Hilbert Space Approach Research Articles

A General Hilbert Space Approach to Framelets

In arbitrary separable Hilbert spaces it is possible to deffine multiscale methods of constructive approximation based on product kernels, restricting their choice in certain ways. These wavelet techniques have already filtering and localization properties and they are applicable in many areas due to their generalized deffinition. But they lack detailed information about their stability and redundancy, which are frame properties. So in this work frame conditions are introduced for approximation methods based on product kernels. In order to provide stability and redundancy the choice of product kernel ansatz function has to be restricted. Taking into account the kernel conditions for multiscale and for frame approximations one is able to deffine wavelet frames (= framelets), inheriting the approximation properties of both techniques and providing a more precise tool for multiscale analysis than the normal wavelets.

Time development in quantum mechanics using a reduced Hilbert space approach

We have created a suite of open source programs that numerically calculate and visualize the evolution of arbitrary initial quantum-mechanical bound states. The calculations are based on the expansion of an arbitrary wave function in terms of basis vectors in a reduced Hilbert space. The approach is stable, fast, and accurate at depicting the long-time dependence of complicated bound states. Several real-time visualizations, such as the position and momentum expectation values and the Wigner quasiprobability distribution for the position and momentum, can be shown. We use these computational tools to study the time-dependent properties of quantum-mechanical systems and discuss the effectiveness of the algorithm.

Sharp and unsharp observables on [formula omitted]-MV algebras—A comparison with the Hilbert space approach

We prove that any observable with a standard value space on a probability MV algebra M can be extended from a mapping defined on measurable subsets with values in M to a mapping defined on bounded real valued measurable functions with values in the set of sharp observables associated with M. Our method of proof is quite different from the method of the proof of analogous result for POV-observables on a Hilbert space. We show that for POV-observables with commuting range both approaches give the same results.

The rigged Hilbert space approach to the Lippmann–Schwinger equation: II. The analytic continuation of the Lippmann–Schwinger bras and kets

The analytic continuation of the Lippmann-Schwinger bras and kets is obtained and characterized. It is shown that the natural mathematical setting for the analytic continuation of the solutions of the Lippmann-Schwinger equation is the rigged Hilbert space rather than just the Hilbert space. It is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution, resulting into a semigroup time evolution. Physically, the semigroup time evolution is simply a (retarded or advanced) propagator.

The rigged Hilbert space approach to the Lippmann–Schwinger equation: Part I

We exemplify the way the rigged Hilbert space deals with the Lippmann-Schwinger equation by way of the spherical shell potential. We explicitly construct the Lippmann-Schwinger bras and kets along with their energy representation, their time evolution and the rigged Hilbert spaces to which they belong. It will be concluded that the natural setting for the solutions of the Lippmann-Schwinger equation--and therefore for scattering theory--is the rigged Hilbert space rather than just the Hilbert space.

Deterministic models of quantum fields

Deterministic dynamical models are discussed which can be described in quantum mechanical terms. - In particular, a local quantum field theory is presented which is a supersymmetric classical model [1]. The Hilbert space approach of Koopman and von Neumann is used to study the classical evolution of an ensemble of such systems. Its Liouville operator is decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. Thus, choosing suitable variables, the classical Liouville equation becomes a functional Schrödinger equation of a genuine quantum field theory. - We briefly mention an U(1) gauge theory with ‘‘varying alpha’’ or dilaton coupling where a corresponding quantized theory emerges in the phase space approach [2]. It is energy-parity symmetric and, therefore, a prototype of a model in which the cosmological constant is protected by a symmetry.

Positivity aspects of the Fantappiè transform

A Hilbert space approach to the classical Fantappiè transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappiè transforms of positive measures on the unit ball inC n relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators.

Propagation of a quantum state in a continuum of coupled oscillators and applications to entanglement

The exponential Hilbert space approach is used to link quantum mechanics in a small system with d-dimensional Hilbert space , with the collective quantum behaviour in a large system comprised of d coupled oscillators with Hilbert space H. In the large system, the expectation value of the mode position operator describes the location of a quantum state within the chain of d oscillators, and the expectation value of the mode momentum operator describes the change of the mode position with time. A continuum of modes (d ? ?) is also considered and in this case these operators obey the commutation relation . A consequence of this is that uncertainties in the location of a state and in its momentum obey an uncertainty relation. Displacements and squeezing in the mode phase space are discussed. For a certain Hamiltonian, the obey equations of motion which are very similar to those of a harmonic oscillator. If the system is in an entangled state, the formalism can be used to quantify concepts like the location of entanglement, and the speed with which the entanglement propagates.

