Conjugation Operator Research Articles

Optimal limiting absorption principle for a Schrödinger type operator on a Lipschitz cylinder

Spectral properties of a Schrodinger type operator on a cylinder is established by a version of Mourre's conjugate operator method. In particular, the limiting absorption principle is proved in an optimal scale of Besov spaces.

Dark matter: A spin one-half fermion field with mass dimension one?

We report an unexpected theoretical discovery of a spin one-half matter field with mass dimension one. It is based on a complete set of eigenspinors of the charge conjugation operator. Because of its unusual properties with respect to charge conjugation and parity, it belongs to a nonstandard Wigner class. Consequently, the theory exhibits nonlocality with $(CPT{)}^{2}=\ensuremath{-}\mathbb{I}$. Its dominant interaction with known forms of matter is via Higgs, and with gravity. This aspect leads us to contemplate it as a first-principle candidate for dark matter.

The maximal conjugate and Hilbert operators on real Hardy spaces

We prove that the maximal conjugate and Hilbert operators are not bounded from the real Hardy space H1 to L1, where the underlying spaces may be over T or R. We also draw corollaries for the corresponding spaces over T2 and R2.

Functional Identification of the Nonlinear Thermal-Conductivity Coefficient by Gradient Methods. I. Conjugate Operators

Consideration is given to the gradient methods of solution of the inverse heat-conduction problem on determination of the nonlinear coefficient λ(T) without its preliminary finite-dimensional approximation.

Spin-half fermions with mass dimension one: theory, phenomenology, and darkmatter

We provide the first details on the unexpected theoretical discovery of a spin-one-halfmatter field with mass dimension one. It is based upon a complete set of dual-helicityeigenspinors of the charge conjugation operator. Due to its unusual properties withrespect to charge conjugation and parity, it belongs to a non-standard Wignerclass. Consequently, the theory exhibits non-locality with . We briefly discuss its relevance to the cosmological ‘horizon problem’. Because theintroduced fermionic field is endowed with mass dimension one, it can carry a quarticself-interaction. Its dominant interaction with known forms of matter is via Higgs, and withgravity. This aspect leads us to contemplate the new fermion as a prime darkmatter candidate. Taking this suggestion seriously we study a supernova-likeexplosion of a galactic-mass dark matter cloud to set limits on the mass of the newparticle and present a calculation on relic abundance to constrain the relevantcross-section. The analysis favours light mass (roughly 20 MeV) and relevant cross-sectionof about 2 pb. Similarities and differences with the WIMP and mirror matterproposals for dark matter are enumerated. In a critique of the theory we reveal ahint on non-commutative aspects of spacetime, and energy–momentum space.

Fermions in Odd Space-Time Dimensions: Back to Basics

It is a well known feature of odd space-time dimensions $d$ that there exist two inequivalent fundamental representations $A$ and $B$ of the Dirac gamma matrices. Moreover, the parity transformation swaps the fermion fields living in $A$ and $B$. As a consequence, a parity invariant Lagrangian can only be constructed by incorporating both the representations. Based upon these ideas and contrary to long held belief, we show that in addition to a discrete exchange symmetry for the massless case, we can also define chiral symmetry provided the Lagrangian contains fields corresponding to both the inequivalent representations. We also study the transformation properties of the corresponding chiral currents under parity and charge conjugation operations. We work explicitly in 2+1 dimensions and later show how some of these ideas generalize to an arbitrary number of odd dimensions.

