Full Operator Research Articles

Inverse bremsstrahlung absorption with full electron-electron collisions operator

A two-dimensional Fokker-Planck program is developed, in which full e-e collisions are taken into account self-consistently with arbitrary anisotropy of electron distribution function (EDF), to investigate inverse bremsstrahlung and the evolution of EDF. The numerical results show that e-e collisions will enhance inverse bremsstrahlung absorption. The evolution of EDF can be divided into two stages distinguished by different absorption rates. During the first stage, an initially Maxwellian EDF transforms rapidly into an anisotropic one with two temperatures. With the increase of anisotropy, the absorption rate decreases dramatically while the contribution ratio of e-e collisions increases rapidly. In particular, we find that with the increase of ion charge state Zi the contribution ratio of e-e collisions increases in the high laser frequency regime, while it decreases in the low frequency regime.

Galloping instability of viscous shock waves

Motivated by physical and numerical observations of time oscillatory “galloping”, “spinning”, and “cellular” instabilities of detonation waves, we study Poincaré–Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional center manifold. We overcome this by direct Lyapunov–Schmidt reduction, using detailed pointwise bounds on the linearized solution operator to carry out a nonstandard implicit function construction in the absence of a spectral gap. The key computation is a space-time stability estimate on the transverse linearized solution operator reminiscent of Duhamel estimates carried out on the full solution operator in the study of nonlinear stability of spectrally stable traveling waves.

INTELLIGENT NEIGHBORHOOD EXPLORATION IN LOCAL SEARCH METHOD

Standard tabu search methods are based on the complete exploration of current solution neighborhood. However, for some problems due to the neighborhood size or to the fitness evaluation complexity, the total exploration of the neighborhood is impractical. The main purpose of this paper is to propose a local search method with no enumeration procedure. In other words, a single solution is examined at each iteration and retained for the future iterations. The idea is to randomly select one solution among a sub-set of the neighborhood of the current one. An adaptive exploration of neighborhood, using extension and restriction mechanisms represented by loop detection and tabu list structure, determines this sub-set. This approach is applied to the K-coloring problem and evaluated on standard benchmarks like DIMACS. The objective is to show how a generic method, without full neighborhood exploration, degradation control and problem-oriented operators, provides a very competitive results comparing to the best dedicated algorithms for graph coloring problems published in the literature.

Stability estimates in stationary inverse transport

We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension $n\geq3$. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [2, 3] and partial stability estimates were obtained in [12] for spatially independent scattering coefficients. We generalize these results and prove an $L^1$-stability estimate for spatially dependent scattering coefficients.

Regulation Mechanism of Spin−Orbit Coupling in Charge-Transfer-Induced Luminescence of Imidazopyrazinone Derivatives

The spin transition in the reactions of the derivatives of imidazo[1,2-a]pyrazin-3(7H)-one (1H) with a triplet molecular oxygen (3O2) has been investigated by the geometry optimization at the B3LYP/6-31+G(d) level and the evaluation of the electronic matrix elements for spin-orbit coupling (SOC) using the full Pauli-Breit SOC operator. The reductive activation for the 3O2 reaction is affected by the proton activity and solvent polarity of a surrounding reaction field. In a polar aprotic solvent, a base-prompted anionic substrate may react with 3O2 in a stepwise manner through complete electron transfer from the substrate anion to 3O2, while the irreversible concerted 3O2 addition via intersystem crossing may become complete in a less polar solvent. SOC in the thermal decomposition of a resulting peroxide adduct can be controlled by the protonation state of the substrate. There exists an optimal protonation state for the suppression of SOC in the charge-transfer-induced luminescence (CTIL) of the peroxide, which is closely related with the ability of a substituent to donate an electron. This will constitute a necessary condition for the high efficiency of chemi- and bioluminescence.

