Showing papers on "Ladder operator published in 2010"
TL;DR: In this article, the authors review the application of the parity operator to the problem of quantum metrology for the detection of small phase shifts with quantum optical interferometry using highly entangled field states such as the so-called N00N states, and states obtained by injecting twin Fock states into a beam splitter.
Abstract: Photon number states are assigned a parity of +1 if their photon number is even and a parity of −1 if odd. The parity operator, which is minus one to the power of the photon number operator, is a Hermitian operator and thus a quantum mechanical observable although it has no classical analogue, the concept being meaningless in the context of classical light waves. In this paper we review work on the application of the parity operator to the problem of quantum metrology for the detection of small phase shifts with quantum optical interferometry using highly entangled field states such as the so-called N00N states, and states obtained by injecting twin Fock states into a beam splitter. With such states and with the performance of parity measurements on one of the output beams of the interferometer, one can breach the standard quantum limit, or shot-noise limit, of sensitivity down to the Heisenberg limit, the greatest degree of phase sensitivity allowed by quantum mechanics for linear phase shifts. Heisenber...
113 citations
TL;DR: In this paper, a method to obtain higher order integrals and polynomial algebras for two-dimensional quantum superintegrable systems separable in Cartesian coordinates from ladder operators was presented.
Abstract: We present a method to obtain higher order integrals and polynomial algebras for two-dimensional quantum superintegrable systems separable in Cartesian coordinates from ladder operators. All systems with a second- and a third-order integral of motion separable in Cartesian coordinates were studied. The integrals of motion of two of them do not generate a cubic algebra. We construct for these Hamiltonians a higher order polynomial algebra from their ladder operators. We obtain quintic and seventh-order polynomial algebras. We also give for the polynomial algebras of order 7 realizations in terms of deformed oscillator algebras. These realizations and finite-dimensional unitary representations allow us to obtain the energy spectrum. We also apply the construction to the caged anisotropic harmonic oscillator and a system involving the fourth Painleve transcendent.
83 citations
TL;DR: For the q-deformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G, this paper constructed an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. The commutator of Dq with a regular function on Gq consists of two parts.
Abstract: For the q-deformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element of Ug Cl(g). The commutator of Dq with a regular function on Gq consists of two parts. One is a twist of a classical commutator and so is automatically bounded. The second is expressed in terms of the commutator of the associator with an extension of D. We show that in the case of the Drinfeld associator the latter commutator is also bounded.
69 citations
TL;DR: An alternative expression for the length operator in loop quantum gravity is presented in this paper, which is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space.
Abstract: An alternative expression for the length operator in loop quantum gravity is presented. The operator is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space. The expression for the regularized length operator can moreover be understood both from a simple geometrical perspective as the average of a formula relating the length to area, volume and flux operators, and also consistently as the result of direct substitution of the densitized triad operator with the functional derivative operator into the regularized expression of the length. Both these derivations are discussed, and the origin of an undetermined overall factor in each case is also elucidated.
64 citations
TL;DR: Nikin et al. as discussed by the authors presented a theoretical approach to study ro-vibrational molecular states from a full nuclear Hamiltonian expressed in terms of normal-mode irreducible tensor operators.
Abstract: A theoretical approach to study ro-vibrational molecular states from a full nuclear Hamiltonian expressed in terms of normal-mode irreducible tensor operators is presented for the first time. Each term of the Hamiltonian expansion can thus be cast in the tensor form in a systematic way using the formalism of ladder operators. Pyramidal XY3 molecules appear to be good candidates to validate this approach which allows taking advantage of the symmetry properties when doubly degenerate vibrational modes are considered. Examples of applications will be given for PH3 where variational calculations have been carried out from our recent potential energy surface [Nikitin et al., J. Chem. Phys. 130, 244312 (2009)].
56 citations
TL;DR: In this article, the authors studied the distribution of eigenvalues of the Schrodinger operator with a complex valued potential and proved that if the Coulomb potential decays faster than the Schroff potential, then all eigen values are in a disc of a finite radius.
Abstract: We study the distribution of eigenvalues of the Schrodinger operator with a complex valued potential V . We prove that if |V | decays faster than the Coulomb potential, then all eigenvalues are in a disc of a finite radius.
47 citations
TL;DR: In this article, the authors studied the Hankel determinant of the generalized Jacobi weight (x − t)γxα(1 − x)β for x [0, 1] with α, β > 0, t < 0 and.