A hierarchical family of three-dimensional potential energy surfaces for He-CO

A hierarchical family of five three-dimensional potential energy surfaces has been developed for the benchmark He-CO system. Four surfaces were obtained at the coupled cluster singles and doubles level of theory with a perturbational estimate of triple excitations, CCSD(T), and range in quality from the doubly augmented double-zeta basis set to the complete basis set (CBS) limit. The fifth corresponds to an approximate CCSDT/CBS surface (CCSD with iterative triples/CBS, denoted CBS+corr). The CBS limit results were obtained by pointwise basis set extrapolations of the individual counterpoise-corrected interaction energies. For each surface, over 1000 interaction energies were accurately interpolated using a reproducing kernel Hilbert space approach with an R-6+R-7 asymptotic form. In each case, both three-dimensional and effective two-dimensional surfaces were developed. In standard Jacobi coordinates, the final CBS+corr surface has a global minimum at rCO=2.1322a0,R=6.418a0, and gamma=70.84 degrees with a well depth of -22.34 cm-1. The other four surfaces have well depths ranging from -14.83 cm-1 [CCSD(T)/d-aug-cc-pVDZ] to -22.02 cm-1 [CCSD(T)/CBS]. For each of these surfaces the infrared spectrum has been accurately calculated and compared to experiment, as well as to previous theoretical and empirical surfaces. The final CBS+corr surface exhibits root-mean-square and maximum errors compared to experiment (4He) of just 0.03 and 0.04 cm-1, respectively, for all 42 transitions and is the most accurate ab initio surface to date for this system. Other quantities investigated include the interaction second virial coefficient, the integral cross sections, and thermal rate coefficients for rotational relaxation of CO by He, and rate coefficients for CO vibrational relaxation by He. All the observable quantities showed a smooth convergence with respect to the quality of the underlying interaction surface.

Contextual viewpoint to quantum stochastics

We study the role of context, complex of physical conditions, in quantum as well as classical experiments. It is shown that by taking into account contextual dependence of experimental probabilities we can derive the quantum rule for the addition of probabilities of alternatives. Thus we obtain quantum interference without applying the wave or Hilbert space approach. The Hilbert space representation of contextual probabilities is obtained as a consequence of the elementary geometric fact: cos-theorem. By using another fact from elementary algebra we obtain complex-amplitude representation of probabilities. Finally, we found contextual origin of noncommutativity of incompatible observables.

A Hilbert space approach to bounded analytic extension in the ball

One proves, using methods of Hilbert spaces with a reproducing kernel, that any bounded analytic function on a complex curve in general position in the unit ball of C$^n$ extends to a function in the Schur class of the ball.

Rigged Hilbert space approach to the Schrödinger equation

It is shown that the natural framework for the solutions of any Schrodinger equation whose spectrum has a continuous part is the Rigged Hilbert Space rather than just the Hilbert space. The difficulties of using only the Hilbert space to handle unbounded Schrodinger Hamiltonians whose spectrum has a continuous part are disclosed. Those difficulties are overcome by using an appropriate Rigged Hilbert Space (RHS). The RHS is able to associate an eigenket to each energy in the spectrum of the Hamiltonian, regardless of whether the energy belongs to the discrete or to the continuous part of the spectrum. The collection of eigenkets corresponding to both discrete and continuous spectra forms a basis system that can be used to expand any physical wave function. Thus the RHS treats discrete energies (discrete spectrum) and scattering energies (continuous spectrum) on the same footing.

Geometric separation of the polarization and charge transfer components of charge sensitivities of open molecular systems

The Hilbert space approach to the internal (isoelectronic) hardness and softness kernels is presented at the global and subsystem resolutions. A separation of components due to the intramolecular polarization (P) and external charge transfer (CT) is performed by projections into the complementary subspaces spanned by the P-vectors, eigenvectors of the molecular linear response operator, and the constant CT-vector. The quantities reflecting the P–CT coupling are discussed and explicit transformations between the (P, CT)-resolved perturbations and responses are reported.

Reproducing Kernel Hilbert Space Methods for wide-sense self-similar Processes

It has recently been observed that wide-sense self-similar processes have a rich linear structure analogous to that of wide-sense stationary processes. In this paper, a reproducing kernel Hilbert space (RKHS) approach is used to characterize this structure. The RKHS associated with a selfsimilar process on a variety of simple index sets has a straightforward description, provided that the scale-spectrum of the process can be factored. This RKHS description makes use of the Mellin transform and linear self-similar systems in much the same way that Laplace transforms and linear time-invariant systems are used to study stationary processes. The RKHS results are applied to solve linear problems including projection, polynomial signal detection and polynomial amplitude estimation, for general wide-sense self-similar processes. These solutions are applied specifically to fractional Brownian motion (fBm). Minimum variance unbiased estimators are given for the amplitudes of polynomial trends in fBm, and two new innovations representations for fBm are presented. 1. Introduction. This paper is concerned with the linear problems associated with wide-sense self-similar processes, that is, with processes whose first and second moments are essentially scale-invariant. In the linear problems we are referring to, the solution is constrained to be a linear functional of an observed random process, and the criterion of optimality is such that the solution is determined by the first and second moments of the observed process. Examples include linear projection with the mean-square error metric, maximum signal-to-noise ratio signal detection and minimum-variance unbiased linear estimation. For Gaussian processes, these linear solutions are still optimal when the linearity constraint is removed. The main finding is that wide-sense self-similar processes have essentially the same structure as wide-sense stationary processes, and that concepts such as autocorrelation and spectral density, used to study stationary processes, have simple analogs useful in the study of self-similar processes. The connection between these two classes was first made by Lamperti (1962), who pointed out a simple invertible transformation which connects any self-similar process with a stationary counterpart, and which is a central idea in this paper. Other recent workmak ing use of Lamperti’s transformation includes Albin (1998),

A Hilbert space approach to instability in semilinear partial differential equations

A method of R. Bellman in the instability theory of autonomous nonlinear ODE is enlarged to incompass semilinear evolution equations of the form $u'+Lu=f(u)$ in the Hilbert space framework. As a consequence it is shown that linearized instability of solutions to parabolic problems with linear damping in a bounded domain implies the Liapounov instability in those topologies for which the question is meaningful.

