Noncanonical Operators Research Articles

On the Asymptotic Behavior of Advanced Differential Equations with a Non-Canonical Operator

In this paper, we aim to study the oscillatory behavior of a class of even-order advanced differential equations with a non-canonical operator. In addition, we present results on the asymptotic behavior of this type of equations and provide an example that illustrates our main results.

Oscillation criteria for second‐order Emden–Fowler delay differential equations with a sublinear neutral term

AbstractNew oscillation criteria for the second‐order Emden–Fowler delay differential equation with a sublinear neutral term are presented. An essential feature of our results is that oscillation of the studied equation is ensured via only one condition. Furthermore, as opposed to the results by Agarwal et al. (Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1861–1875), Li and Rogovchenko (Math. Nachr. 288 (2015), no. 10, 1150–1162; Monatsh. Math. 184 (2017), no. 3, 489–500), and Xu (Monatsh. Math. 150 (2007), no. 2, 157–171), new criteria can be applied to Emden–Fowler delay differential equations with noncanonical operators and a sublinear neutral term. Our results essentially improve, extend, and simplify some known ones reported in the literature. The results are illustrated with examples.

Oscillatory Properties of Third-Order Neutral Delay Differential Equations with Noncanonical Operators

In the paper, we study the oscillatory and asymptotic properties of solutions to a class of third-order linear neutral delay differential equations with noncanonical operators. Via the application of comparison principles with associated first and second-order delay differential inequalities, we offer new criteria for the oscillation of all solutions to a given differential equation. Our technique essentially simplifies the process of investigation and reduces the number of conditions required in previously known results. The strength of the newly obtained results is illustrated on the Euler type equations.

On oscillation of third-order noncanonical delay differential equations

In the paper, we give new criteria for the oscillation of third-order delay differential equations with noncanonical operators. Our results essentially improve and simplify the ones in Dzurina et al. (2018). The results are illustrated by a Euler-type delay differential equation.

The fourth order strongly noncanonical operators

AbstractIt is shown that the strongly noncanonical fourth order operator$$\begin{array}{} \displaystyle \mathcal {L}\,y=\left(r_3(t)\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'\right)' \end{array}$$can be written in essentially unique canonical form as$$\begin{array}{} \displaystyle \mathcal {L}\,y = q_4(t)\left(q_3(t)\left(q_2(t)\left(q_1(t)\left(q_0(t)y(t)\right)'\right)'\right)'\right)'. \end{array}$$The canonical representation essentially simplifies examination of the fourth order strongly noncanonical equations$$\begin{array}{} \displaystyle \left(r_3(t)\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'\right)'+p(t)y(\tau(t))=0. \end{array}$$

Oscillation of third-order differential equations with noncanonical operators

New oscillation criteria for third-order delay differential equations with noncanonical operators are presented. Contrary to existing results, oscillation of the studied equation is attained via only two conditions. Our criteria not only improve, extend and significantly simplify existing ones, but the newly proposed approach could hopefully serve as a reference in the less-developed theory of noncanonical equations of higher-order. The importance of the results obtained is illustrated via Euler-type equations.

Oscillation of Second-Order Nonlinear Neutral Dynamic Equations with Noncanonical Operators

The study of half-linear differential equations has become an important area of research due to the fact that such equations occur in a variety of real-world problems such as in the study of \(p\)-Laplace equations, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium. On the basis of these background details, we study oscillatory behavior of a class of second-order neutral functional dynamic equations on a time scale. New criteria improve and complement related results reported in the literature. Some examples are included to illustrate the results obtained. In particular, an example regarding the second-order neutral differential equation is also provided to show that these theorems improve those in the continuous case.

ASYMPTOTIC BEHAVIOR OF FOURTH-ORDER NEUTRAL DYNAMIC EQUATIONS WITH NONCANONICAL OPERATORS

This paper is concerned with asymptotic behavior of a class of fourth-order neutral delay dynamic equations with a noncanonical operator on an arbitrary time scale. A new asymptotic criterion and an illustrative example are included.

Properties of higher-order half-linear functional differential equations with noncanonical operators

Some new results are presented for the oscillatory and asymptotic behavior of higher-order half-linear differential equations with a noncanonical operator. We study the delayed and advanced equations subject to various conditions.MSC:34C10, 34K11.

Equation of motion method for composite field operators

The Green's function formalism in Condensed Matter Physics is reviewed within the equation of motion approach. Composite operators and their Green's functions naturally appear as building blocks of generalized perturbative approaches and require fully self-consistent treatments in order to be properly handled. It is shown how to unambiguously set the representation of the Hilbert space by fixing both the unknown parameters, which appear in the linearized equations of motion and in the spectral weights of non-canonical operators, and the zero-frequency components of Green's functions in a way that algebra and symmetries are preserved. To illustrate this procedure some examples are given: the complete solution of the two-site Hubbard model, the evaluation of spin and charge correlators for a narrow-band Bloch system, the complete solution of the three-site Heisenberg model, and a study of the spin dynamics in the Double-Exchange model.