Abstract: We study the Hankel determinant of the generalized Jacobi weight (x − t)γxα(1 − x)β for x [0, 1] with α, β > 0, t < 0 and . Based on the ladder operators for the corresponding monic orthogonal polynomials Pn(x), it is shown that the logarithmic derivative of the Hankel determinant is characterized by a Jimbo–Miwa–Okamoto σ-form of the Painleve VI system.
46 citations
TL;DR: In this paper, the authors construct integrals of motion for multidimensional classical systems from ladder operators of one-dimensional systems and show how these integrals generate a polynomial Poisson algebra.
Abstract: We construct integrals of motion for multidimensional classical systems from ladder operators of one-dimensional systems. This method can be used to obtain new systems with higher order integrals. We show how these integrals generate a polynomial Poisson algebra. We consider a one-dimensional system with third order ladder operators and found a family of superintegrable systems with higher order integrals of motion. We obtain also the polynomial algebra generated by these integrals. We calculate numerically the trajectories and show that all bounded trajectories are closed.
44 citations
TL;DR: In this article, the authors propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrodinger equation using point canonical transformation method and obtain ground state eigenfunction.
Abstract: In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrodinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schrodinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new.
39 citations
01 Nov 2010
TL;DR: In this article, a continuous kernel function for Hermitian operators is defined, which is a generalization of the Lowner kernel function, and it is shown that it is positive definite if and only if whenever.
Abstract: Let be a real continuous function on an interval, and consider the operator function defined for Hermitian operators . We will show that if is increasing w.r.t. the operator order, then for the operator function is convex. Let and be functions defined on an interval . Suppose is non-decreasing and is increasing. Then we will define the continuous kernel function by , which is a generalization of the Lowner kernel function. We will see that it is positive definite if and only if whenever for Hermitian operators , and we give a method to construct a large number of infinitely divisible kernel functions.
39 citations
TL;DR: In this article, the authors consider the inverse problem of reconstructing the convolution operator from the spectrum and prove the global solvability of this nonlinear integral equation with a singularity.
Abstract: We consider the sum of the Sturm-Liouville operator and a convolution operator. We study the inverse problem of reconstructing the convolution operator from the spectrum. This problem is reduced to a nonlinear integral equation with a singularity. We prove the global solvability of this nonlinear equation, which permits one to show that the asymptotics of the spectrum is a necessary and sufficient condition for the solvability of the inverse problem. The proof is constructive.
TL;DR: In this paper, the dynamical symmetries of the Kratzer-type molecular potentials (generalized kratzer potentials) were studied by using the factorization method.
Abstract: The dynamical symmetries of the Kratzer-type molecular potentials (generalized Kratzer molecular potentials) are studied by using the factorization method. The creation and annihilation (ladder) operators for the radial eigenfunctions satisfying quantum dynamical algebra SU(1,1) are established. Factorization method is a very simple method of calculating the matrix elements from these ladder operators. The matrix elements of different functions of r, \(r\frac{d}{dr}\), their sum Γ1 and difference Γ2 are evaluated in a closed form. The exact bound state energy eigenvalues En,l and matrix elements of r, \(r\frac{d}{dr}\), their sum Γ1 and difference Γ2 are calculated for various values of n and l quantum numbers for CO and NO diatomic molecules for the two potentials. The results obtained are in very good agreement with those obtained by other methods.
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TL;DR: In this paper, a reduced operator system S ⊆ B(H) is defined as a system whose boundary ideal is at most a constant in the Choquet boundary of the Hilbert space.
Abstract: We classify operator systems S ⊆ B(H ) that act on finite dimensional Hilbert spaces H by making use of the noncommutative Choquet boundary. S is said to be reduced when its boundary ideal is {0}. In the category of operator systems, that property functions as semisimplicity does in the category of complex Banach algebras. We construct explicit examples of reduced operator systems using sequences of “parameterizing maps” � k : C r → B(Hk), k = 1 ,...,N . We show that every reduced operator system is isomorphic to one of these, and that two sequences give rise to isomorphic operator systems if and only if they are “unitarily equivalent” parameterizing sequences. Finally, we construct nonreduced operator systems S that have a given boundary ideal K and a given reduced image in C ∗ (S)/K, and show that these constructed examples exhaust the possibilities.
TL;DR: In this paper, the authors derived the distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite size in terms of a solution of the fifth Painleve transcendent.
TL;DR: In this paper, the Gohberg-Rouche theorem was used to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space.
Abstract: This paper is related to the spectral stability of traveling wave solutions of partial dif- ferential equations. In the first part of the paper we use the Gohberg-Rouche Theorem to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of the corresponding Birman-Schwinger type operator pencil. In the second part of the paper we apply this result to discuss three particular classes of problems: the Schr¨ odinger operator, the operator obtained by linearizing a degenerate system of reaction diffusion equations about a pulse, and a general high order differential operator. We study relations between the algebraic multiplicity of an isolated eigenvalue for the respective operators, and the order of the eigenvalue as the zero of the Evans function for the corresponding first order system.
