Showing papers on "Ladder operator published in 2003"

Exact PT-symmetry is equivalent to Hermiticity

Abstract: We show that a quantum system possessing an exact antilinear symmetry, in particular PT-symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We apply our general results to PT-symmetry in finite dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field theories.

Shapelets: I. a method for image analysis

Abstract: We present a new method for the analysis of images, a fundamental task in observational astronomy. It is based on the linear decomposition of each object in the image into a series of localized basis functions of different shapes, which we call ‘shapelets’. A particularly useful set of complete and orthonormal shapelets is that consisting of weighted Hermite polynomials, which correspond to perturbations around a circular Gaussian. They are also the eigenstates of the two-dimensional quantum harmonic oscillator, and thus allow us to use the powerful formalism developed for this problem. One of their special properties is their invariance under Fourier transforms (up to a rescaling), leading to an analytic form for convolutions. The generator of linear transformations such as translations, rotations, shears and dilatations can be written as simple combinations of raising and lowering operators. We derive analytic expressions for practical quantities, such as the centroid (astrometry), flux (photometry) and radius of the object, in terms of its shapelet coefficients. We also construct polar basis functions which are eigenstates of the angular momentum operator, and thus have simple properties under rotations. As an example, we apply the method to Hubble Space Telescope images, and show that the small number of shapelet coefficients required to represent galaxy images lead to compression factors of about 40 to 90. We discuss applications of shapelets for the archival of large photometric surveys, for weak and strong gravitational lensing and for image deprojection.

Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation

Abstract: A parametrization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed. Using this parametrization we find the region of permissible vectors which represent a density operator. The inequalities which specify the region are shown to involve the Casimir invariants of the group. In particular cases, this allows the determination of degeneracies in the spectrum of the operator. The identification of the Casimir invariants also provides a method of constructing quantities which are invariant under local unitary operations. Several examples are given which illustrate the constraints provided by the positivity requirements and the utility of the coherence vector parametrization.

The factorization method in inverse scattering from periodic structures

Abstract: The application of the factorization method, the refined version of the linear sampling method, to scattering by a periodic surface is considered. Central to this method is the near field operator N, mapping incident fields to the corresponding scattered fields on a horizontal line. A factorization of N forms the basis for the method. It is shown that the middle operator in this factorization, the adjoint of the single-layer operator, is coercive if the wavenumber is small enough. Thus this operator can, for general wavenumber, always be written as the sum of a coercive and a compact operator. We use this property to define an auxiliary positive operator N# which can be constructed directly from N and which makes it possible to reconstruct the scattering surface directly using a simple numerical algorithm.

Harmonic oscillator with minimal length uncertainty relations and ladder operators

Abstract: We construct creation and annihilation operators for deformed harmonic oscillators with minimal length uncertainty relations. We discuss a possible generalization to a large class of deformations of canonical commutation relations. We also discuss the dynamical symmetry of a noncommutative harmonic oscillator.

Calculation of the hidden symmetry operator in -symmetric quantum mechanics

Abstract: In a recent paper it was shown that if a Hamiltonian H has an unbroken symmetry, then it also possesses a hidden symmetry represented by the linear operator . The operator commutes with both H and . The inner product with respect to is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator for the non-Hermitian -symmetric Hamiltonian H = 1/2p2 + 1/2x2 + ix3 using perturbative techniques. It is also shown how to construct the operator for H = 1/2p2 + 1/2x2 − x4 using nonperturbative methods.

The rogers–ramanujan recursion and intertwining operators

Abstract: We use vertex operator algebras and intertwining operators to study certain substructures of standard -modules, allowing us to conceptually obtain the classical Rogers–Ramanujan recursion. As a consequence we recover Feigin–Stoyanovsky's character formulas for the principal subspaces of the level 1 standard -modules.

Data-sparse approximation to the operator-valued functions of elliptic operator

Abstract: llIn previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator L. The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent (zI - L) -1 , z ∈ C. In the present paper, we consider various operator functions, the operator exponential e -tL , negative fractional powers L -α , the cosine operator function cos(t√L) L-k and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents (z k I - L) -1 mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.

Floquet-Bloch operator for the Bose-Hubbard model with static field.

