigenvalues.     

Iterationsverfahren höherer Ordnung in Banach-Räumen

The Newton process for operator equations in say a linear normed complete space converges under certain hypothesis about the Fréchet-derivatives of the operator with at least the order two. There are different ways to improve this Newton process. For instance you obtain a process of order three if you add a correction element containing the second Fréchet-derivative of the operator [1]. In the following note we will generalize this idea. In a recursive manner — by adding higher derivatives — we will construct iterative processes of any orderk (k > 1). A general theorem due toCollatz provides us error estimates for this processes. Last we will illustrate the processes by several examples.     

On Fliess operators driven by L 2-Itô processes

Fliess operators, which are a type of functional series expansion, have been used to describe a broad class of nonlinear input–output maps driven by deterministic inputs. But in most applications, a system's inputs have noise components. This paper has three objectives. The first objective is to show that the notion of a Fliess operator can be generalized to admit a class of L ₂-Itô stochastic input processes. The next objective is to show that they converge absolutely over an arbitrarily large but finite time interval when a certain coefficient growth condition is met. However, a significant number of systems fail to meet this condition. Thus, the final objective is to consider an interval of convergence having a random length so that a Fliess operator might converge under less restrictive growth conditions.     

Asymptotic estimates of wave functions and the existence of wave operators

The existence of wave operators is proved for the case, where the unperturbed operator is the operator of multiplication by a smooth function in momentum space and the perturbation is an arbitrary operator satisfying a fall off condition near infinity or a weighted L_p-estimate in configuration space. Under somewhat more restrictive conditions the invariance principle is also proved.     

The equations of the thermodynamic model of magnetoelasticity of dielectric ferromagnetic bodies

We develop a thermodynamic approach to the mathematical modeling of magnetoelastic processes in dielectric ferromagnetic bodies that are subject to the action of force loading, heating, and an external electromagnetic field. The basis for the construction of the physical relations is the principle of local thermodynamic state. In the computations of the momentum balance equation of the magnetization process account is taken of its tensor character.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 30–34.     

Analytic Study of Magnetohydrodynamic Flow and Boundary Layer Control Over a Wedge

A genuine variational principle developed by Gyarmati, in the field of thermodynamics of irreversible processes unifying the theoretical requirements of technical, environmental and biological sciences is employed to study the effects of uniform suction and injection on MHD flow adjacent to an isothermal wedge with pressure gradient in the presence of a transverse magnetic field. The velocity distribution inside the boundary layer has been considered as a simple polynomial function and the variational principle is formulated. The Euler-Lagrange equation is reduced to a simple polynomial equation in terms of momentum boundary layer thickness. The velocity profiles, displacement thickness and the coefficient of skin friction are calculated for various values of wedge angle parameter m, magnetic parameter ξ and suction/injection parameter H. The present results are compared with known available results and the comparison is found to be satisfactory. The present study establishes high accuracy of results obtained by this variational technique.     

Estimation of Mean and Covariance Operator of Autoregressive Processes in Banach Spaces

The autoregressive model in a Banach space (ARB) contains many continuous time processes used in practice, for example, processes that satisfy linear stochastic differential equations of order k, a very particular case being the Ornstein–Uhlenbeck process. In this paper we study empirical estimators for ARB processes. In particular we show that, under some regularity conditions, the empirical mean is asymptotically optimal with respect to a.s. convergence and convergence of order 2. Limit in distribution and the law of the iterated logarithm are also presented. Concerning the empirical covariance operator we note that, if (X _n, n ∈ &Zopf;) is ARB then (X _n ⊗ X _n, n ∈ &Zopf;) is AR in a suitable space of linear operators. This fact allows us to interpret the empirical covariance operator as a sample mean of an AR and to derive similar results for it. This revised version was published online in August 2006 with corrections to the Cover Date.     

Singular unitary dilations

We study the problem of determining which bounded linear operator on a Hilbert space can be dilated to a singular unitary operator. Some of the partial results we obtained are (1) every strict contraction has a diagonal unitary dilation, (2) everyC ₀ contraction has a singular unitary dilation, and (3) a contraction with one of its defect indices finite has a singular unitary dilation if and only if it is the direct sum of a singular unitary operator and aC ₀(N) contraction. Such results display a scenario which is in marked contrast to that of the classical case where we have the absolute continuity of the minimal unitary power dilation of any completely nonunitary contraction.     

An n-dimensional Wiener-Plancherel formula

The n-dimensional analogues of Wiener's “s-function” and difference operator associated to rectangular limit processes are defined. The Wiener-Plancherel formula is proved in this context for functions having bounded quadratic means.     

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.     

On the correspondence between potential operators and semi-groups associated with Markov processes

Summary Hunt's theory of potentials has recently been treated from the operator-theoretical viewpoint by K. Yosida in a series of his papers. The aim of the treatment is to analyze the mechanism of Hunt's theory and moreover to contribute new and sharpened results to the theory. In the present paper we shall develop his work into showing that the closure of what has been called so far potential operator with its domain restricted to the space of continuous functions with compact supports does deserve the title of the potential operator by verifying its various properties. There will also be given generalizations and straightforward proofs of the theorems proved by K. Yosida concerning pseudoresolvents and the principle of majoration. It will enable us in the near future to treat the whole theory not only in the function spaces such as c (S), but also in the abstract Banach lattices.