Showing papers on "Ladder operator published in 2005"
TL;DR: In this article, a ladder operator formalism applied to orthogonal polynomials with discontinuous weights is presented. But the ladder operator is not suitable for the case where the weight is a product of an absolutely continuous reference weight w0 and a standard jump function.
Abstract: In this letter we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, An(z) and Bn(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w0 and a standard jump function, then An(z) and Bn(z) have apparent simple poles at these jumps. We exemplify the approach by taking w0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painleve IV.
64 citations
TL;DR: Using the analytic representation of the so-called Gazeau-klauder coherent states (CSs), a new class of generalized CSs, namely the family of dual states associated with the states, can be constructed through viewing these states as temporally stable nonlinear CSs as discussed by the authors.
Abstract: Using the analytic representation of the so-called Gazeau–Klauder coherent states (CSs), we shall demonstrate that how a new class of generalized CSs, namely the family of dual states associated with theses states, can be constructed through viewing these states as temporally stable nonlinear CSs Also we find that the ladder operators, as well as the displacement type operator corresponding to these two pairs of generalized CSs, may be easily obtained using our formalism, without employing the supersymmetric quantum mechanics (SUSYQM) techniques Then, we have applied this method to some physical systems with known spectrum, such as Poschl–Teller, infinite well, Morse potential and hydrogenlike spectrum as some quantum mechanical systems Finally, we propose the generalized form of the Gazeau–Klauder CS and the corresponding dual family
62 citations
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TL;DR: In this article, it was shown that the ideal of a general operator space X is the intersection of X with an ideal in any C *-algebra or C*-module.
Abstract: In recent work of the second author, a technical result was proved establishing a bijective correspondence between certain open projections in a C*-algebra containing an operator algebra A, and certain one-sided ideals of A. Here we give several remarkable consequences of this result. These include a generalization of the theory of hereditary subalgebras of a C*-algebra, and the solution of a ten year old problem on the Morita equivalence of operator algebras. In particular, the latter gives a very clean generalization of the notion of Hilbert C*-modules to nonselfadjoint algebras. We show that an `ideal' of a general operator space X is the intersection of X with an `ideal' in any containing C*-algebra or C*-module. Finally, we discuss the noncommutative variant of the classical theory of `peak sets'.
60 citations
TL;DR: The spectrum of the Schrodinger operator in a quantum waveguide is known to be unstable in two and three dimensions as discussed by the authors, and any local enlargement of the waveguide produces eigenvalues beneath the continuo
Abstract: The spectrum of the Schrodinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuo ...
58 citations
TL;DR: In this article, a new anharmonic oscillator was proposed and the exact solutions of the Schrodinger equation with this oscillator were presented, and the ladder operators were established directly from the normalized radial wave functions and used to evaluate the closed expressions of matrix elements for some related functions.
51 citations
TL;DR: In this paper, the properties of the frequency operator for an infinite number of systems and disproved claims in the literature that the quantum probability postulate can be derived from these properties.
50 citations
TL;DR: Two new two dimensional (2-D) complex operators for estimating the energy and orientation of 2-D oriented patterns are proposed and both the local and nonlocal operators give improved performance when compared with a simple gradient based estimator.
Abstract: Two new two dimensional (2-D) complex operators for estimating the energy and orientation of 2-D oriented patterns are proposed. The starting point for our work is a new 2-D extension of the Teager-Kaiser energy operator incorporating orientation estimation. The first new energy operator is based on partial derivatives and can be considered a local (point-based) estimator. Using a nonlocal (pseudo-differential) operator we derive a second and more general energy operator. A scale invariant nonlocal operator is derived from the recently proposed spiral phase quadrature (or Riesz) transform. The Teager-Kaiser energy operator and the phase congruency local energy are unified in a single equation for both 1-D and 2-D. Robust orientation estimation, important for isotropic demodulation of fringe patterns is demonstrated. Theoretical error analysis of the local operator is greatly simplified by a logarithmic formulation. Experimental results using the operators on noisy images are shown. In the presence of Gaussian additive noise both the local and nonlocal operators give improved performance when compared with a simple gradient based estimator.
47 citations
TL;DR: In this article, a model-independent, nonperturbative proof of operator product expansions in quantum field theory is presented, based on a recently proposed phase space condition that allows a precise description of point field structures.
Abstract: The paper presents a model-independent, nonperturbative proof of operator product expansions in quantum field theory. As an input, a recently proposed phase space condition is used that allows a precise description of point field structures. Based on the product expansions, we also define and analyze normal products (in the sense of Zimmermann).
