Showing papers on "Multiplication operator published in 1986"

Reconstruction of dynamics from an eigenstate

Abstract: For Hamiltonians that have a formal (canonical) decomposition H=− 1/2 Δ+V(x), V(x) being a multiplication operator, the definition of the dynamics by a ground state measure leads to an energy (Dirichlet) form formulation of quantum mechanics that is more general than the operator Schrodinger approach. Here, the question of reconstruction of the dynamics from an eigenstate when the potential is not restricted to the class of multiplication operators is analyzed. By explicit analysis of several examples, it is found that, once a particular operator class is chosen, the potential and the energy form are, to some extent, determined by the eigenstate. However, depending on the type of operator the potential is chosen to be, many distinct dynamics can be associated to the same fixed eigenstate. The nature of the stochastic processes associated with each reconstructed dynamic is also discussed, as well as a generalization of the stochastic dynamics formalism allowing for nonlocal potentials.

The Spectral Shift Function for a Dissipative and a Selfadjoint Operator, and Trace Formulas for Resonances

Abstract: A trace formula is given for an abstract pair consisting of a dissipative operator and a selfadjoint operator, and a connection is established between the spectral shift function of this pair and the corresponding scattering matrix. As a consequence, trace formulas are obtained for a specific dissipative operator arising in the problem of resonance scattering of plane waves on a one-dimensional semi-infinite crystal.Bibliography: 8 titles.

Stochastic integration for operator valued processes on hilbert spaces and on nuclear spaces

Abstract: The representation of a nuclear space valued square integrable martingale by means of another nuclear space valued square integrable martingale is given in terms of stochastic inegrals of operator valued processes. The construction of the stochastic integral goes through that of operator valued processes on Hilbert spaces. A new approach is given for the Hilbertian case, so that only the integration of Hilbert-Schmidt operator valued processes is needed to represent square integrable martingales

Lorentz transformations in terms of initial and final vectors

Abstract: Given arbitrary initial vector(s) and their final vector(s) in a Lorentz transformation, the problem is to determine the operator of the transformation. The solution presented here consists of expressing the tensor contained in the Lorentz operator in terms of the given vectors and then the operator itself follows by the exponentiation of a generating matrix. This is possible in certain special cases. In the general case a number of simultaneous nonlinear equations have to be solved. The analytical solution of these equations is elusive while attempts at numerical solution indicate that supplementary information is required. The corresponding procedure for a singular operator is also presented.

Invariants for Wiener-Hopf Equivalence of Analytic Operator Functions

Abstract: Necessary conditions for Wiener-Hopf equivalence are established in terms of the incoming and outgoing subspaces associated with realizations of the given analytic operator functions. Other results about the behaviour of the incoming and outgoing subspaces under certain elementary operations are also included.

A Kind of New Operator Identities Related to SO3 Group

Abstract: In terms of the technique of performing integration within ordered product we show the integration appearing in the rotation operator can be explicitly carried out and leads to a kind of new operator identities, their coherent state matrix elements are then easy to be computed.

Analytic equivalence of the boundary eigenvalue operator function and its characteristic matrix function

Abstract: The boundary eigenvalue operator function associated to an analytic family of boundary value problems is shown to be analytically equivalent to a simple extension of its characteristic matrix function. Explicit formulas for the operator functions that establish the equivalence are given.

An implementation of operators for symbolic algebra systems

Abstract: In this paper we propose a design and implementation of operators and their associated functionality for symbolic algebra systems. We believe that operators should blend harmoniously (syntactically and semantically) with the underlying language, in such a way that users will find them convenient and appealing to use. It is “vox populi” that operators are needed in a symbolic algebra system, although there is little consensus on what these should be, what the semantics should be, allowable operations, syntax, etc. All of these ideas, and examples, have been implemented and work as described in our current version of Maple Cha85.During the first Maple retreat83 we established a basic design for operators. The implementation of this design was delayed until some remaining crucial details were finally solved during the 1985 Maple retreat (Sept 1985). In this sense, this paper is the result of the collective work of all the participants of these two retreats.What is an operator? We would like to define an operator to be an abstract data type which describes (at various possible degrees: totally, partially or minimally) an operation to be performed on its arguments. This abstract data type is closely associated with the operations of application and composition, but will also allow most (or all) of the other algebraic operations.We it found useful to have some “witness” examples that we want to solve in an elegant and general form. The two main examples were:(a) How to represent the first derivative of ƒ(x) at 0, i.e. ƒ′(0) (the above really boils down to an effective representation of the differentiation operator)(b) How to represent and to operate with a non-communative multiplication operator, for example matrix multiplication.Of course many systems solve the above problems, but in some cases (in particular for the first example) as an ad-hoc solution. By an ad-hoc solution we mean that, for the differentiation example, this operator cannot be written in terms of the primitives given by the language.It is important to note that there are three issues to resolve: a purely representational/syntactic argument: how to input/output these operators.a purely functional argument: how to perform all the operations we want performed.an integrational argument: how to join operators harmoniously with a symbolic algebra system.