TL;DR: In this article, the nonuniqueness of the factorization is exploited to derive new isospectral nonsingular potentials, which may have many possible applications in atomic and molecular physics.
Abstract: Factorization of quantum-mechanical potentials has a long history extending back to the earliest days of the subject. In the present article, the nonuniqueness of the factorization is exploited to derive new isospectral nonsingular potentials. Many one-parameter families of potentials can be generated from known potentials using a factorization that involves superpotentials defined in terms of excited states of a potential. For these cases an operator representation is available. If ladder operators are known for the original potential, then a straightforward procedure exists for defining such operators for its isospectral partners. The generality of the method is illustrated with a number of examples which may have many possible applications in atomic and molecular physics.
TL;DR: In this article, a guiding-center Fokker-planck operator is derived in a coordinate system that is well suited for the implementation in a numerical code, which is transformed such that it can commute with the orbit-averaging operation.
Abstract: A guiding-center Fokker-Planck operator is derived in a coordinate system that is well suited for the implementation in a numerical code. This differential operator is transformed such that it can commute with the orbit-averaging operation. Thus, in the low-collisionality approximation, a three-dimensional Fokker-Planck evolution equation for the orbit-averaged distribution function in a space of invariants is obtained. This transformation is applied to a collision operator with nonuniform isotropic field particles. Explicit neoclassical collisional transport diffusion and convection coefficients are derived, and analytical expressions are obtained in the thin orbit approximation. To illustrate this formalism and validate our results, the bootstrap current is analytically calculated in the Lorentz limit.
TL;DR: In this article, the authors extend the Hijazi inequality involving the energy-momentum tensor to the eigenvalues of the Dirac operator on Spin c manifolds without boundary.
TL;DR: In this paper, the spectral properties of a 3 × 3 block operator matrix with unbounded entries and domain consisting of vectors which satisfy certain relations between their components are studied, and the essential spectra of this operator are determined.
Abstract: In this paper we study spectral properties of a 3 × 3 block operator matrix with unbounded entries and with domain consisting of vectors which satisfy certain relations between their components. It is shown that, under certain conditions, this block operator matrix defines a closed operator, and the essential spectra of this operator are determined. These results are applied to a three-group transport equation.
TL;DR: In this article, the position of the spectrum of a self-adjoint operator L with respect to its spectrum of an anti-commuting operator was investigated and norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L were obtained in terms of the norm of V and the distances between pairs of disjoint spectral sets associated with the operator L and/or the operator A.
Abstract: Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L. All the bounds are given in terms of the norm of V and the distances between pairs of disjoint spectral sets associated with the operator L and/or the operator A. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed. The sharp norm bounds obtained for the operator angles generalize the celebrated Davis-Kahan trigonometric theorems to the case of J-self-adjoint perturbations.
TL;DR: In this article, the spectrum generating algebra (SGA) for a free particle in the three dimensional sphere was constructed for both, classical and quantum descriptions, including the ladder operators that give the eigenstates of the free Hamiltonian.
Abstract: We construct the spectrum generating algebra (SGA) for a free particle in the three dimensional sphere $S^3$ for both, classical and quantum descriptions. In the classical approach, the SGA supplies time-dependent constants of motion that allow to solve algebraically the motion. In the quantum case, the SGA include the ladder operators that give the eigenstates of the free Hamiltonian. We study this quantum case from two equivalent points of view.
TL;DR: In this paper, the singular operator limit of thin tubes with Dirichlet boundary conditions was studied, and a procedure to obtain the effective Schrodinger operator in different subspaces was proposed, generalizing recent results in case of compact tubes.
Abstract: The $\Gamma$-convergence of lower bounded quadratic forms is used to study the singular operator limit of thin tubes (i.e., the vanishing of the cross section diameter) of the Laplace operator with Dirichlet boundary conditions; a procedure to obtain the effective Schrodinger operator (in different subspaces) is proposed, generalizing recent results in case of compact tubes. Finally, after scaling curvature and torsion the limit of a broken line is briefly investigated.
TL;DR: In this article, the Schrodinger factorization was applied to construct the ladder operators for the hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator in arbitrary dimensions.
Abstract: We apply the Schrodinger factorization to construct the ladder operators for the hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator in arbitrary dimensions. By generalizing these operators we show that the dynamical algebra for these problems is the su(1, 1) Lie algebra.
