Conjugation Operator Research Articles

On deformation of certain functional classes in the spaces and

The problem of violation of the invariance of the functional classes and under a multidimensional conjugation operator is studied in the case when the moduli of continuity satisfy Zygmund's condition. Direct estimates are obtained and sharpness of these estimates is established.

Phase conjugation of continuous quantum variables

The phase conjugation of an unknown Gaussian state cannot be realized perfectly by any physical process. A semi-classical argument is used to derive a tight lower bound on the noise that must be introduced by an approximate phase conjugation operation. A universal transformation achieving the optimal imperfect phase conjugation is then presented, which is the continuous counterpart of the universal-NOT transformation for quantum bits. As a consequence, it is also shown that more information can be encoded into a pair of conjugate Gaussian states than using twice the same state.

Mass-symmetric leptonium and quarkonium equations

Abstract The energy levels of leptonium and quarkonium are mass-symmetric, but in relativistic two-fermion equations, this symmetry is explicit only in a recent eight-component form. The spin asymmetry of that equation is shown here to reflect a new group of unitary transformations. The eight-component charge conjugation operator is also constructed and applied to quarkonium.

CONJUGATE OPERATORS FROM DISPERSION CURVES FOR PERTURBATIONS OF FIBERED SYSTEMS IN CYLINDERS1†

We study the spectral properties of elliptic systems on perturbed fibered cylinders. This general framework covers the acoustic and elastic cylinders. Although the electromagnetic case does not exactly fit into this framework, we prove similar results for it. Our approach is based on constructing an appropriate conjugate operator using general properties of the dispersion curves. † This work was supported by INRIA, France, and partially by NSF grant DMS-96-22310.

A MOURRE ESTIMATE FOR A SCHRÖDINGER OPERATOR ON A BINARY TREE

Let G be a binary tree with vertices V and H be a Schrödinger operator acting on ℓ2 (V). A decomposition of the space ℓ2 (V) into invariant subspaces is exhibited yielding a conjugate operator A for use in the Mourre estimate. We show that for potentials q satisfying a first order difference decay condition, a Mourre estimate for H holds.

Noncommutative and gauge theories

We study the generalization of noncommutative gauge theories to the case of orthogonal and symplectic groups. We find out that this is possible, since we are allowed to define orthogonal and symplectic subgroups of noncommutative unitary gauge transformations even though the gauge potentials and gauge transformations are not valued in the orthogonal and symplectic subalgebras of the Lie algebra of antihermitean matrices. Our construction relies on an antiautomorphism of the basic noncommutative algebra of functions which generalizes the charge conjugation operator of ordinary field theory. We show that the corresponding noncommutative picture from low energy string theory is obtained via orientifold projection in the presence of a non-trivial NSNS B-field.

Why two qubits are special

We analyze some special properties of a system of two qubits, and in particular of the so-called Bell basis for this system, and discuss the possibility of extending these properties to higher dimensional systems. We give a general construction for orthonormal bases of maximally entangled vectors, which works in any dimension, and is based on Latin squares and complex Hadamard matrices. However, for none of these bases the special properties of the operation of complex conjugation in Bell basis hold, namely that maximally entangled vectors have up-to-a-phase real coefficients and that factorizable unitaries have real matrix elements.

Linear quasi-differential operators in locally integrable spaces on the real line

The theory of ordinary linear quasi-differential expressions and operators has been extensively developed in integrable-square Hilbert spaces. There is also an extensive theory of ordinary linear differential expressions and operators in integrable-p Banach spaces.However, the basic definition of linear quasi-differential expressions involves Lebesgue locally integrable spaces on intervals of the real line. Such spaces are not Banach spaces but can be considered as complete locally convex linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete locally convex linear topological space, but now with the topology derived as a strict inductive limit.This paper develops the properties of linear quasi-differential operators in a locally integrable space and the first conjugate space. Conjugate and preconjugate operators are defined in, respectively, dense and total domains.

Numerical solution of boundary integral equations by means of attenuation factors

We consider first-kind boundary integral equations with logarithmic kernel such as those arising from solving Dirichlet problems for the Laplace equation by means of single-layer potentials. The first-kind equations are transformed into equivalent equations of the second kind which contain the conjugation operator and which are then solved with a degenerate-kernel method based on Fourier analysis and attenuation factors. The approximations we consider, among them spline interpolants, are linear and translation invariant. In view of the particularly small kernel, the linear systems resulting from the discretization can be solved directly by fixed-point iteration.

On boundedness of the hardy and bellman operators in the spaces h and bmo

Hardy (1919) proved that the space Lp(T), 1 ⩽ p ⩽ ∞, is invariant under (C, 1)-transformation of Fourier coefficients. This transformation may be considered as a linear integral operator H: Lp(T) → Lp(T), 1 ⩽ p ⩽ ∞. In this paper we show that the operator H is unbounded in the space BMO(T) of periodic functions of bounded mean oscillation. For the conjugate operator B introduced by Bellman (1944) B: Lq(T) → Lq(T), 1/ρp + 1/rho;q = 1, the boundedness of B in BMO(T) is proved directly without duality argument. An example is given to show that B is unbounded in H(T).