Double‐square‐root one‐way wave equation prestack tau migration in heterogeneous media

ABSTRACTIn this paper, source‐receiver migration based on the double‐square‐root one‐way wave equation is modified to operate in the two‐way vertical traveltime (τ) domain. This tau migration method includes reasonable treatment for media with lateral inhomogeneity. It is implemented by recursive wavefield extrapolation with a frequency‐wavenumber domain phase shift in a constant background medium, followed by a phase correction in the frequency‐space domain, which accommodates moderate lateral velocity variations. More advanced τ‐domain double‐square‐root wave propagators have been conceptually discussed in this paper for migration in media with stronger lateral velocity variations. To address the problems that the full 3D double‐square‐root equation prestack tau migration could meet in practical applications, we present a method for downward continuing common‐azimuth data, which is based on a stationary‐phase approximation of the full 3D migration operator in the theoretical frame of prestack tau migration of cross‐line constant offset data. Migrations of synthetic data sets show that our tau migration approach has good performance in strong contrast media. The real data example demonstrates that common‐azimuth prestack tau migration has improved the delineation of the geological structures and stratigraphic configurations in a complex fault area.Prestack tau migration has some inherent robust characteristics usually associated with prestack time migration. It follows a velocity‐independent anti‐aliasing criterion that generally leads to reduction of the computation cost for typical vertical velocity variations. Moreover, this τ‐domain source‐receiver migration method has features that could be of help to speed up the convergence of the velocity estimation.

Determination of spin‐orbit coupling contributions in the framework of density functional theory

We present a noniterative method to calculate spin-orbit coupling by means of a theoretical approach that provides the use of the full Breit-Pauli operator. This method was applied to compute one and two-electron spin-orbit coupling contributions between singlet and triplet, and doublet and doublet states, respectively. These states have been represented by monodeterminantal wave functions and optimized using the PW91 gradient-corrected exchange-correlation functional and the hybrid B3LYP one. They have been supplied by the conventional density functional theory packages, and thus coupled by our spin-orbit coupling code. Different size basis sets have been employed and the obtained results have been compared with the corresponding ones provided by some of the already existing methods and with the experimental data. They have been found to be in good quantitative agreement.

Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone

In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For {\it arbitrary} self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism encoding the self-adjoint extension chosen. For particular examples, like the Friedrich's extension, the answer is easily extracted from the general result. In combination with \cite{BKD}, a closed expression for the determinant of an arbitrary self-adjoint extension of the full Laplace-type operator on the generalized cone can be obtained.

Compressed wavefield extrapolation with curvelets

An explicit algorithm for the extrapolation of one-way wavefields is proposed which combines recent developments in information theory and theoretical signal processing with the physics of wave propagation. Because of excessive memory requirements, explicit formulations for wave propagation have proven to be a challenge in 3-D. By using ideas from “compressed sensing”, we are able to formulate the (inverse) wavefield extrapolation problem on small subsets of the data volume, thereby reducing the size of the operators. According to compressed sensing theory, signals can successfully be recovered from an imcomplete set of measurements when the measurement basis is incoherent with the representation in which the wavefield is sparse. In this new approach, the eigenfunctions of the Helmholtz operator are recognized as a basis that is incoherent with curvelets that are known to compress seismic wavefields. By casting the wavefield extrapolation problem in this framework, wavefields can successfully be extrapolated in the modal domain via a computationally cheaper operatoion. A proof of principle for the “compressed sensing” method is given for wavefield extrapolation in 2-D. The results show that our method is stable and produces identical results compared to the direct application of the full extrapolation operator.

Molecular Spectroscopy beyond the Born−Oppenheimer Approximation: A Computational Study of the CF3O and CF3S Radicals

This paper addresses some advances in the theoretical description of molecular spectroscopy beyond the Born-Oppenheimer adiabatic approximation. A solution of the nuclear dynamics problem complicated by the EE Jahn-Teller effect and spin-orbit coupling is considered for the case of the CF3O and CF3S radicals, all the model parameters being obtained solely from ab initio calculations without any adjustment to experimental numbers. Vibrational and vibronic model parameters were calculated at the equation-of-motion coupled cluster level of theory with basis sets of triple-zeta quality. The spin-orbit coupling in X 2E CF3O and CF3S was parametrized by means of a perturbative solution of the full Breit-Pauli spin-orbit operator. Spin-vibronic eigenvalues and eigenfunctions were computed in a basis set of products of electronic, electron spin, and vibrational functions. Results demonstrate the importance of explicit inclusion of the spin-orbit coupling and at least cubic Jahn-Teller terms in the model Hamiltonian for the high precision evaluation of spin-vibronic energy levels of CF3O and CF3S. The theoretical results support and complement the spectroscopic data observed for these species.