42 citations
TL;DR: In this paper, it was shown that a non-Hermitian Hamiltonian has a real positive spectrum and exhibits unitary time evolution if the Hamiltonian possesses an unbroken space-time reflection symmetry.
Abstract: A non-Hermitian Hamiltonian has a real positive spectrum and exhibits unitary time evolution if the Hamiltonian possesses an unbroken $\mathcal{P}\mathcal{T}$ (space-time reflection) symmetry. The proof of unitarity requires the construction of a linear operator called $\mathcal{C}$. It is shown here that $\mathcal{C}$ is the complex extension of the intrinsic parity operator and that the $\mathcal{C}$ operator transforms under the Lorentz group as a scalar.
42 citations
TL;DR: In this article, a class of time-dependent quantum systems with non-Hermitian Hamiltonians, particularly those whose Hermitian counterparts are important for the comprehension of posed problems in quantum optics and quantum chemistry, are studied.
Abstract: In this work we intend to study a class of time-dependent quantum systems with non-Hermitian Hamiltonians, particularly those whose Hermitian counterparts are important for the comprehension of posed problems in quantum optics and quantum chemistry. They consist of an oscillator with time-dependent mass and frequency under the action of a time-dependent imaginary potential. The wave functions are used to obtain the expectation value of the Hamiltonian. Although it is neither Hermitian nor PT symmetric, the Hamiltonian under study exhibits real values of energy.
TL;DR: In this paper, the structure of the many-boson wavefunctions of k-clustered quantum Hall droplets appear naturally as matrix elements of ladder operators in integrable representations of the affine Lie algebra.
Abstract: We explore the structure of the bosonic analogues of the k-clustered 'parafermion' quantum Hall states. We show how the many-boson wavefunctions of k-clustered quantum Hall droplets appear naturally as matrix elements of ladder operators in integrable representations of the affine Lie algebra . Using results of Feigin and Stoyanovsky, we count the dimensions of spaces of symmetric polynomials with given k-clustering properties and show that as the droplet size grows the partition function of its edge excitations evolves into the character of the representation. This confirms that the Hilbert space of edge states coincides with the representation space of the edge-current algebra. We also show that a spin-singlet, two-component k-clustered boson fluid is similarly related to integrable representations of . Parafermions are not necessary for these constructions.
TL;DR: In this article, it was shown that if the theory is integrable, the addition of a requirement of factorization at high energies can lead to exact determination of the generic matrix element of this operator on the asymptotic states.
TL;DR: In this article, the connection coefficients between two polynomial sets using their corresponding lowering operators are expressed explicitly and some well-known results as particular cases including some duplication and addition formulas.
TL;DR: In this article, a systematic search of different types of ladder operators for the Jaynes-Cummings model in the rotating-wave approximation is realized using algebraic techniques, and the link between their results and previous studies on the diagonalization of the associated Hamiltonian is established.
Abstract: Using algebraic techniques, we realize a systematic search of different types of ladder operators for the Jaynes-Cummings model in the rotating-wave approximation. The link between our results and previous studies on the diagonalization of the associated Hamiltonian is established. Using some of the ladder operators obtained before, examples are given on the possibility of constructing a variety of interesting coherent states for this Hamiltonian.
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TL;DR: Weierstrass representations of surfaces in 3D and 4D spaces have been studied in this paper, and their applications to the theory of the Willmore functional and related problems of spectral theory of two-dimensional Dirac operator with periodic coefficients.
Abstract: We give a survey on the Weierstrass representations of surfaces in three- and four-dimensional spaces, their applications to the theory of the Willmore functional and on related problems of spectral theory of the two-dimensional Dirac operator with periodic coefficients
TL;DR: The 3-transposition groups that act on vertex operator algebras in the way described by Miyamoto in [Mi] are classified under the assumption that the group is centerfree and the VOA carries a positive-definite invariant Hermitian form as discussed by the authors.
Abstract: The 3-transposition groups that act on vertex operator algebras in the way described by Miyamoto in [Mi] are classified under the assumption that the group is centerfree and the VOA carries a positive-definite invariant Hermitian form.
TL;DR: In this paper, the authors construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles.
Abstract: We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
TL;DR: In this paper, the structure of the lower edge of the spectrum of a periodic magnetic Schrödinger operator is investigated, and it is shown that the energy depends quadratically on the quasimomentum in a neighborhood of a lower edge.
Abstract: The structure of the lower edge of the spectrum of a periodic magnetic Schrödinger operator is investigated. It is known that in the nonmagnetic case the energy depends quadratically on the quasimomentum in a neighborhood of the lower edge of the spectrum of the operator. An example of a magnetic Schrödinger operator is constructed for which energy is partially degenerate with respect to one component of the quasimomentum.