ENTANGLED EIGENSTATES AND SQUEEZING OPERATOR FOR COMPLEX SCALAR FIELDS

We derive the entangled eigenstate ‖ξ> of complex scalar fields ϕ and ϕ† in the Fock space. The ‖ξ> state is found to embed the entanglement possessed by the Einstein–Podolsky–Rosen states in quantum mechanics. The ‖ξ> set spans a complete and orthonormal representation. The advantage of the new <ξ‖-representation helps us to derive the normally ordered forms of the squeezing and charge conjugate operators for complex scalar fields rather easily.

ON THE SPECTRAL PROPERTIES OF DISCRETE SCHRÖDINGER OPERATORS

We use the method of the conjugate operator to prove the limiting absorption principle and the absence of the singular continuous spectrum for the discrete Schrödinger operator. We also obtain local decay estimates. Our results apply to a large class of perturbating potentials V tending arbitrarily slowly to zero at infinity.

Absolutely continuous spectrum of perturbed Stark operators

Absolutely continuous spectrum of perturbed Stark operators

At Last We Have an Undisputed Observation of ‘Direct’ CP Violation in Kaon Decay

A long-awaited experimental result, recently announced by the KTeV collaboration at the Fermilab Tevatron, does much to clarify the mechanism by which the decay of the neutral K mesons violates CP symmetry. Despite 35 years of painstaking investigation since the discovery of CP-violating K0 decay by James Cronin, Val Fitch, James Christenson, and René Turlay, the important issue of what causes this asymmetry in nature had remained disturbingly unsettled. CP denotes the combined operation of charge conjugation (C), that is to say, the replacement of particles by their antiparticles, and parity inversion (P). After the rude overthrow of parity conservation in 1957, CP offered a refuge for believers in mirror symmetry, but only until the next rude overthrow, by Fitch, Cronin and company, in 1964.

Maximal monotonicity of dense type, local maximal monotonicity, and monotonicity of the conjugate are all the same for continuous linear operators

The concept of a monotone operator --- which covers both linear positive semi-definite operators and subdifferentials fo convex functions --- is fundamental in various branches of mathematics. Over the last few decades, several stronger notions of monotonicity have been introduced: Gossez's maximal monotonicity of dense type, Fitzpatrick and Phelps's local maximal monotonicity, and Simons's monotonicity of type (NI). While these monotonicities are automatic for maximal monotone operators in reflexive Banach spaces and for subdifferentials of convex functions, their precise relationship is largely unknown. Here, it is shown --- within the beautiful framework of Convex Analysis --- that for continuous linear monotone operators, \emph{all these notions coincide and are equivalent to the monotonicity of the conjugate operator}. This condition is further analyzed and illustrated by examples.

Steganography based on region segmentation with a complexity measure

This paper describes a new steganographic technique using image segmentation based on a local complexity measure. The key to this approach is that a binary image can be divided into “informative” and “noiselike” regions by using a “complexity measure.” This property allows us to embed secret data in the noiselike regions if the secret data have a random pattern. If the secret data are simple, then we apply an image conjugate operation that transforms a simple pattern into a complex pattern. In the experiment, a JPEG file was embedded into a gray scale dummy image (256 × 256, 8 bits/pixel) without losing any visual information. © 1999 Scripta Technica, Syst Comp Jpn, 30(3): 1–9, 1999

Anisotropic Jacobi matrices with absolutely continuous spectrum

We describe a class of anisotropic Jacobi matrices with absolutely continuous spectrum and discrete singular spectrum. The discrete Schrödinger operator perturbed by a potential having different behaviors near +∞ and -∞, belongs to this class. We use the method of the conjugate operator and techniques similar to those for the 3-body problem.

Charged particles in a -curved background

The coupling to a 2+1 background geometry of a quantized charged test particle in a strong magnetic field is analyzed. Canonical operators adapting to the fast and slow freedoms produce a natural expansion in the inverse square root of the magnetic field strength. The fast freedom is solved to the second order. At any given time, space is parameterized by a couple of conjugate operators and effectively behaves as the `phase space' of the slow freedom. The slow Hamiltonian depends on the magnetic field norm, its covariant derivatives, the scalar curvature and presents a peculiar coupling with the spin-connection.

𝓟𝓣-symmetric quantum mechanics

This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H‡=H, where ‡ represents combined parity reflection and time reversal 𝒫𝒯, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p2+x2(ix)ε of the harmonic oscillator Hamiltonian, where ε is a real parameter. The system exhibits two phases: When ε⩾0, the energy spectrum of H is real and positive as a consequence of 𝒫𝒯 symmetry. However, when −1<ε<0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because 𝒫𝒯 symmetry is spontaneously broken. The phase transition that occurs at ε=0 manifests itself in both the quantum-mechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H=p2+x2N(ix)ε with N integer and ε>−N; each of these complex Hamiltonians exhibits a phase transition at ε=0. These 𝒫𝒯-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.

Discrete symmetries andS-matrix of theXXZchain

We formulate the notion of parity for the periodic XXZ spin chain within the Quantum Inverse Scattering Method. We also propose an expression for the eigenvalues of the charge conjugation operator. We use these discrete symmetries to help classify low-lying S^z=0 states in the critical regime, and we give a direct computation of the S matrix.