Abstract: This paper deals with the spectral properties of the one-dimensional Bose-Hubbard Hamiltonian amended by an external static field-a model for cold spinless atoms loaded in a quasi-one-dimensional optical lattice and subject to an additional static (for example, gravitational) force. The analysis is performed in terms of the Floquet-Bloch operator, defined as the evolution operator of the system over one Bloch period. Depending on the particular choice of parameters, the spectrum is found to be either regular or chaotic. Moreover, in the chaotic case, the matrix of the Floquet-Bloch operator is well characterized as a random matrix of the circular orthogonal ensemble.

Gaussian quantum operator representation for bosons

Abstract: We introduce a Gaussian quantum operator representation, using the most general possible multimode Gaussian operator basis. The representation unifies and substantially extends existing phase-space representations of density matrices for Bose systems and also includes generalized squeezed-state and thermal bases. It enables first-principles dynamical or equilibrium calculations in quantum many-body systems, with quantum uncertainties appearing as dynamical objects. Any quadratic Liouville equation for the density operator results in a purely deterministic time evolution. Any cubic or quartic master equation can be treated using stochastic methods.

Gaps in the Spectrum of the Maxwell Operator with Periodic Coefficients

Abstract: The periodic Maxwell operator is considered. Piecewise constant coefficients are constructed in such a way that the spectrum of operator has the gaps.

POSITIVITY OF THE INTERTWINING OPERATOR AND HARMONIC ANALYSIS ASSOCIATED WITH THE JACOBI-DUNKL OPERATOR ON ${\mathbb R}$

Abstract: We consider a differential-difference operator Λα,β, $\alpha > -\frac{1}{2}$, $\beta\in {\mathbb R}$ on ${\mathbb R}$. The eigenfunction of this operator equal to 1 at zero is called the Jacobi–Dunkl kernel. We give a Laplace integral representation for this function and we prove that for $\alpha\ge\beta\ge -\frac{1}{2}$, $\alpha e -\frac{1}{2}$, the kernel of this integral representation is positive. This result permits us to prove that the Jacobi–Dunkl intertwining operator and its dual are positive. Next we study the harmonic analysis associated with the operator Λα,β (Jacobi–Dunkl transform, Jacobi–Dunkl translation operators, Jacobi–Dunkl convolution product, Paley–Wiener and Plancherel theorems…).

Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

Abstract: We consider the Hamiltonian Hμ(K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator Hμ0(K) lying below z ≤ 0 with respect to the total quasimomentum K → 0 and the spectral parameter z → −0.

Dirac operator and a twisted cyclic cocycle on the standard Podles quantum sphere

Abstract: A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant differential calculus on the standard Podles sphere. The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.

The Bloch wave operator: generalizations and applications: Part I. The time-independent case

Abstract: This is part 1 of a two-part review on wave operator theory and methods. The basic theory of the time-independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers, with a discussion of the links between the matrix and projection operator approaches. The matrix approach is shown to lead to simple derivations of the wave operators and effective Hamiltonians of Lowdin, Bloch, Des Cloizeaux and Kato as well as to some associated variational forms. The principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator, leading to the construction of the effective Hamiltonians of Bloch and of Des Cloizeaux. Several mathematical techniques which are useful in implementing this approach are explained, some of them being relatively little known in the area of wave operator calculations. The theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods. The main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms, with simple Pade approximant methods being found sufficient to deal with most of the cases of divergence which are encountered. The use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy-dependent effective Hamiltonian of Lowdin. In general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the Bloch effective Hamiltonian appears as a separate entity. The presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics, chemistry and engineering. Some of the techniques described in this part 1 will be further extended and applied in part 2 of the review, which deals with the changes which are required to extend wave operator theory to the case of a time-dependent Hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule.

Efimov's Effect in a Model of Perturbation Theory of the Essential Spectrum

Abstract: A model operator similar to the energy operator of a system with a nonconserved number of particles is studied. The essential spectrum of the operator is described, and under some natural conditions on the parameters it is shown that there are infinitely many eigenvalues lying below the bottom of the essential spectrum.

On the Helmholtz operator for Euler morphisms

Abstract: The variational sequence describes the Helmholtz conditions for local variationality in terms of the Helmholtz map, which is defined on a factor space. We study a tensor modification called the Helmholtz operator. For the first and second order cases we prove that, up to a multiplicative constant, the Helmholtz operator is the unique natural operator of the type in question.