Totally hereditarily normaloid operators and Weyl's theorem for an elementary operator

A Hilbert space operator T ∈ B ( H ) is hereditarily normaloid (notation: T ∈ HN ) if every part of T is normaloid. An operator T ∈ HN is totally hereditarily normaloid (notation: T ∈ THN ) if every invertible part of T is normaloid. We prove that THN-operators with Bishop's property ( β ) , also THN-contractions with a compact defect operator such that T −1 ( 0 ) ⊆ T ∗ −1 ( 0 ) and non-zero isolated eigenvalues of T are normal, are not supercyclic. Take A and B in THN and let d A B denote either of the elementary operators in B ( B ( H ) ) : Δ A B and δ A B , where Δ A B ( X ) = A X B − X and δ A B ( X ) = A X − X B . We prove that if non-zero isolated eigenvalues of A and B are normal and B −1 ( 0 ) ⊆ B ∗ −1 ( 0 ) , then d A B is an isoloid operator such that the quasi-nilpotent part H 0 ( d A B − λ ) of d A B − λ equals ( d A B − λ ) −1 ( 0 ) for every complex number λ which is isolated in σ ( d A B ) . If, additionally, d A B has the single-valued extension property at all points not in the Weyl spectrum of d A B , then d A B , and the conjugate operator d A B ∗ , satisfy Weyl's theorem.

Relations between multiresolution analysis and quantum mechanics

We discuss a procedure to construct multiresolution analyses (MRA) of L2(R) starting from a given seed function h(s) which should satisfy some conditions. Our method, originally related to the quantum mechanical Hamiltonian of the fractional quantum Hall effect, is shown to be model independent. The role of a canonical map between certain canonically conjugate operators is discussed. This clarifies our previous procedure and makes much easier most of the original formulas, producing a convenient framework to produce examples of MRA.

Charged Representations of the Infinite Fermi and Clifford Algebras

The real and quaternionic charge conjugation operators invariant under the infinite-dimensional Clifford algebra, or compatible with the Fermi algebra, are determined. There results a maze of inequivalent irreducible charged representations, all of which are non-Fock. The representation vectors and their charges admit two interpretations besides those of spinors or states of quantum fields: as wavelets on the circle, with charge conjugations acting via ordinary complex conjugation; and as infinite-dimensional numbers, with charge conjugations acting by automorphisms.

On the duality problem for weakly compact, strictly singular operators

We show that there exists a weakly compact, strictly singular operator T such that the conjugate operator T * is not strictly cosingular. This example gives a negative answer to an old question of Pełczyński.

Isometries, Fock spaces, and spectral analysis of Schrödinger operators on trees

We construct conjugate operators for the real part of a completely non-unitary isometry and we give applications to the spectral and scattering theory of a class of operators on (complete) Fock spaces, natural generalizations of the Schrödinger operators on trees. We consider C * -algebras generated by such Hamiltonians with certain types of anisotropy at infinity, we compute their quotient with respect to the ideal of compact operators, and give formulas for the essential spectrum of these Hamiltonians.

Scattering theory for the Schrödinger equation with repulsive potential

We consider the scattering theory for the Schrödinger equation with − Δ − | x | α as a reference Hamiltonian, for 0 < α ⩽ 2 , in any space dimension. We prove that, when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of α, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity and describe its spectrum. Some results are generalized to the case where − | x | α is replaced by a general second order polynomial.

Rabi oscillations in a two-level atomic system with a pseudo-Hermitian Hamiltonian

Rabi oscillations of a two-level atomic system with damping effects are described by the use of a pseudo-Hermitian Hamiltonian. The Rabi frequency and the amplitudes for being in the upper and lower level depend in the present system on both the Hermitian resonant interaction between the em field and the atomic dipole and on damping effects. The time development of the two-level system is studied by using the metric of the pseudo-Hermitian Hamiltonian and its bi-orthonormal basis of states. The special characteristics of the present system are related to PT (parity-time reversal) and C (generalized conjugation operator) invariant properties of the pseudo-Hermitian Hamiltonian.

