Showing papers on "Ladder operator published in 1999"
TL;DR: Using an iterative construction of the first-order intertwining technique, this article found k-parametric families of exactly solvable anharmonic oscillators whose spectra consist of a part isospectral to the oscillator plus k additional levels at arbitrary positions below E0D 1.
Abstract: Using an iterative construction of the first-order intertwining technique, we find k- parametric families of exactly solvable anharmonic oscillators whose spectra consist of a part isospectral to the oscillator plus k additional levels at arbitrary positions below E0D 1 . It is seen that the 'natural' ladder operators for these systems give place to polynomial nonlinear algebras, and it is shown that these algebras can be linearized. The coherent states construction is performed in the nonlinear and linearized cases.
151 citations
TL;DR: In this paper, a natural extension of the notion of the contragredient module for vertex operator algebras is given, and Zhu's C 2 -finiteness condition holds, fusion rules (for any three irreducible modules) are finite and the vertex operator algebra themselves are finitely generated.
139 citations
TL;DR: In this article, the authors show that any single-mode quantum state can be generated from the vacuum by alternate application of the coherent displacement operator and the creation operator, and propose an experimental implementation of the scheme for traveling optical fields, which is based on field mixings and conditional measurements.
Abstract: We show that any single-mode quantum state can be generated from the vacuum by alternate application of the coherent displacement operator and the creation operator. We propose an experimental implementation of the scheme for traveling optical fields, which is based on field mixings and conditional measurements in a beam-splitter array, and calculate the probability of state generation.
137 citations
TL;DR: In this article, the authors construct Baxter's Q operator for the homogeneous XXX model as an integral operator in the standard representation of SL(2) and show that Lipatov's duality symmetry operator arises naturally as the leading term of the asymptotic expansion of the Q operator.
Abstract: Applying the Pasquier-Gaudin procedure we construct Baxter's Q operator for the homogeneous XXX model as an integral operator in the standard representation of SL(2). The connection between the Q operator and the local Hamiltonians is discussed. We show that Lipatov's duality symmetry operator arises naturally as the leading term of the asymptotic expansion of the Q operator for large values of the spectral parameter.
104 citations
TL;DR: In this paper, the shape-invariant Lie algebra spanned by the supersymmetric ladder operators plus the identity operator is used to generate a discrete complete orthonormal basis for the quantum treatment of the one-dimensional Morse potential.
Abstract: By introducing the shape-invariant Lie algebra spanned by the supersymmetric ladder operators plus the identity operator, we generate a discrete complete orthonormal basis for the quantum treatment of the one-dimensional Morse potential. In this basis, which we call the pseudo-number-states, the Morse Hamiltonian is tridiagonal. Then we construct algebraically the continuous overcomplete set of coherent states for the Morse potential in close analogy with the harmonic oscillator. These states coincide with a class of states constructed earlier by Nieto and Simmons [Phys. Rev. D 20, 1342 (1979)] by using the coordinate representation. We also give the unitary displacement operator creating these coherent states from the ground state.
85 citations
TL;DR: In this article, the generalized decomposition of unity (or positive-operator-valued measures) from any self-adjoint extension of the time-of-arrival operator is discussed.
Abstract: We reappraise and clarify the contradictory statements found in the literature concerning the time-of-arrival operator introduced by Aharonov and Bohm in Phys. Rev. 122, 1649 (1961). We use Naimark's dilation theorem to reproduce the generalized decomposition of unity (or positive-operator-valued measures) from any self-adjoint extension of the operator, emphasizing a natural one, which arises from the analogy with the momentum operator on the half-line. General time operators are set within a unifying perspective. It is shown that they are not in general related to the time of arrival, even though they may have the same form.
83 citations
TL;DR: In this article, the action operator is defined for the consistent histories formalism, as the quantum analog of the classical action functional for the simple harmonic oscillator case, and it is shown that action operators are the generator of time transformations and are associated with two types of time evolution of the standard quantum theory: the wavepacket reduction and the unitary time evolution.
Abstract: We define the action operator for the consistent histories formalism, as the quantum analog of the classical action functional, for the simple harmonic oscillator case. We conclude that the action operator is the generator of time transformations, and is associated with the two types of time evolution of the standard quantum theory: the wave-packet reduction and the unitary time evolution. We construct the corresponding classical histories and demonstrate the relevance with the quantum histories. Finally, we show the relation of the action operator to the decoherence functional.
67 citations
TL;DR: In this paper, the cosine of an angle is defined via a scalar product density operator and the area operator, and conditions to ensure that semiclassical geometry states replicate classical angles are investigated.
