Abstract: Modern scanning techniques, such as computed tomography, have begun to produce true three-dimensional imagery of internal structures. The first stage in finding structure in these images, like that for standard two-dimensional images, is to evaluate a local edge operator over the image. If an edge segment in two dimensions is modeled as an oriented unit line segment that separates unit squares (i.e., pixels) of different intensities, then a three-dimensional edge segment is an oriented unit plane that separates unit volumes (i.e., voxels) of different intensities. In this correspondence we derive an operator that finds the best oriented plane at each point in the image. This operator, which is based directly on the 3-D problem, complements other approaches that are either interactive or heuristic extensions of 2-D techniques.

Abstract: The coupled-cluster approach to obtaining the bond-state wave functions of many-electron systems is extended, with a set of physically reasonable approximations, to admit a multiconfiguration reference state. This extension permits electronic structure calculations to be performed on correlated closed- or open-shell systems with potentially uniform precision for all molecular geometries. Explicit coupled cluster working equations are derived using a multiconfiguration reference state for the case in which the so-called cluster operator is approximated by its one- and two-particle components. The evaluation of the requisite matrix elements is facilitated by use of the unitary group generators which have recently received wide attention and use in the quantum chemistry community.

Abstract: The role of time reversal symmetry in natural and magnetic optical activity is discussed. Natural optical rotation is shown to be generated by an anti-hermitian odd parity time-even operator and magnetic optical rotation by an anti-hermitian even parity time-odd operator. This shows that lack of time reversal invariance is not the source of natural optical rotation and that free atoms can show natural optical rotation without violating reversality, which leads to a fundamental distinction between the conditions necessary for natural optical rotation and a permanent space-fixed electric dipole moment. General transition optical activity and polarizability tensors between components of degenerate states are discussed with reference to possible new Raman experiments and new contributions to discriminating intermolecular forces between chiral molecules. Time reversal symmetry also leads to a new criterion for chiral objects and to the concept that natural optical activity provides an example of spontaneous sy...

Abstract: By applying the matrix-exponential operator technique to the radiative-transfer equation in discrete form, new analytical solutions are obtained for the transmission and reflection matrices in the limiting cases x ≪ 1 and x ≫ 1, where x is the optical depth of the layer. Orthogonality of the eigenvectors of the matrix exponential apparently yields new conditions for determining Chandrasekhar’s characteristic roots. The exact law of reflection for the discrete eigenfunctions is also obtained. Finally, when used in conjunction with the doubling method, the matrix exponential should result in reductions in both computation time and loss of precision.

Abstract: This paper is concerned with the placement of the spectrum of the closed-loop operator $A + BK$ resulting from use of a linear feedback control law $u = Kx$ in the infinite dimensional linear control system $x' = Ax + Bu$. For a class of systems in Hilbert space with certain assumptions on the spectrum of the operator A, a complete characterization of the achievable spectra is obtained. The proofs are carried out in an operator-theoretic context.

Abstract: The Zakharov and Shabat equation for the scattering problem is studied: The estimates, analytical properties, and asymptotic expansions of the Jost solution are presented for a general class of the potentials Q(x) not vanishing at infinity. The existence of the similarity transformation is also shown. For Q(x) vanishing at infinity, the continuous part of the spectrum doubly degenerates. However, nonvanishing (finite) asymptotic values of Q(x) dissolve the degeneracy completely. The expansion theorem is given in C 0 2(R) and for a class of Q(x) we prove that the Zakharov and Shabat equation yields a non‐self‐adjoint spectral operator in the Hilbert space in the sense of Dunford and Schwartz.

Abstract: We consider a problem of optimal control of the stochastic evolution equation $d\xi = (A(t)\xi + B(t)u)dt + \sigma (t)dw$, on a separable Hilbert space, where $\{ A(t),B(t),\sigma (t),t \geqq 0\} $ are progressively measurable operator-valued random processes with A generally unbounded. We prove the existence and uniqueness of (weak) solutions of the evolution equation. Then we present the existence of optimal controls and necessary conditions of optimality for a quadratic (random) cost function. For optimal feedback controls we solve a random operator Riccati equation and a backward stochastic evolution equation. The backward equation is solved by transposing a random isomorphism generated from a forward evolution equation. The optimal feedback control is given by a random affine transformation of the state. Some examples are presented to indicate usefulness of the results. This work is a partial extension of the results of Bismut [SIAM J. Control Optim., 14 (1976), pp. 419–444; 15 (1977), pp. 1–4] and B...

