Canonical Operator Research Articles

Quantization of nonlocal fractional field theories via the extension problem

We use the extension problem proposed by Caffarelli and Silvestre to study the quantization of a scalar nonlocal quantum field theory built out of the fractional Laplacian. We show that the quantum behavior of such a nonlocal field theory in $d$-dimensions can be described in terms of a local action in $d+1$ dimensions which can be quantized using the canonical operator formalism though giving up local commutativity. In particular, we discuss how to obtain the two-point correlation functions and the vacuum energy density of the nonlocal fractional theory as a brane limit of the bulk correlators. We show explicitly how the quantized extension problem reproduces exactly the same particle content of other approaches based on the spectral representation of the fractional propagator. We also briefly discuss the inverse fractional Laplacian and possible applications of this approach in general relativity and cosmology.

Uniform Asymptotic Solution in the Form of an Airy Function for Semiclassical Bound States in One-Dimensional and Radially Symmetric Problems

We consider stationary scalar and vector problems for differential and pseudodifferential operators leading to the appearance of asymptotic solutions of one-dimensional problems localized in a neighborhood of intervals (“bound states”). Based on the semiclassical approximation and the Maslov canonical operator, we develop a constructive algorithm that allows writing an asymptotic solution globally under certain conditions using an Airy function of complex argument.

Conic Lagrangian Varieties and Localized Asymptotic Solutions of Linearized Equations of Relativistic Gas Dynamics

We study asymptotic solution of the Cauchy problem for the linearized system of relativistic gas dynamics. We assume that initial condiditiopns are strongly localized near a space-like surface in the Minkowsky space. We prove that the solution can be decomposed into three modes, corresponding to different routsb of the equations of characteristics. One of these roots is twice degenerate and the there are no focal points in the corresponding miode. The other two roots are simple; in order to describe the corresponding modes, we use the modificication of the Maslov’s canonical operator which was obtained recently.

On the Application of Asymptotic Formulae Based on the Modified Maslov Canonical Operator to the Modeling of Acoustic Pulses Propagation in Three-Dimensional Shallow-Water Waveguides

In this study a technique for the modeling of propagation of acoustic pulses in shallow-water waveguides with three-dimensional bottom inhomogeneities is described. The described approach is based on the ray theory of sound propagation and the method of modified Maslov canonical operator. Representation of acoustical field in terms of the canonical operator gives several important advantages in practical computations. In particular, it is possible to compute the time series of a pulse at a reception point located on the caustics of a family of rays. Besides, a significant part of calculations within the proposed approach can be performed analytically; therefore, overall computational costs are substantially reduced. As an example, sound propagation in a wedge-shaped waveguide representing a shelf area near the coast line is considered. The ray geometry in such a waveguide is discussed both in the isovelocity case and in the presence of the thermocline in the water column. For both cases, the time series of an acoustical pulse propagating along the track aligned along the isobaths (parallel to the apex edge of the wedge) is calculated.

The Uniqueness Theorems in the Inverse Problems for Dirac Operators

We introduce new supplementary data to the set of the eigenvalues, to determine uniquely potential matrix in the inverse problem for Dirac canonical operator. Besides, we obtain others uniqueness theorems in inverse problems, which are the analogues of well-known Borg, Marchenko and McLaughlin-Rundell theorems in inverse Sturm-Liouville problems.

Disentangling the IR spectra of 2,3,3,3-tetrafluoropropene using an ab initio description of vibrational polyads by means of canonical Van Vleck perturbation theory

The vibrational spectra of 2,3,3,3-tetrafluoropropene (2333TFP) are studied in the infrared experimentally and theoretically by the canonical second-order Van Vleck operator perturbation theory (CVPT2) and full quartic potential energy surface (PES). 2333TFP belongs to hydrofluoroolefins (HFOs) considered among the most promising alternatives to the hydrofluorocarbons presently in use in refrigeration. The medium resolution infrared spectra of gaseous 2333TFP were recorded in the range 9500–30 cm−1. The integrated IR intensities for all bands falling between 6300–400 cm−1 were accurately determined. Theoretical descriptions of up to four-quanta anharmonic vibrational states of 2333TFP are performed using the state-of-the-art numerical-analytic implementation of CVPT2. High quality harmonic frequencies, obtained at different coupled-cluster levels of theory up to the CCSD(T*)-F12c/VTZ-F12 model were combined with the full quartic MP2/cc-pVTZ anharmonic force field to form the hybrid PES. All Fermi and Darling-Dennison resonances were detected using a universal criterion and treated with the variational CVPT2/VCI stage by using two-, three-, and four-quanta harmonic oscillator basis sets. General recommendations for choosing such basis sets in future studies are formulated. In addition to all fundamentals and their resonance satellites, the vibrational analysis led to the assignment of virtually all observed first/second overtone and binary/ternary combination bands. Computed dipole moment first derivatives in the principal axis system yielded reliable predictions of band shapes for fundamental transitions in good agreement with observed counterparts. Computed anharmonic IR intensities showed an excellent agreement with observed data in the measured ranges. The fundamental spectroscopic effect of excited resonances is observed experimentally and explained theoretically as a universal mechanism of formation of vibrational polyads. The efficiency of the CVPT2/VCI approach employed is clearly demonstrated and can be considered as a benchmark for modeling vibrational states and interpretation of vibrational spectra of semi-rigid molecules.

