Showing papers on "Ladder operator published in 2012"

The Weyl Operator and its Generalization

Abstract: Introduction.- The Fundamental Idea, Terminology, and Operator Algebra.- The Weyl Operator.- The Algebra of the Weyl Operator.- Product of Operators, Commutators, and the Moyal Sin Bracket.- Some Other Ordering Rules.- Generalized Operator Association.- The Fourier, Monomial, and Delta Function Associations.- Transformation Between Associations.- Path Integral Approach.- The Distribution of a Symbol and Operator.- The Uncertainty Principle.- Phase-Space Distributions.- Amplitude, Phase, Instantaneous Frequency, and the Hilbert Transform.- Time - Frequency Analysis.- The Transformation of Differential Equations into Phase Space.- The Representation of Functions.- The N Operator Case.

The geodesic X-ray transform with fold caustics

Abstract: We give a detailed microlocal study of X-ray transforms over geodesics-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation associated to the Fourier integral operator. In two dimensions, for the geodesic transform, we show that there is always a cancellation of singularities to some order, and we give an example where that order is infinite; therefore the normal operator is not microlocally invertible in that case. In the case of three dimensions or higher if the canonical relation is a local canonical graph we show microlocal invertibility of the normal operator. Several examples are also studied.

Families of superintegrable Hamiltonians constructed from exceptional polynomials

Abstract: We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable.

Radial coherent and intelligent states of paraxial wave equation

Abstract: Ladder operators for the radial index of the paraxial optical modes in the cylindrical coordinates are calculated. The operators obey the su(1,1) algebra commutation relations. Based on this Lie algebra, we found that coherent modes constructed as eigenstates of the destruction operator or resulting from the action of the displacement operator on the fundamental mode are different. Some properties of these two kinds of radial coherent modes are studied in detail.

Operator product expansion in QCD in off-forward kinematics: separation of kinematic and dynamical contributions

Abstract: We develop a general approach to the calculation of target mass and finite t = (p′ − p)2 corrections in hard processes which can be studied in the framework of the operator product expansion and involve momentum transfer from the initial to the final hadron state. Such corrections, which are usually referred to as kinematic, can be defined as contributions of operators of all twists that can be reduced to total derivatives of the leading twist operators. As the principal result, we provide a set of projection operators that pick up the “kinematic” part of an arbitrary flavor-nonsinglet twist-four operator in QCD. A complete expression is derived for the time-ordered product of two electromagnetic currents that includes all kinematic corrections to twist-four accuracy. The results are immediately applicable to the studies of deeply-virtual Compton scattering, transition γ * → Mγ form factors and related processes. As a byproduct of this study, we find a series of “genuine” twist-four flavor-nonsinglet quark-antiquark-gluon operators which have the same anomalous dimensions as the leading twist quark-antiquark operators.

Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution

Abstract: We consider the spectral problem for a model second-order differential operator with an involution. The operator is given by the differential expression Lu = −u″(−x) and boundary conditions of general form. We obtain a criterion for the basis property of the systems of eigenfunctions of this operator in terms of the coefficients in the boundary conditions.

Exact Solutions of the Morse-like Potential, Step-Up and Step-Down Operators via Laplace Transform Approach

Abstract: We intend to realize the step-up and step-down operators of the potential V(x) = V1e2βx + V2eβx. It is found that these operators satisfy the commutation relations for the SU(2) group. We find the eigenfunctions and the eigenvalues of the potential by using the Laplace transform approach to study the Lie algebra satisfied the ladder operators of the potential under consideration. Our results are similar to the ones obtained for the Morse potential (β → −β).

Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators

Abstract: The type III Hermite $X_m$ exceptional orthogonal polynomial family is generalized to a double-indexed one $X_{m_1,m_2}$ (with $m_1$ even and $m_2$ odd such that $m_2 > m_1$) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painleve IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial Heisenberg algebra of order $m_2-m_1+1$, which may alternatively be interpreted in terms of a special type of $(m_2-m_1+2)$th-order shape invariance property.

Families of superintegrable Hamiltonians constructed from exceptional polynomials

Abstract: We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable.

Effective multibody-induced tunneling and interactions in the Bose-Hubbard model of the lowest dressed band of an optical lattice

Abstract: We construct the effective lowest-band Bose-Hubbard model incorporating interaction-induced on-site correlations. The model is based on ladder operators for local correlated states, which deviate from the usual Wannier creation and annihilation, allowing for a systematic construction of the most appropriate single-band low-energy description in the form of the extended Bose-Hubbard model. A formulation of this model in terms of ladder operators not only naturally contains the previously found effective multibody interactions, but also contains multibody-induced single-particle tunneling, pair tunneling, and nearest-neighbor interaction processes of higher orders. An alternative description of the same model can be formulated in terms of occupation-dependent Bose-Hubbard parameters. These multiparticle effects can be enhanced using Feshbach resonances, leading to corrections which are well within experimental reach and of significance to the phase diagram of ultracold bosonic atoms in an optical lattice. We analyze the energy-reduction mechanism of interacting atoms on a local lattice site and show that this cannot be explained only by a spatial broadening of Wannier orbitals on a single-particle level, which neglects correlations.