How the geometric calculus resolves the ordering ambiguity of quantum theory in curved space

Abstract: The long standing problem of the ordering ambiguity in the definition of the Hamilton operator for a point particle in curved space is naturally resolved by using the powerful geometric calculus based on Clifford algebra. The momentum operator is defined to be the vector derivative (the gradient) multiplied by ?i; it can be expanded in terms of basis vectors ?? as p = ?i????. The product of two such operators is unambiguous, and such is the Hamiltonian which is just the d'Alembert operator in curved space; the curvature scalar term is not present in the Hamiltonian if we confine our consideration to scalar wavefunctions only. It is also shown that p is Hermitian and a self-adjoint operator: the presence of the basis vectors ?? compensates the presence of ?|g| in the matrix elements and in the scalar product. The expectation value of such an operator follows the classical geodetic line.

An algebraic q-deformed form for shape-invariant systems

Abstract: A quantum deformed theory applicable to all shape-invariant bound-state systems is introduced by defining q-deformed ladder operators. We show that these new ladder operators satisfy new q-deformed commutation relations. In this context we construct an alternative q-deformed model that preserves the shape-invariance property presented by the primary system. q-deformed generalizations of Morse, Scarf and Coulomb potentials are given as examples.

Hubble operator in isotropic loop quantum cosmology

Abstract: We present a construction of the Hubble operator for the spatially flat isotropic loop quantum cosmology. This operator is a Dirac observable on a subspace of the space of physical solutions. This subspace gets selected dynamically, requiring that its action be invariant on the physical solution space. As a simple illustrative application of the expectation value of the operator, we do find a generic phase of (super)inflation, a feature shown by Bojowald from the analysis of effective Friedmann equation of loop quantum cosmology.

A realization of the q-deformed harmonic oscillator: rogers-Szegö and Stieltjes-Wigert polynomials

Abstract: We discuss some results from q-series that can account for the foundations for the introduction of orthogonal polynomials on the circle and on the line, namely the Rogers-Szego and Stieltjes-Wigert polynomials. These polynomials are explicitly written and their orthogonality is verified. Explicit realizations of the raising and lowering operators for these polynomials are introduced in analogy to those of the Hermite polynomials that are shown to obey the q-commutation relations associated with the q-deformed harmonic oscillator.

Thermal state of the general time-dependent harmonic oscillator

Abstract: Taking advantage of dynamical invariant operator, we derived quantum mechanical solution of general time-dependent harmonic oscillator. The uncertainty relation of the system is always larger than ħ/2 not only in number but also in the thermal state as expected. We used the diagonal elements of density operator satisfying Leouville-von Neumann equation to calculate various expectation values in the thermal state. We applied our theory to a special case which is the forced Caldirola-Kanai oscillator.

Time in quantum mechanics and quantum field theory

Abstract: W Pauli pointed out that the existence of a self-adjoint time operator is incompatible with the semi-bounded character of the Hamiltonian spectrum. As a result, there has been much argument about the time–energy uncertainty relation and other related issues. In this paper, we show a way to overcome Pauli's argument. In order to define a time operator, by treating time and space on an equal footing and extending the usual Hamiltonian Ĥ to the generalized Hamiltonian Ĥμ (with Ĥ0 = Ĥ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrodinger equation i∂μψ = Ĥμψ and Heisenberg equation d/dxμ = ∂μ + i[Ĥμ, ] are obtained, from which we have: (1) t is to Ĥ0 as xj is to Ĥj (j = 1, 2, 3); likewise, t is to i∂0 as xj is to i∂j; (2) the proposed time operator is canonically conjugate to i∂0 rather than to Ĥ0, therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual ΔxμΔpμ ≥ 1/2 and the Mandelstam–Tamm treatment ΔxμΔHμ ≥ 1/2, have been formulated.

Normally ordered expansion of the inverse of the coordinate operator and the momentum operator

Abstract: We derive the normally ordered expansion formulae of 1/n and 1/n by virtue of the method of integral within an ordered product of operators in the sense of the principal value integral, where , are the coordinate and momentum operator, respectively. Application of the new formula is briefly discussed.