Abstract: Inspired by the spin geometry theorem, two operators are defined which measure angles in the quantum theory of geometry. One operator assigns a discrete angle to every pair of surfaces passing through a single vertex of a spin network. This operator, which is effectively the cosine of an angle, is defined via a scalar product density operator and the area operator. The second operator assigns an angle to two `bundles' of edges incident to a single vertex. While somewhat more complicated than the earlier geometric operators, there are a number of properties that are investigated including the full spectrum of several operators and, using results of the spin geometry theorem, conditions to ensure that semiclassical geometry states replicate classical angles.
66 citations
TL;DR: In this paper, the irreducible modules for the fixed point vertex operator subalgebra of the vertex operator algebra associated to the Heisenberg algebra with central charge 1 under the −1 automorphism were classified.
62 citations
TL;DR: In this article, minimal generating subspaces of weak PBW type for vertex operator algebras are studied and a procedure is developed for finding such subspace, and some results on generalized modules are obtained for vertex operators that satisfy a certain condition.
62 citations
TL;DR: In this article, a photon position operator with commuting components is constructed, and it is proved that it equals the Pryce operator plus a term that compensates for the adiabatic phase.
Abstract: A photon position operator with commuting components is constructed, and it is proved that it equals the Pryce operator plus a term that compensates for the adiabatic phase. Its eigenkets are transverse and longitudinal vectors, and thus states can be selected that have definite polarization or helicity. For angular momentum and boost operators defined in the usual way, all of the commutation relations of the Poincar\'e group are satisfied. This new position operator is unitarily equivalent to the Newton-Wigner-Pryce position operator for massive particles.
TL;DR: In this article, the relativistic quantum mechanics is considered in the framework of the nonstandard synchronization scheme for clocks, which preserves Poincar{'e} covariance but distinguishes an inertial frame, which enables to avoid the problem of noncausal transmision of information related to breaking of the Bell's inequalities in QM.
Abstract: In this paper the relativistic quantum mechanics is considered in the framework of the nonstandard synchronization scheme for clocks. Such a synchronization preserves Poincar{\'e} covariance but (at least formally) distinguishes an inertial frame. This enables to avoid the problem of a noncausal transmision of information related to breaking of the Bell's inequalities in QM. Our analysis has been focused mainly on the problem of existence of a proper position operator for massive particles. We have proved that in our framework such an operator exists for particles with arbitrary spin. It fulfills all the requirements: it is Hermitean and covariant, it has commuting components and moreover its eigenvectors (localised states) are also covariant. We have found the explicit form of the position operator and have demonstrated that in the preferred frame our operator coincides with the Newton--Wigner one. We have also defined a covariant spin operator and have constructed an invariant spin square operator. Moreover, full algebra of observables consisting of position operators, fourmomentum operators and spin operators is manifestly Poincar\'e covariant in this framework. Our results support expectations of other authors (Bell, Eberhard) that a consistent formulation of quantum mechanics demands existence of a preferred frame.
TL;DR: In this article, a toppling invariant for the stochastic sandpile model was constructed, and lower and upper bounds for the minimum number of particles in a recurrent configuration were derived for finite-size rectangles.
Abstract: We present some analytical results for the stochastic sandpile model studied earlier by Manna. In this model, the operators corresponding to particle addition at different sites commute. The eigenvalues of operators satisfy a system of coupled polynomial equations. For an L×L square, we construct a nontrivial toppling invariant, and hence a ladder operator which acting on eigenvectors of the evolution operator gives new eigenvectors with different eigenvalues. For periodic boundary conditions in one direction, one more toppling invariant can be constructed. We show that there are many forbidden subconfigurations, and only an exponentially small fraction of all stable configurations are recurrent. We obtain rigorous lower and upper bounds for the minimum number of particles in a recurrent configuration, and conjecture a formula for its exact value for finite-size rectangles.
TL;DR: In this paper, the authors derived the common eigenvector |q,k〉 of the charge operator Q = a "a−b " b and (a " − b)(a − b " ) as charge raising (lowering) operators.
TL;DR: In this paper, the quantum dynamical evolution of atomic and molecular aggregates, from their compact to their fragmented states, is parametrized by a single collective radial parameter, and all the remaining particle coordinates in d dimensions democratically, as a set of angles orthogonal to this collective radius or by equivalent variables, bypasses all independent-particle approximations.
Abstract: The quantum dynamical evolution of atomic and molecular aggregates, from their compact to their fragmented states, is parametrized by a single collective radial parameter. Treating all the remaining particle coordinates in d dimensions democratically, as a set of angles orthogonal to this collective radius or by equivalent variables, bypasses all independent-particle approximations. The invariance of the total kinetic energy under arbitrary d-dimensional transformations which preserve the radial parameter gives rise to novel quantum numbers and ladder operators interconnecting its eigenstates at each value of the radial parameter. We develop the systematics and technology of this approach, introducing the relevant mathematics tutorially, by analogy to the familiar theory of angular momentum in three dimensions. The angular basis functions so obtained are treated in a manifestly coordinate-free manner, thus serving as a flexible generalized basis for carrying out detailed studies of wavefunction evolution in multi-particle systems.