Theoretical study on ground and low-lying excited electronic states of BrCl: Ab initio MRCI(SD)+Q calculations

Ab initio calculations on the ground and low-lying electronic states of the BrCl molecule have been carried out within the framework of the internally contracted multireference singles and doubles configuration interaction method by using the all-electronic basis sets cc-pVTZ-DK with Davidson size-extensively correction and Douglas-Kroll-Hess scalar relativistic correction. The spin-orbit coupling interactions are further taken into account through the state interaction approach with the full Breit-Pauli operator. The 23 Ω states generated from the 12 Λ − S states are studied. The spectroscopic constants of the bound states are derived by fitting the calculated potential energy curves. The calculated potential barrier of B 3 Π 0 + is located at the internuclear distance R of 3.261 Å and lies on 236.8 cm −1 above the first dissociation limit, consistent with experimentally the potential barrier height (275 cm −1) and the intersection location R (3.26 Å). The spin-orbit coupling between the B 3 Π 0 + state and the essentially unbound state Y0 + is investigated. It is shown that the B 3 Π 0 + state is significantly mixed with another 3 Π 0 + state as well as 3 Σ 0 + - state, and the former is dominant. On the basis of the selection rule of the spin-orbit coupling and analysis of the Ω state components, the intersection of the repulsive C 1 Π 1 state with the attractive branch of the B 3 Π 0 + potential curve near the level of the first molecular dissociation limit is confirmed to be forbidden for the spin-orbit coupling, and theoretically it is further suggested that the possible rotation-electronic coupling between B 3 Π 0 + and C 1 Π 1 is responsible for the spontaneous predissociation of the B 3 Π 0 + state. The basis set superposition errors for the calculated spectroscopic constants are examined. Also the effects of the basis set and the theoretical methods on the calculated dissociation energy of the ground state are briefly discussed.

Consistency check on volume and triad operator quantization in loop quantum gravity: I

The volume operator plays a pivotal role for the quantum dynamics of loop quantum gravity (LQG). It is essential to construct triad operators that enter the Hamiltonian constraint and which become densely defined operators on the full Hilbert space, even though in the classical theory the triad becomes singular when classical GR breaks down. The expression for the volume and triad operators derives from the quantization of the fundamental electric flux operator of LQG by a complicated regularization procedure. In fact, there are two inequivalent volume operators available in the literature and, moreover, both operators are unique only up to a finite, multiplicative constant which should be viewed as a regularization ambiguity. Now on the one hand, classical volumes and triads can be expressed directly in terms of fluxes and this fact was used to construct the corresponding volume and triad operators. On the other hand, fluxes can be expressed in terms of triads and triads can be replaced by Poisson brackets between the holonomy and the volume operators. Therefore one can also view the holonomy operators and the volume operator as fundamental and consider the flux operator as a derived operator. In this paper we mathematically implement this second point of view and thus can examine whether the volume, triad and flux quantizations are consistent with each other. The results of this consistency analysis are rather surprising. Among other findings we show the following. (1) The regularization constant can be uniquely fixed. (2) One of the volume operators can be ruled out as inconsistent. (3) Factor ordering ambiguities in the definition of triad operators are immaterial for the classical limit of the derived flux operator. The results of this paper show that within full LQG triad operators are consistently quantized. In this paper we merely present ideas and the results of the consistency check. In a companion paper we supply detailed proofs.

Entropy and correlation functions of a driven quantum spin chain

We present an exact solution for a quantum spin chain driven through its critical points. Our approach is based on a many-body generalization of the Landau-Zener transition theory, applied to a fermionized spin Hamiltonian. The resulting nonequilibrium state of the system, while being a pure quantum state, has local properties of a mixed state characterized by finite entropy density associated with Kibble-Zurek defects. The entropy and the finite spin correlation length are functions of the rate of sweep through the critical point. We analyze the anisotropic $XY$ spin-$1∕2$ model evolved with a full many-body evolution operator. With the help of Toeplitz determinant calculus, we obtain an exact form of correlation functions. The properties of the evolved system undergo an abrupt change at a certain critical sweep rate, signaling the formation of ordered domains. We link this phenomenon to the behavior of complex singularities of the Toeplitz generating function.