Analysis of \mathbb{C}P^{N-1} sigma models via projective structures

Abstract: This paper represents a study of projector solutions to the Euclidean sigma model in two dimensions and their associated surfaces immersed in the su(N) Lie algebra. Any solution for the sigma model defined on the extended complex plane with finite action can be written as a raising operator acting on a holomorphic one. Here the proof is formulated in terms rank-1 projectors so it is explicitly gauge invariant. We apply these results to the analysis of surfaces associated with the models defined using the generalized Weierstrass formula for immersion. We show that the surfaces are conformally parametrized by the Lagrangian density, with finite area equal to the action of the model, and express several other geometrical characteristics of the surface in terms of the physical quantities of the model. Finally, we provide necessary and sufficient conditions that a surface be related to a sigma model.

Perfect state transfer in PT -symmetric non-Hermitian networks

Abstract: We systematically study the parity- and time-reversal ($\mathcal{PT}$) symmetric non-Hermitian version of a quantum network proposed in the paper of Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)]. The nature of this model shows that it is a paradigm to demonstrate the complex relationship between the pseudo-Hermitian Hamiltonian and its Hermitian counterpart as well as a candidate in the experimental realization to simulate $\mathcal{PT}$-symmetry breaking. We also show that this model allows a conditional perfect state transfer within the unbroken $\mathcal{PT}$-symmetry region but not an arbitrary one. This is due to the fact that the evolution operator at a certain period is equivalent to the $\mathcal{PT}$ operator for the real-valued wave function in the elaborate $\mathcal{PT}$-symmetric Hilbert space.

Eigenvalue density of the non-Hermitian Wilson Dirac operator.

Abstract: We find the lattice spacing dependence of the eigenvalue density of the non-Hermitian Wilson Dirac operator in the ϵ domain. The starting point is the joint probability density of the corresponding random matrix theory. In addition to the density of the complex eigenvalues we also obtain the density of the real eigenvalues separately for positive and negative chiralities as well as an explicit analytical expression for the number of additional real modes.

Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems

Abstract: We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. They are the classical equivalent of quantum systems involving the fourth and fifth Painleve transcendents. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of Lissajous's figures.

Finite-volume scaling of the Wilson-Dirac operator spectrum

Abstract: The microscopic spectral density of the Hermitian Wilson-Dirac operator is computed numerically in quenched lattice QCD. We demonstrate that the results given for fixed index of the Wilson-Dirac operator can be matched by the predictions from Wilson chiral perturbation theory. We test successfully the finite volume and the mass scaling predicted by Wilson chiral perturbation theory at fixed lattice spacing.

Exact Solutions of the Morse-like Potential, Step-Up and Step-Down Operators via Laplace Transform Approach

Abstract: We intend to realize the step-up and step-down operators of the potential $V(x)=V_{1}e^{2\beta x}+V_{2}e^{\beta x}$. It is found that these operators satisfy the commutation relations for the SU(2) group. We find the eigenfunctions and the eigenvalues of the potential by using the Laplace transform approach to study the Lie algebra satisfied the ladder operators of the potential under consideration. Our results are similar to the ones obtained for the Morse potential ($\beta \rightarrow -\beta$).

Squeezed coherent states and the one-dimensional Morse quantum system

Abstract: The Morse potential one-dimensional quantum system is a realistic model for studying vibrations of atoms in a diatomic molecule. This system is very close to the harmonic oscillator one. We thus propose a construction of squeezed coherent states similar to the one of the harmonic oscillator using ladder operators. The properties of these states are analysed with respect to the localization in position, minimal Heisenberg uncertainty relation, the statistical properties and illustrated with examples using the finite number of states in a well-known diatomic molecule.

Operator Splittings and Their Applications

Abstract: Introduction Operator splitting: history and motivation Description and analysis of the operator splittings Further investigations of the operator splittings Numerical solution of the split sub-problems The application of operator splitting to real-life problems Index.

New fundamental quantum mechanical operator-ordering identities for the coordinate and momentum operators

Abstract: In quantum mechanics theory one of the basic operator orderings is Q - P and P - Q ordering, where Q and P are the coordinate operator and the momentum operator, respectively. We derive some new fundamental operator identities about their mutual reordering. The technique of integration within Q - P ordering and P - Q ordering is introduced. The Q - P ordered and P - Q ordered formulas of the Wigner operator are also deduced which makes arranging the operators in either Q - P or P - Q ordering much more convenient.

Qwn-euler operator and associated cauchy problem

Abstract: In this paper the quantum white noise (QWN)-Euler operator is defined as the sum , where and NQ stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that has an integral representation in terms of the QWN-derivatives as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.