TL;DR: In this paper, the Al-Salam-Chihara polynomials were studied when q > 1 and several solutions of the associated moment problem were found, and the orthogonality relations lead to explicit evaluations of several integrals.
Abstract: We study the Al-Salam-Chihara polynomials when q > 1. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with respect to which the Ismail-Masson system of rational functions is biorthogonal. An integral representation of the right inverse of a divided difference operator is also obtained.
TL;DR: In this paper, exact solutions of the one-dimensional Schrodinger equation with a harmonic oscillator plus an inverse square potential are presented, and the ladder operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1).
Abstract: We present exact solutions of the one-dimensional Schrodinger equation with a harmonic oscillator plus an inverse square potential. The ladder operators are constructed by the factorization method. We find that these operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1). Based on those ladder operators, we obtain the analytical expressions of matrix elements for some related functions ρ and $\rho\frac{d}{d\rho}$ with ρ=x2. Finally, we make some comments on the Barut–Girardello coherent states and the hidden symmetry between E(x) and E(ix) by substituting x→ix.
TL;DR: In this paper, the authors derived an expression for the commutator of functions of operators with constant commutations relations in terms of the partial derivatives of these functions and extended the well-known commutation relation between one operator and a function of another operator.
Abstract: We derive an expression for the commutator of functions of operators with constant commutations relations in terms of the partial derivatives of these functions. This result extends the well-known commutation relation between one operator and a function of another operator. We discuss the range of applicability of the formula with examples in quantum mechanics.
TL;DR: In this article, a Taylor formula with integral remainder was established for the Bessel-Struve operator, and the translation operator associated with lα was extended to Taylor series.
Abstract: In this paper, we consider a differential operator lα, α > −1/2, called Bessel–Struve operator. We construct and study transmutation operator between lα and the second derivative operator d2/dx 2. We establish for the Bessel–Struve operator a Taylor formula with integral remainder. We apply these results to expand as Taylor series the translation operator associated with lα. We provide an analyticity criterion for functions on ℝ involving lα.
TL;DR: In this article, a model operator acting in a subspace of a Fock space and obtaining a symmetrized analogue of the Faddeev equation was considered, and the position and the structure of its essential spectrum were described.
Abstract: We consider a model operator acting in a subspace of a Fock space and obtain a symmetrized analogue of the Faddeev equation. For the operator considered, we describe the position and the structure of its essential spectrum.
TL;DR: In this paper, the authors derived the quantum dynamics of a particle in the modified Poschl-Teller potential from the group SL (2, R) by applying a group approach to quantization (GAQ).
Abstract: The quantum dynamics of a particle in the modified Poschl-Teller potential is derived from the group SL (2, R) by applying a group approach to quantization (GAQ). The explicit form of the Hamiltonian as well as the ladder operators is found in the enveloping algebra of this basic symmetry group. The present algorithm provides a physical realization of the non-unitary, finite-dimensional, irreducible representations of the SL(2, R) group. The non-unitarity manifests itself in that only half of the states are normalizable, in contrast with the representations of SU(2) where all the states are physical.
TL;DR: In this paper, the second divergence and the differential properties of the gradient operator are used to demonstrate new integral transformations for vector and scalar fields on curved surfaces, such as second divergence theorem, the second gradient theorem, second curl theorem, and the second circulation theorem.
TL;DR: In this paper, the bound eigenfunctions and spectrum of a Dirac hydrogen atom are found taking advantage of the SU ( 1, 1 ) Lie algebra in which the radial part of the problem can be expressed.
TL;DR: In this article, a comprehensive theory of phase for finite-dimensional quantum systems is developed, where the only physical requirement is that phase is complementary to amplitude, and the notion of mutually unbiased bases is used to implement this complementarity.
Abstract: We develop a comprehensive theory of phase for finite-dimensional quantum systems. The only physical requirement we impose is that phase is complementary to amplitude. To implement this complementarity we use the notion of mutually unbiased bases, which exist for dimensions that are powers of a prime. For a d-dimensional system (qudit) we explicitly construct d+1 classes of maximally commuting operators, each one consisting of d?1 operators. One of these classes consists of diagonal operators that represent amplitudes (or inversions). By finite Fourier transformation, it is mapped onto ladder operators that can be appropriately interpreted as phase variables. We discuss examples of qubits and qutrits, and show how these results generalize previous approaches.
TL;DR: The notion of vertex operator coalgebra was introduced in this paper, which corre- sponds to the family of correlation functions modeling one string propagating in space-time splitting into n strings in conformal field theory.