Schrödinger uncertainty relation and squeezed state

In a quantum optical model, we demonstrate both analytically and numerically that if the measurement of physical observables corresponds to non-canonical operators, the Schrödinger uncertainty relation may be used to define the squeezing, where the Schrödinger lower limit sets a higher bound on quantum fluctuations than the Heisenberg one does. The effect of the second-order correction to Rayleigh scattering on the squeezing is also discussed.

Puri's criterion and higher-order squeezing.

Recently, Puri revisited the usual uncertainty relation arguments. Utilizing a well-known and sharpened version of the uncertainty relation, he has proposed an alternative squeezing criterion for the case of noncanonical operators. We investigated this in the hope that Puri's proposal might allow one to define a higher-order squeezing criterion that would avoid an anomaly found with the standard Hong-Mandel (HM) scheme, i.e., the possibility of simultaneous-quadrature squeezing. However, it is found that higher-order squeezing based on Puri's squeezing criterion has what one might judge to be an even worse and unphysical defect than the HM scheme, in that even at the squeezing ``trigger level,'' squeezing in one quadrature is necessarily accompanied by considerable antisqueezing in the other quadrature.

Minimum-uncertainty states for noncanonical operators.

The usual uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}〈C${\mathrm{〉}}^{2}$/4 between two Hermitian operators A and B satisfying the noncanonical commutation relation [A,B]=iC, where C is not a constant multiple of the unit operator, fails to give a nontrivial lower bound on the product of the variances \ensuremath{\Delta}${\mathit{A}}^{2}$ and \ensuremath{\Delta}${\mathit{B}}^{2}$ when 〈C〉=0. For those operators, therefore, the general uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}[〈C${\mathrm{〉}}^{2}$+〈F${\mathrm{〉}}^{2}$]/4 where 〈F〉=〈AB+BA〉-2〈A〉〈B〉 is better suited to determine the lower bound on \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$. The implications of the general uncertainty relation and the properties of the minimum-uncertainty states, i.e., the states for which the general uncertainty relation is satisfied with equality, are discussed. The minimum-uncertainty states are found to fall in two different classes. One is the usually studied class for which 〈F〉=0, that is, the case when the usual uncertainty relation holds. The other is the hitherto unnoticed class for 〈C〉=0, that is, when the usual uncertainty relation is redundant. The squeezing properties of the present class of minimum-uncertainty states is discussed by defining squeezing in the light of the general uncertainty relation.

The Weyl Quantization of Phase Angle

We suggest a candidate for a non-canonical Hermitian operator corresponding to phase angle, based on the Weyl correspondence rule. The matrix elements of this operator, for harmonic oscillator number states, coincide, in the correspondence limit, with those of a phase operator proposed by Barnett and Pegg, when the dimension of their defining space becomes infinite.

Derivation of equations of motion of noncanonical operators from an action principle

The Lagrangian description of a quantum system with a finite number of noncanonical operators is presented. Using this description the dynamics of the operators of the system are derived. The class of variations, commutation rules, and the form of Lagrangian are shown to be self-consistent.

Axial-Vector Current in Spinor Electrodynamics

Adler has shown that in perturbation theory for spinor electrodynamics, suitably regularized, the divergence of the unrenormalized axial-vector current contains, in addition to the expected mass term, a term of the form ${\ensuremath{\epsilon}}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}{F}_{\ensuremath{\alpha}\ensuremath{\beta}}{F}_{\ensuremath{\gamma}\ensuremath{\delta}}$. Motivated by this result, we study here the renormalized axial-vector current ${{j}_{\ensuremath{\mu}}}^{5}$. We explicitly construct the essentially unique finite local ${{j}_{\ensuremath{\mu}}}^{5}$ and find it to be invariant under local gauge transformations. Taking the divergence ${\ensuremath{\partial}}_{\ensuremath{\mu}}{{j}_{\ensuremath{\mu}}}^{5}$ gives the renormalized analog of Adler's additional term. Similar noncanonical operator terms are seen to occur in equal-time commutators involving ${{j}_{\ensuremath{\mu}}}^{5}$. Although all matrix elements of ${\ensuremath{\partial}}_{\ensuremath{\mu}}{{j}_{\ensuremath{\mu}}}^{5}$ are finite, we find that the off-shell Green's function $〈T{\ensuremath{\partial}}_{\ensuremath{\mu}}{{j}_{\ensuremath{\mu}}}^{5}(x)\ensuremath{\psi}(y)\overline{\ensuremath{\psi}}(z)〉$ is divergent and, correspondingly, that equal-time commutators involving ${{j}_{\ensuremath{\mu}}}^{5}$ are in general singular. We show that Ward identities can nevertheless be given a meaning. The algebraic properties of ${{j}_{\ensuremath{\mu}}}^{5}$ are seen to be reflected in the scattering amplitudes of the theory. Renormalized integral representations of axial-vector vertices are constructed, and radiative corrections to weak interaction are discussed. We conclude with a discussion of the axial-vector currents in other spinor models.