General Lorentz transformations and applications

Abstract: It is known that the most general proper orthochronous vector Lorentz (transformation) operator can be generated by a skew‐symmetric 4×4 matrix containing an antisymmetric tensor of the second rank. The corresponding Lorentz operator for the two‐component spinor is presented and, as can be expected, it contains the same tensor as the vector operator. Since the Pauli matrices of the spinor operator have very simple multiplication properties, the behavior of the tensor under multiplication of spinor operators is easily obtained. By comparison the corresponding properties of the tensor in vector operators can be obtained without multiplying 4×4 matrices. The physical meaning of the tensor contained in a Lorentz operator is discussed. Apart from the usual or regular operator a singular operator is discussed. Still other types of Lorentz operators are possible.

Dieudonné operators on C(K,E)

Abstract: A Banach space operator is called a Dieudonne operator if it maps weakly Cauchy sequences to weakly convergent sequences. A space E is said to have property (D) if, whenever K is a compact Hausdorff space and T is an operator from C(K,E) into a space F , T is a Dieudonne operator if and only if its representing measure is both strongly additive and has for its values Dieudonne operators from E into F . The purpose of this paper is to show that if E ∗ has the Radon-Nikodým property then E has (D) if and only if E ∗∗ has the Radon-Nikodým property.

The choice of regularization parameter when solving a linear operator equation

Abstract: A method of solving a certain class of linear operator equations of the 1st kind in Hilbert space is proved. The method consists in finding the extremal of the Tikhonov smoothing functional and subsequently choosing the regularization parameter on the basis of the principles of quasi-optimality or ratio in “global” form.

On the poincaré theta operator

Abstract: It is well known that the norm of the Poincare theta operator does not exceed one. Recently, the problem of determining necessary and sufficient conditions guaranteeing that the norm is strictly le...

Asymptotic and approximate formulas in the inverse scattering problem for the Schrödinger operator

Abstract: with positive constants C O and B satisfying B > i, (1.3) i.e., Q(x) is a short-range potential. H and H 0 denote uniaue self-adjoint extensions of h and h 0 : -A restricted to C0OR3) in L2~R3), respectively. Since the multiplication operator Q(x)x is a bounded linear operator on L2OR3), the domain D(H) of H coincides with the domain D(H0) of H 0 and they are equal to H2~.3), the Sobolev space of the second order on ~3 (see, e.g., Kato [i0], Chapter 5).

On an integral operator for convex univalent functions

Abstract: Let K (m,M) denote the class of functions regular and satisfying |1 + zf″(z)/f′(z)− m| < M in |z| < 1, where |m−1| < M ≤ m. Recently, R.K. Pandey and G. P. Bhargava have shown that if f ε K (m,M), then the function du also belongs to K (m,M) provided α is a complex number satisfying the inequality |α| ≤ (1−b)/2, where b = (m-1)/M. In this paper we show by a counterexample that their inequality is in general wrong, and prove a corrected version of their result. We show that F ε K (m,M) provided that α is a real number satisfying −φ ≤ α ≤1, φ = (M−|m−1|)/(M + |m−1|), or a complex number satisfying |α| ≤ φ. In both cases the bounds for α are sharp.

Optical reflection from planetary surfaces as an operator-eigenvalue problem

Abstract: The understanding of quantum mechanical phenomena has come to rely heavily on theory framed in terms of operators and their eigenvalue equations. This paper investigates the utility of that technique as related to the reciprocity principle in diffuse reflection. The reciprocity operator is shown to be unitary and Hermitian; hence, its eigenvectors form a complete orthonormal basis. The relevant eigenvalue is found to be infinitely degenerate. A superposition of the eigenfunctions found from solution by separation of variables is inadequate to form a general solution that can be fitted to a one-dimensional boundary condition, because the difficulty of resolving the reciprocity operator into a superposition of independent one-dimensional operators has yet to be overcome. A particular lunar application in the form of a failed prediction of limb-darkening of the full Moon from brightness versus phase illustrates this problem. A general solution is derived which fully exploits the determinative powers of the reciprocity operator as an unresolved two-dimensional operator. However, a solution based on a sum of one-dimensional operators, if possible, would be much more powerful. A close association is found between the reciprocity operator and the particle-exchange operator of quantum mechanics, which may indicate the direction for further successful exploitation of the approach based on the operational calculus.