Maslov's Canonical Operator in Problems on Localized Asymptotic Solutions of Hyperbolic Equations and Systems

An analog of Maslov's canonical operator is defined for functions localized in a neighborhood of subsets of positive codimension.

Nonstandard Lagrangian Singularities and Asymptotic Eigenfunctions of the Degenerating Operator $$ - {d \over {dx}}D\left( x \right){d \over {dx}}$$

We express the asymptotic eigenfunctions of the operator $$ - {d \over {dx}}D\left( x \right){d \over {dx}}$$ that degenerates at the endpoints of an interval in terms of the modified Maslov canonical operator introduced in our previous studies.

ДВУХФАЗОВЫЕ АСИМПТОТИКИ В ВИДЕ ФУНКЦИЙ ЭЙРИ ДЛЯ ЗАДАЧИ ОБ ОБРУШЕНИИ ВОЛНЫ ДЛЯ УРАВНЕНИЯ I-БЮРГЕРСА

We consider wave breaking problems for the Burgers equation with a small “imaginary viscosity,” which in fact plays the role of small dispersion. Although this equation has no apparent physical meaning, the problem in question is an interesting analog of the famous Gurevich-Pitaevsky problem on the onset of an oscillation zone as the breaking of a simple wave occurs for the Korteweg-de Vries equation. In contrast to the latter, the solution of the i-Burgers equation in the oscillation zone can be described explicitly and has a two-phase structure. This was indicated more than 25 years ago in (Dobrokhotov et al., 1992), where the solution was constructed in the form of a function of Maslov’s canonical operator. Now we use the recent results in (Dobrokhotov, Nazaikinskii, 2018) to present the solutions in the more efficient form of uniform asymptotics represented as the logarithmic derivative of the Airy function of a composite argument. The research was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. AAAA-A17-117021310377-1).

Continuous wavelet transform involving canonical convolution

The main objective of this paper is to study the continuous wavelet transform in terms of canonical convolution and its adjoint. A relation between the canonical convolution operator and inverse linear canonical transform is established. The continuity of continuous wavelet transform on test function space is discussed.

Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models

SummaryThe parameters in the governing system of partial differential equations of multiple‐network poroelasticity models typically vary over several orders of magnitude, making its stable discretization and efficient solution a challenging task. In this paper, we prove the uniform Ladyzhenskaya–Babuška–Brezzi (LBB) condition and design uniformly stable discretizations and parameter‐robust preconditioners for flux‐based formulations of multiporosity/multipermeability systems. Novel parameter‐matrix‐dependent norms that provide the key for establishing uniform LBB stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ but also to all the other model parameters, such as the permeability coefficients Ki; storage coefficients ; network transfer coefficients βi j,i,j = 1,…,n; the scale of the networks n; and the time step size τ. Moreover, strongly mass‐conservative discretizations that meet the required conditions for parameter‐robust LBB stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm‐equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.

A Hardy–Littlewood Maximal Operator for the Generalized Fourier Transform on $${\mathbb {R}}$$R

In this paper, we define and study a canonical Hardy–Littlewood-type maximal operator associated with the one-dimensional generalized Fourier transform. For this operator to which covering methods do not apply, we construct a geometric maximal operator, which controls pointwise the canonical maximal operator, and for which we can use the machinery of real analysis to obtain a maximal theorem.

ENFORCEABLE OPERATOR ALGEBRAS

AbstractWe adapt the classical notion of building models by games to the setting of continuous model theory. As an application, we study to what extent canonical operator algebras are enforceable models. For example, we show that the hyperfinite II1 factor is an enforceable II1 factor if and only if the Connes Embedding Problem has a positive solution. We also show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian $\text{C}^{\ast }$-algebras and use this to show that it is the prime model of its theory.