The connection between the rigged Hilbert space and the complex scaling approaches for resonances. The Friedrichs model

The rigged Hilbert space approach and the complex scaling method are compared for the Friedrichs model. It is shown that they define the same resonance pole. The choice of the test space for the rigged Hilbert space approach is discussed.

Reaction dynamics of S(1D)+H2/D2 on a new ab initio potential surface

A new ab initio potential energy surface is generated for the chemical reaction, S(1D)+H2. The quantum chemistry calculations were carried out at the multi-reference configuration interaction (MRCI) level with multi-configuration self-consistent field (MCSCF) reference wave functions. The 1A′, 2A′, 3A′, 1A″, and 2A″ singlet surfaces were computed on a uniform spatial grid of over 2000 points to simulate the full reaction pathway. The results indicate a barrierless insertion pathway along the T-shaped geometry and an 8 kcal/mol barrier to abstraction along the collinear geometry. The lowest surface was fit to a smooth analytical function form based on the reproducing kernel Hilbert space approach and a Carter–Murrell-type expansion. The dynamics of the S(1D)+H2/D2 reactions were simulated using the quasi-classical trajectory method. The results are generally consistent with an insertion mechanism mediated through capture dynamics in the entrance channel followed by the statistical decay of a long-lived complex. Comparison to recent molecular beam experiments shows agreement in the broad pattern of results but also exhibits significant differences in the more finely resolved quantities.

Dielectronic recombination satellite transitions in dense plasmas

Radiative and dielectronic recombination of multiply charged many-electron ions in high-temperature plasmas are usually treated as independent, non-interfering processes. A projection-operator and resolvent-operator approach has been developed to provide a fundamental, unified quantum-mechanical description of the combined electron–ion photo-recombination process. This ordinary Hilbert-space approach provides a valid description in low-density electron–ion beam interactions or in low-density plasmas. By means of the Hebrew University Lawrence Livermore Atomic Code (HULLAC), cross sections for photo-recombination of He-like ions through autoionizing states in Li-like ions have been obtained. For certain transitions, interference effects are predicted in the form of radiatively modified, asymmetric satellite cross-section profiles (or spectral line shapes). A density–matrix formulation has been developed for high-density plasmas, for which collisional and radiative relaxation processes can play an important role. Using Liouville-space projection-operator techniques, collisional and radiative relaxation processes (including cascades) are incorporated on an equal footing and in a self-consistent manner with autoionization and radiative emission. Both time-independent (resolvent-operator) and time-dependent (equation-of-motion) formulations are developed. The density–matrix approach provides a comprehensive description of the broadening of dielectronic satellite spectral lines due to autoionization processes, radiative transitions, electron–ion collisions, and electric and magnetic fields. We plan to adapt HULLAC so that the various elementary electron–ion collisional and radiative transition amplitudes can be used as input data in a generalized collisional-radiative model based on the density–matrix formulation. By means of the density–matrix approach, radiation processes involving resonant and non-resonant transitions in a diverse class of quantized electronic systems can be treated within the context of a single quantum statistical formulation.

A general hilbert space approach to wavelets and its application in geopotential determination

A general approach to wavelets is presented within a framework of a separable functional Hilbert space H. Basic tool is the construction of H-product kernels by use of Fourier analysis with respect to an orthonormal basis in H. Scaling functions and wavelet are defined in terms of H-product kernels. Wavelts are shown to be ‘building blocks’ that decorrelate the data. A pyramid scheme provides fast computation. Finally, the determination of the earth's gravitational potential from single and multipole expressions is organzied as an example of wavelet approximation in Hilbert space structure.

Quantization and scattering of massive scalar fields on exterior domains

A quantum theory is developed for the scattering of massive scalar fields on a class of non-globally hyperbolic spacetimes represented by foliations of Minkowski spacetime with a fixed compact set removed from each Cauchy surface. The field is restricted to the exterior of this set (exterior domain). At the classical level, the boundary value problem is recast as an abstract Cauchy problem in a Hilbert space, and the field solution is obtained as a unitary mapping of Cauchy data. The scattering theory is treated using a two Hilbert space approach, and the wave operators are constructed and shown to be asymptotically complete. At the quantum level, a field operator is constructed yielding a representation of the CCRs on a Fock space. Representations for the `in' and `out' asymptotic fields are developed, and a scattering operator is constructed and shown to be unitarily implementable.