Excitation spectrum for an inhomogeneously dipole-field-coupled superconducting qubit chain

Abstract: When a chain of superconducting qubits couples to a coplanar resonator in a cavity, each of its N qubits (equally-spaced with distance l) experiences a different dipole-field coupling strength due to the waveform of the cavity field. We find that this inhomogeneous coupling leads to a pair of l-dependent ladder operators for the angular momentum of the spin chain. Varying the qubit spacing l changes the transition amplitudes between the angular momentum levels. We derive an exact diagonalization of the general N-qubit Hamiltonian and, through the N=4 case, demonstrate how the l-dependent operators lead to a denser one-excitation spectrum and a probability redistribution of the eigenstates. Moreover, it will be shown that the variation of l between its two limiting values coincides with the crossover between Frenkel- and Wannier-type excitons in the superconducting spin chain.

Investigation of PT-symmetric Hamiltonian Systems from an Alternative Point of View

Abstract: Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.

Factorization of Laplace Operators on Higher Spin Representations

Abstract: This paper deals with the problem of factorizing integer powers of the Laplace operator acting on functions taking values in higher spin representations. This is a far-reaching generalization of the well-known fact that the square of the Dirac operator is equal to the Laplace operator. Using algebraic properties of projections of Stein–Weiss gradients, i.e. generalized Rarita–Schwinger and twistor operators, we give a sharp upper bound on the order of polyharmonicity for functions with values in a given representation with half-integral highest weight.

Construction of Models in low-dimensional Quantum Field Theory using Operator Algebraic Methods

Abstract: Part II. Construction of Longo–Witten unitaries and models in BQFT 47 Chapter 4. Longo–Witten unitaries for conformal nets associated with lattices and loop group models. . Introduction In algebraic quantum field theory (AQFT) [Haa96] one studies local nets (e.g. von Neumann algebras) that assign to a space-time region the algebra of observables localized in it. These nets are asked to fulfill certain axioms coming from basic physical principles; we mention as examples the locality principle—which asks that the algebras assigned to causally disjoint regions should commute (local nets)—and the covariant assignment with respect to some " symmetry group " of the space-time. This approach brought many conceptional and model independent results, but recently it seems to be useful for construction and classification of models. In the AQFT approach also conformal quantum field theory (CQFT) has been treated successfully by considering nets on two dimensional Minkowski space and its chiral parts, which can be regarded as nets on the real line or as nets on the circle. Some important achievements are classification results [KL04, Xu05] and new constructions [Xu07], which were not obtained in other approaches. An important tool is the the relation to subfactors: inclusions of von Neumann algebras with trivial centers. For example there is a notion of index of subfactors giving relations to the statistical dimensions of representations. For finite index subfactors and there are several invariants with combinatorial nature making classifications possible. A conformal net on the real line (or the circle) is a net (precosheaf), which assigns to each proper interval a von Neumann algebra on a fixed Hilbert space H. Locality is encoded by asking that the algebras of disjoint intervals commute, covariance that there is a positive energy representation of the Möbius group (PSL(2, R) PSU(1, 1)) such that A(gI) = U(g)A(I)U(g) *. Often these nets turn out to be also covariant with respect to Diff + (S 1) (the orientation preserving diffeomorphisms of the circle) and there is a subnet Vir giving a representation of the Virasoro algebra and one can associates a central charge c. The von Neumann algebras turn out to be type III 1 factor in Connes classification. Superselection sectors or Doplicher–Haag–Roberts (DHR) representations can be described by localized endomorphisms giving a connection to Jones theory subfactors. In contrast to the result of higher dimensions, where just Fermions and Bosons can exist, in low dimensions more general so-called anyons exist …

Raising and lowering operators for the Dirac-Woods-Saxon potential in the presence of spin and pseudospin symmetry

Abstract: In the case of spin symmetry and pseudospin symmetry, raising and lowering operators and the bound state solutions of the Dirac equation for the spherically Woods-Saxon potential are presented within the context of Supersymmetric Quantum Mechanics. The energy equation and corresponding two-component spinors of the two Dirac particles are obtained in the closed form for arbitrary spin-orbit quantum number k by using the Pekeris approximation. The Hamiltonian hierarchy method and the shape invariance property are used in the calculations.

Soliton surfaces associated with sigma models: differential and algebraic aspects

Abstract: In this paper, we consider both differential and algebraic properties of surfaces associated with sigma models. It is shown that surfaces defined by the generalized Weierstrass formula for immersion for solutions of the sigma model with finite action, defined in the Riemann sphere, are themselves solutions of the Euler?Lagrange equations for sigma models. On the other hand, we show that the Euler?Lagrange equations for surfaces immersed in the Lie algebra with conformal coordinates, that are extremals of the area functional, subject to a fixed polynomial identity, are exactly the Euler?Lagrange equations for sigma models. In addition to these differential constraints, the algebraic constraints, in the form of eigenvalues of the immersion functions, are systematically treated. The spectrum of the immersion functions, for different dimensions of the model, as well as its symmetry properties and its transformation under the action of the ladder operators are discussed. Another approach to the dynamics is given, i.e. description in terms of the unitary matrix which diagonalizes both the immersion functions and the projectors constituting the model.