Asymptotics of Wave Functions of the Stationary Schrödinger Equation in the Weyl Chamber

We study stationary solutions of the Schrodinger equation with a monotonic potential U in a polyhedral angle (Weyl chamber) with the Dirichlet boundary condition. The potential has the form \(U\left( x \right) = \sum _{j = 1}^nV\left( {{x_j}} \right),x = \left( {{x_1}, \ldots ,{x_n}} \right) \in {\mathbb{R}^n}\) , with a monotonically increasing function V (y). We construct semiclassical asymptotic formulas for eigenvalues and eigenfunctions in the form of the Slater determinant composed of Airy functions with arguments depending nonlinearly on xj. We propose a method for implementing the Maslov canonical operator in the form of the Airy function based on canonical transformations.

Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials

We discuss two approaches that can be used to obtain the asymptotics of Hermite polynomials. The first, well-known approach is based on the representation of Hermite polynomials as solutions of a spectral problem for the harmonic oscillator Schrodinger equation. The second approach is based on a reduction of the finite-difference equation for the Hermite polynomials to a pseudodifferential equation. Associated with each of the approaches are Lagrangian manifolds that give the asymptotics of Hermite polynomials via the Maslov canonical operator.

Efficient Formulas for the Maslov Canonical Operator near a Simple Caustic

We revive well-known old formulas for the asymptotics of rapidly oscillating integrals with two nearly coincident saddle points and present an efficient approach to the computation of functions represented by Maslov’s canonical operator near a simple caustic in an algorithmic fashion convenient for implementation on software like Wolfram Mathematica. The exposition is elementary and assumes that no preliminary information about the canonical operator is be known to the reader.

Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem

Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to theHamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain.

Electronic optics in graphene in the semiclassical approximation

We study above-barrier scattering of Dirac electrons by a smooth electrostatic potential combined with a coordinate-dependent mass in graphene. We assume that the potential and mass are sufficiently smooth, so that we can define a small dimensionless semiclassical parameterh≪1. This electronic optics setup naturally leads to focusing and the formation of caustics, which are singularities in the density of trajectories. We construct a semiclassical approximation for the wavefunction in all points, placing particular emphasis on the region near the caustic, where the maximum of the intensity lies. Because of the matrix character of the Dirac equation, this wavefunction contains a nontrivial semiclassical phase, which is absent for a scalar wave equation and which influences the focusing. We carefully discuss the three steps in our semiclassical approach: the adiabatic reduction of the matrix equation to aneffective scalar equation, the construction of the wavefunction using the Maslov canonical operator and the application of the uniform approximation to the integral expression for the wavefunction in the vicinity of a caustic. We consider several numerical examples and show that our semiclassical results are in very good agreement with the results of tight-binding calculations. In particular, we show that the semiclassical phase can have a pronounced effect on the position of the focus and its intensity.

Wick theorem for all orderings of canonical operators

Wick’s theorem, known for yielding normal ordered from time-ordered bosonic fields, may be generalized for a simple relationship between any two orderings that we define over canonical variables, in a broader sense than before. In this broad class of orderings, the general Wick theorem follows from the Baker–Campbell–Hausdorff identity. We point out that, generally, the characteristic function does not induce an unambigous scheme to order the multiple products of the canonical operators although the value of the ordered product is unique. We construct a manifold of different schemes for each value of s of s-orderings of Cahill and Glauber.

Kinetic theory and classical limit for real scalar quantum field in curved spacetime

Starting from a real scalar quantum field theory with quartic self-interactions and non-minimal coupling to classical gravity, we define four equal-time, spatially covariant phase-space operators through a Wigner transformation of spatially translated canonical operators within a 3+1 decomposition. A subset of these operators can be interpreted as fluctuating particle densities in phase-space whenever the quantum state of the system allows for a classical limit. We come to this conclusion by expressing hydrodynamic variables through the expectation values of these operators and moreover, by deriving the dynamics of the expectation values within a spatial gradient expansion and a 1-loop approximation which subsequently yields the Vlasov equation with a self-mass correction as a limit. We keep an arbitrary classical metric in the 3+1 decomposition which is assumed to be determined semi-classically. Our formalism allows to systematically study the transition from quantum field theory in curved space-time to classical particle physics for this minimal model of self-interacting, gravitating matter. As an application we show how to include relativistic and self-interaction corrections to existing dark matter models in a kinetic description by taking into account the gravitational slip, vector perturbations and tensor perturbations.