Classification of contractively complemented Hilbertian operator spaces

We construct some separable infinite-dimensional homogeneous Hilbertian operator spaces H ∞ m , R and H ∞ m , L , which generalize the row and column spaces R and C (the case m = 0 ). We show that a separable infinite-dimensional Hilbertian JC ∗ -triple is completely isometric to one of H ∞ m , R , H ∞ m , L , H ∞ m , R ∩ H ∞ m , L , or the space Φ spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that H ∞ m , L (respectively H ∞ m , R ) is completely isometric to the space of creation (respectively annihilation) operators on the m (respectively m + 1 ) anti-symmetric tensors of the Hilbert space. Together with the finite-dimensional case studied in [M. Neal, B. Russo, Representation of contractively complemented Hilbertian operator spaces on the Fock space, Proc. Amer. Math. Soc. 134 (2006) 475–485], this gives a full operator space classification of all rank-one JC ∗ -triples in terms of creation and annihilation operator spaces. We use the above structural result for Hilbertian JC ∗ -triples to show that all contractive projections on a C ∗ -algebra A with infinite-dimensional Hilbertian range are “expansions” (which we define precisely) of normal contractive projections from A * * onto a Hilbertian space which is completely isometric to R, C, R ∩ C , or Φ. This generalizes the well-known result, first proved for B ( H ) by Robertson in [A.G. Robertson, Injective matricial Hilbert spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991) 183–190], that all Hilbertian operator spaces that are completely contractively complemented in a C ∗ -algebra are completely isometric to R or C. We use the above representation on the Fock space to compute various completely bounded Banach–Mazur distances between these spaces, or Φ.

Explicit 3D depth migration with a constrained operator

Numerical anisotropy is one of the main problems in the design of explicit 3D depth-extrapolation operators. This paper introduces a new method based on constraining the number of independent coefficients for the full 3D extrapolation operator. The extrapolation operator is divided into two regions. The coefficients for the inner part of the extrapolation operator are treated the same as the full 3D extrapolation operator. The coefficients for the outer part of the extrapolation operator are constrained to be constant as a function of azimuth for a given radius. This strategy reduces the number of floating-point operations because, for each extrapolation step, the number of complex multiplications are reduced and replaced by complex additions. The numerical workload of this alternative scheme is comparable to the Hale-McClellan scheme. Impulse responses are compared with finite-difference solutions for the two-way acoustic-wave equation. It is demonstrated that the numerical anisotropy for the proposed scheme is negligible and that the constrained-depth-extrapolation operator can be used in media with large lateral velocity contrasts. The design of constrained-depth-extrapolation operators with different maximum propagation angles in inline and crossline directions is explained and exemplified. These types of operators can be used to suppress the propagation of aliased energy in the crossline direction during depth extrapolation while reducing numerical cost.

A NEW SEMICLASSICAL DYNAMICS FROM THE INTERACTION REPRESENTATION

This paper develops a new semiclassical mechanics from an exact quantum prescription. In this formulation, a zeroth order propagation, rather than Hamiltonian, is specified. The exact, full evolution operator is then given from a specific interaction representation of the evolved "perturbation" Hamiltonian. We then investigate a variety of approximate, semiclassical, and mixed Quantum/Classical methods, along with exact methodologies to evaluate this time dependent interaction Hamiltonian. The approximate full evolution operator can be described in a variety of ways including an iterated Lippman-Schwinger like equation, and an expansion of the perturbation propagator generated from the time evolved Hamiltonian.