Abstract: It is shown that the existence of Aharonov–Bohm scattering depends upon the criteria used for establishing the stationary states. If one applies Pauli’s criterion, there is no scattering. It is shown further that applying the usual criteria that the wave functions be continuous and single valued, as was done by Aharonov and Bohm, leads to stationary state wave functions which, with two exceptions, are eigenstates of the acceleration operator corresponding to eigenvalue zero. The acceleration operator is undefined for the remaining two states. Thus, only the eigenfunctions satisfying the Pauli criterion lead to well‐defined, sensible physics.

Abstract: 1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l by where y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L 0 in the Hilbert space ℒ m 2 (J; w) of all complex, m-dimensional vector-valued functions y on J satisfying with inner product where . All selfadjoint extensions of L 0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

Abstract: The operator description of Fourier optics is extended and applied to holography. The existing lens models for ideal holographic processes appear as a self-evident intermediate result; generalization to include apertures, recording-material modulation transfer function, and extended source effects is straightforward. The extended source effect is generally shown to be equivalent to a modification of the actual holographic apertures. The final result is a compact expression for the description of the holographically reconstructed field distribution at an arbitrary plane. A useful, comprehensive list of operator relations is given in two appendixes.

Abstract: : For a class of gradient evolutionary equations, we prove that the w-limit set of a bounded orbit is an equilibrium point if the dimension of the null space of the linear variational operator is no more than one. This implies the result of Matano concerning a parabolic equation in one space dimension with separated boundary conditions. The statement about gradient systems is a consequence of a more general property which has applications, for example, to the stability of traveling waves. (Author)

Abstract: The calculation of the radial coupling between the nuclear motion and the electronic structure which induces charge transfer at low energy is discussed, and the utility of the force operator form for computing the coupling between large configuration-interaction wave functions is demonstrated. Many properties suggest that such couplings will be useful in constructing the diabatic scattering states; the agreement with available matrix elements computed numerically is excellent.

Abstract: The formal structure of the time dependence of the state of an atom in a perturber bath in an intense classical monochromatic radiation field is studied. The authors allow the cross sections of binary collisions to be affected by the radiation field. The result is used to derive an expression for the collision-broadened saturated fluorescence excitation spectrum and absorption rate in terms of a collisional relaxation operator. This operator can be related to the rate of collisionally aided radiative excitation.

Abstract: The Fokker-Planck equation governing the relaxation of the electron speed (energy) distribution in gases is solved in a number of cases of special interest. The solution is given in terms of eigenfunctions of the Fokker-Planck operator, satisfying an orthonormalization condition in which the steady-state distribution is the weight function. The real cross-sections of the noble gases He, Ne, Ar, Kr and Xe, together with model collision frequencies of the form ν(υ)=αυn withn=0.5, 1, 1.5, 3 and 3.5, are used to calculate eigenvalues and eigenfunctions. The first fifteen eigenvalues are obtained in each case both in the absence and in the presence of a d.c. electric field and, in the latter case, both with atoms at rest and atoms in motion. Calculations of relaxation times and examples of evolutions towards their steady-state forms of given initial distributions are reported in several particular cases.

Abstract: In this note it is shown that the Cauchy problem associated with the infinitesimal generatorA of a strongly continuous operator cosine function remains uniformly well-posed under bounded time-dependent perturbations ofA.

Abstract: This paper describes a novel approximate SU(2) symmetry for valence‐shell π‐electron correlation in unsaturated hydrocarbons CnHn+2. The symmetry is related to accidental near‐degeneracy and vibrationlike excitations in a Huckel resonance picture. Diagonalization of the CI energy matrix within each manifold splits the degeneracy and gives configuration‐mixed states, including collective one‐ and two‐electron excitations thought to be important in the first excited 1A−g level of butadiene. The approximate SU(2) symmetry for the electron correlation which causes the splitting is found with a Pariser–Parr–Pople Hamiltonian. Coupled SU(2) representations have Casimir invariants that are near‐constants of the motion, and CI mixing coefficients are identified as Clebsch–Gordan coefficients. Exact SU(2) symmetries are found for a two‐electron zero‐differential‐overlap repulsion operator, and for a many‐electron PPP Hamiltonian with harmonic two‐body repulsion terms. Polyene spectra computed with an Ohno–Coulomb ...

Abstract: The combinatorics of Gel'fand states which are useful in the graphical unitary group approach to many electron correlation problem and spin free quantum chemistry is considered. Using operator theoretic methods it is shown that the generators of Gel'fand states are S-functions.

Abstract: The influence of non-parabolicity on the velocity operator is derived, and it is argued that the expression used previously by Arora and Jaafarian (1976) is not correct. With the correct form of the velocity operator it is shown that in the limit of vanishing scattering the Hall effect is not influenced by non-parabolicity.