TL;DR: In this article, a necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. And a theorem of Williams is generalized to obtain a translatable radius of an operator in the direction of another operator.
Abstract: One of the couple of translatable radii of an operator in the direction of another operator introduced in earlier work [PAUL, K.: Translatable radii of an operator in the direction of another operator, Scientae Mathematicae 2 (1999), 119–122] is studied in details. A necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. Finally a theorem of Williams ([WILLIAMS, J. P.: Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136]) is generalized to obtain a translatable radius of an operator in the direction of another operator.
TL;DR: In this article, a Lindblad master equation for a harmonic oscillator is formally solved, which yields the spectral resolution of the Liouvillian, that is, all eigenvalues and eigenprojections are obtained.
Abstract: A Lindblad master equation for a harmonic oscillator, which describes the dynamics of an open system, is formally solved. The solution yields the spectral resolution of the Liouvillian, that is, all eigenvalues and eigenprojections are obtained. This spectral resolution is discussed in depth in the context of the biorthogonal system and the rigged Hilbert space, and the contribution of each eigenprojection to expectation values of physical quantities is revealed. We also construct the ladder operators of the Liouvillian, which clarify the structure of the spectral resolution.
TL;DR: In this paper, the Dirac operator is constructed by lifting the affine metric connection with torsion 1/3 to the spin structure, which is a symmetric elliptic differential operator acting on sections of the spinor bundle and can be identified in special cases with Kostant's cubic Dirac or the Dolbeault operator.
Abstract: On a Riemannian spin manifold (M
n
, g), equipped with a non-integrable geometric structure and characteristic connection ▽
c
with parallel torsion ▽
c
T
c
= 0, we can introduce the Dirac operator D
1/3, which is constructed by lifting the affine metric connection with torsion 1/3 T
c
to the spin structure. D
1/3 is a symmetric elliptic differential operator, acting on sections of the spinor bundle and can be identified in special cases with Kostant’s cubic Dirac operator or the Dolbeault operator. For compact (M
n
, g), we investigate the first eigenvalue of the operator $${\left(D^{1/3} \right)^{2}}$$
. As a main tool, we use Weitzenbock formulas, which express the square of the perturbed operator D
1/3 + S by the Laplacian of a suitable spinor connection. Here, S runs through a certain class of perturbations. We apply our method to spaces of dimension 6 and 7, in particular, to nearly Kahler and nearly parallel G
2-spaces.
TL;DR: In this paper, the authors considered the discrete Schrodinger operator on the graph obtained in the strong-coupling approximation from the standard electron Schroffinger operator in the system composed of a quantum wire and quantum dot.
Abstract: We consider the discrete Schrodinger operator on the graph obtained in the strong-coupling approximation from the standard electron Schrodinger operator in the system composed of a quantum wire and quantum dot. We investigate the general spectral properties of this operator and the problem of the existence and behavior of the eigenvalues and resonances depending on the small coupling constant. We study the scattering problem for weak potentials in the stationary approach.
TL;DR: The approach involves bosonic ladder operators that generate sets of transverse optical modes and provides a universal geometric interpretation of the phase shifts acquired by nonastigmatic modes under typical experimental conditions.
Abstract: We study geometric phases that arise from (cyclic) transformations of the transverse spatial structure of paraxial optical modes. Our approach involves bosonic ladder operators that, in the spirit of the quantum-mechanical harmonic oscillator, generate sets of transverse optical modes. It applies to modes of all orders in a very natural way and provides a universal geometric interpretation of the phase shifts acquired by nonastigmatic modes under typical experimental conditions.
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TL;DR: In this paper, the concepts of stable and unstable C-symmetry are introduced in the extension theory framework and the main results are the following: if A is a J-self-adjoint extension of S, then either R or C = C, and if A has a real spectrum, then A has stable C symmetry.
Abstract: AoaN J-self-adjoint extensions of the Phillips symmetric operator S are studied. The concepts of stable and unstable C-symmetry are introduced in the extension theory framework. The main results are the following: if A is a J-self-adjoint extension of S, then either �(A) = R or �(A) = C; if A has a real spectrum, then A has a stable C-symmetry and A is similar to a self-adjoint operator; there are no J-self-adjoint extensions of the Phillips operator with unstable C-symmetry.
TL;DR: In this paper, the authors apply the Schr\"odinger factorization to construct the ladder operators for hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator in arbitrary dimensions.
Abstract: We apply the Schr\"odinger factorization to construct the ladder operators for hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator in arbitrary dimensions. By generalizing these operators we show that the dynamical algebra for these problems is the $su(1,1)$ Lie algebra.