Spin and statistics in classical mechanics

The spin–statistics connection is obtained for classical particles. The connection holds within pseudomechanics, a theory of particle motion that extends classical physics to include anticommuting Grassmann variables, and that exhibits classical analogs of both spin and statistics. Classical realizations of Lie groups are constructed in a canonical formalism generalized to include Grassmann variables. The theory of irreducible canonical realizations of the Poincaré group is developed in this framework, with particular emphasis on the rotation subgroup. The behavior of irreducible realizations under time inversion and charge conjugation is obtained. The requirement that the Lagrangian retain its form under the combined operation of charge conjugation and time reversal leads directly to the spin–statistics connection by an adaptation of Schwinger’s 1951 proof to irreducible canonical realizations of the Poincaré group of spin j: Generalized spin coordinates and momenta satisfy fundamental Poisson bracket relations for 2j even, and fundamental Poisson antibracket relations for 2j odd.

Minimal Escape Velocities for Unitary Evolution Groups

Starting from a strict Mourre inequality, the minimal escape velocity for a unitary evolution group in a Hilbert space is derived under some minimal conditions. If the self-adjoint generator H of this evolution is a Schrodinger operator and if the conjugate operator is the generator of dilations, then it follows that H has very good and easily understandable propagation properties. The striking fact is that no proof of the absence of singularly continuous spectrum of H is available yet under such weak conditions.

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C 0-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [ H,i A] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.

Ideals of compact operators

AbstractWe give an example of a Banach spaceXsuch thatK(X, X) is not an ideal inK(X, X**). We prove that ifz* is a weak* denting point in the unit ball of Z* and ifXis a closed subspace of a Banach spaceY, then the set of norm-preserving extensionsH B(x* ⊗z*) ⊆(Z*,Y)* of a functionalx* ⊗Z* ∈ (Z⊗X)* is equal to the setH B(x*) ⊗ {z*}. Using this result, we show that ifXis anM-ideal inYandZis a reflexive Banach space, thenK(Z, X) is anM-ideal inK(Z, Y) wheneverK(Z, X) is an ideal inK(Z, Y). We also show thatK(Z, X) is an ideal (respectively, anM-ideal) inK(Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, anM-ideal) inYandX* has the compact approximation property with conjugate operators.

Green Functions of the Stationary Schrödinger Equation and Tomographic Representation of Quantum Mechanics

Invertible maps of operators of quantum observables onto functions of c-number arguments and their associative products are reviewed. In particular, the symplectic tomography map is discussed and an expression connecting an arbitrary operator and its tomographic symbol is written down. This formula is applied to obtain explicit expressions for tomographic symbols, which are symplectic tomograms of Green functions of stationary and nonstationary Schrodinger equations written for the case of harmonic oscillator. The connection between the so-called classical propagator Π(X,μ,ν,t,X′,μ′,ν′,0) and the tomographic symbol of the evolution operator of nonstationary Schrodinger equation is found. The spin tomography is presented as a map of operators acting in spinor space onto functions of c-arguments. As an example, the spin located in a magnetic field is considered and the tomographic symbol of resolventa is obtained. Tomographic symbols of hermitian conjugate operators are shown to be complex conjugate functions.

Spectral phase conjugation with cross-phase modulation compensation

Spectral phase conjugation with short pump pulses in a third-order nonlinear material is analyzed in depth. It is shown that if signal amplification is considered, the conversion efficiency can be significantly higher than previously considered, while the spectral phase conjugation operation remains accurate. A novel method of compensating for cross-phase modulation, the main parasitic effect, is also proposed. The validity of our theory and the performance of the spectral phase conjugation scheme are studied numerically.

Singularities of the resolvent at the thresholds of a stratified operator: a general method

AbstractOur problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto‐acoustic operator H defined in ℝ2 × I where I is a bounded interval of ℝ with coefficients depending only on z∈I. One applies the ‘conjugate operator method’ to an operator obtained by a spectral decomposition of the partial Fourier transform Ĥ of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator may be constructed which ensures the ‘good properties’ of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues). If the point is a threshold, the limiting absorption principle is valid in a closed subspace of the usual one (namely L, with s&gt;½) and we are interested by the behaviour of R(z), z close to a threshold, applying in the usual space L, with s&gt;½ when z tends to the threshold. Copyright © 2004 John Wiley &amp; Sons, Ltd.