Theoretical study of the C3S molecule

For the most stable linear isomer of C3S in its X1Σ+ state a six-dimensional potential energy surface (PES) has been calculated ab initio by coupled cluster – connected triples (CCSD(T)) method. The analytic form of the PES has been transformed in a quartic force field in dimensionless normal coordinates and employed in calculations of spectroscopic constants using second-order perturbation theory. The PES and the full kinetic energy operator in internal coordinates have been used to calculate variationally the anharmonic ro-vibrational energies for J=0 and J=1. The two experimental band origins of C3S observed in the gas phase, ν1 and ν1+ν5−ν5, agree very well with the theoretical values. The anharmonic ro-vibrational levels, including the bending modes up to 2200 cm−1, are reported. The singlet ground state PES has a saddle point at about 1.25 eV above the linear minimum and two other higher lying cyclic local minima. The only dipole- and spin-allowed electronic transition between 0 and 5 eV is calculated to be the 1Π−X1Σ+ transition with a vertical transition energy of 353.2 nm in good agreement with the matrix value of 378 nm. The dissociative paths C + C2S, C2 + CS and C3 + S of low lying singlet and triplet states have been investigated.

NONLINEAR MONOTONIZATION OF THE BABENKO SCHEME FOR THE QUASI‐LINEAR ADVECTION EQUATION

The paper is devoted to construction and development of new method for numerical solution of hyperbolic type equations [14, 17]. In the previous papers [4, 5, 6, 7, 8, 9] authors have investigated theoretically and tested experimentally 26 different finite‐difference schemes on 4 point patterns for the simplest hyperbolic equation: linear advection equation. This equation has the main features of every hyperbolic equation and is the important part of many mathematical models. In other cases the advection operator is the important part of the full operator of the problem. All 26 schemes have been compared experimentally on the special representative set of tests. Nevertheless to simplicity of the equation, almost all schemes have different disadvantages. They are discussed in detail in the cited papers. So, the investigation of new schemes for this equation is still an important task. In [4, 5, 6, 7, 8, 9] some new schemes were constructed for solving this advection equation. The nonlinear monotone Babenko scheme ("square") proved to be the best among all 26 schemes. So, it is a big interest to generalize this scheme to more difficult equations. The important example is a quasi‐linear advection equation. In this paper our basic aim is to construct a quasi‐monotone nonlinear Babenko scheme for solving the quasi‐linear advection equation and to test it experimentally. The monotonisation of the scheme is done by adding the artificial diffusion with limiters. We also present advanced results of comparative analysis of the new scheme with other known schemes. We have considered explicit and implicit upwind approximation schemes [4, 6, 13, 16] which is firstorder accurate in time and space, the Lax‐Wendroff scheme [4] which is the first order accurate in time and second order accurate in space. We also analyze the monotonised “Cabaret” scheme proposed in [10, 11]. It is second order accurate in time and space, and its monotonisation is based on apriori knowledge of the dependence region of the exact solution. The authors of this scheme called it by “jumping advection”. The considered schemes are compared numerically by using a set of tests, which is similar to one used in [4, 5, 6, 8]. Šiame straipsnyje pasiūlyta kvazi‐monotonie netiesine Babenkos skirtumu schema kvazitiesinei pernešimo lygčiai spresti. Schemos monotoniškumas pasiekiamas pridedant dirbtine difuzija su apribojimais. Pateiktas šios schemos palyginimas su kitomis schemomis. Taip pat analizuojama antros eiles pagal laika ir erdve monotonine “Cabaret” schema. Pateikti testu rezultatai.

Electrodisintegration of 3He below and above deuteron breakup threshold

Recent advances in the study of electrodisintegration of 3He are presented and discussed. The pair-correlated hyperspherical harmonics method is used to calculate the initial and final state wave functions, with a realistic Hamiltonian consisting of the Argonne v18 two-nucleon and Urbana IX three-nucleon interactions. The model for the nuclear current and charge operators retains one- and many-body contributions. Particular attention is made in the construction of the two-body current operators arising from the momentum-dependent part of the two-nucleon interaction. Three-body current operators are also included so that the full current operator is strictly conserved. The present model for the nuclear current operator is tested comparing theoretical predictions and experimental data of pd radiative capture cross section and spin observables.

Almost sharp quantum effects

Quantum effects are represented by operators on a Hilbert space satisfying 0⩽A⩽I, and sharp quantum effects are represented by projection operators. We say that an effect A is almost sharp if A=PQP for projections P and Q. We give simple characterizations of almost sharp effects. We also characterize effects that can be written as longer products of projections. For generality we first work in the formalism of von Neumann algebras. We then specialize to the full operator algebra B(H) and to finite dimensional Hilbert spaces.