TL;DR: In this article, it was shown that one can use only four large vectors at the cost of executing the core conjugate algorithm twice, which is less than by a factor of 2, depending on the architecture of the computer one uses.
Abstract: The overlap lattice-Dirac operator contains the sign function ∊(H). Recent practical implementations replace ∊(H) by a ratio of polynomials, H Pn(H2)/Qn(H2), and require storage of 2n+2 large vectors. Here I show that one can use only four large vectors at the cost of executing the core conjugate algorithm twice. The slow-down might be less than by a factor of 2, depending on the architecture of the computer one uses.
TL;DR: In this paper, the eigenvalues of the Dirac operator on an isometrically immersed surface M 2 ↪ R 3 were proved for two-dimensional compact manifolds of genus zero and genus one.
TL;DR: In this paper, a generalization of the factorization technique is shown to be a powerful algebraic tool to discover further properties of a class of integrable systems in Quantum Mechanics.
Abstract: A generalization of the factorization technique is shown to be a powerful algebraic tool to discover further properties of a class of integrable systems in Quantum Mechanics. The method is applied in the study of radial oscillator, Morse and Coulomb potentials to obtain a wide set of raising and lowering operators, and to show clearly the connection that link these systems.
TL;DR: In this article, the operator integral of a complex valued measurable function f with respect to a positive operator measure E is considered, and sufficient and necessary conditions for the equality pr(∫f ǫdF)=∫ ǔ f Ãǫ dE are obtained.
Abstract: The operator integral ∫f dE of a complex valued measurable function f with respect to a positive operator measure E is considered. If F is a Neumark dilation of E into a projection measure, then the “projected” operator integral pr(∫ fdF) is a restriction of the operator ∫ f dE. Necessary and sufficient conditions for the equality pr(∫f dF)=∫ f dE are obtained. The results are applied to determine the moment operators of the phase space observables generated by the number states.
TL;DR: In this article, it was shown that the 3D Boltzmann linear operator without angular cutoff may be decomposed into an elliptic and negative like operator and a lower order one.
Abstract: We show that the 3D Boltzmann linear operator, without angular cutoff, may be written as the sum of an elliptic and negative like operator and a lower order one. This result holds true for s > 2, on the assumption that intermolecular laws behave like 1/rs . We apply this decomposition to the existence and regularity issues of weak solutions for the corresponding (non) homogeneous Boltzmann equation.
Dissertation•
01 Jan 1999
TL;DR: In this paper, the spectral Nevanlinna-pick problem was studied from an operator theoretic perspective, and necessary conditions for the existence of a polynomial solution were derived.
Abstract: We establish necessary conditions, in the form of the positivity of Pick-matrices, for the existence of a solution to the spectral Nevanlinna-Pick problem: Let k and n be natural numbers. Choose n distinct points zj in the open unit disc, D, and n matrices Wj in Mk(C), the space of complex k × k matrices. Does there exist an analytic function φ : D → Mk(C) such that φ(zj) =Wj for j = 1, ...., n and σ(φ(z)) ⊂ D for all z ∈ D? We approach this problem from an operator theoretic perspective. We restate the problem as an interpolation problem on the symmetrized polydisc Γk, Γk = {(c1(z), . . . , ck(z)) | z ∈ D} ⊂ C k where cj(z) is the j th elementary symmetric polynomial in the components of z. We establish necessary conditions for a k-tuple of commuting operators to have Γk as a complete spectral set. We then derive necessary conditions for the existence of a solution φ of the spectral NevanlinnaPick problem. The final chapter of this thesis gives an application of our results to complex geometry. We establish an upper bound for the Caratheodory distance on int Γk. Chapter 1 Interpolation Problems This thesis is concerned with establishing necessary conditions for the existence of a solution to the spectral Nevanlinna-Pick problem. In the sections of this chapter which follow, we define a number of interpolation problems beginning with the classical Nevanlinna-Pick problem. After presenting a full solution to this classical mathematical problem, we give a brief summary of some results in linear systems theory. These results demonstrate how the classical NevanlinnaPick problem arises as a consequence of robust control theory. We then slightly alter the robust stabilization problem and show that this alteration gives rise to the spectral Nevanlinna-Pick problem. Chapter 1 is completed with the introduction of a new interpolation problem which is closely related to both versions of the Nevanlinna-Pick problem. Although the problems discussed all have relationships with linear systems and control theory, they are interesting mathematical problems in their own right. The engineering motivation presented in Section 1.2 is not essential to the work which follows but allows the reader a brief insight into the applications of our results. Chapter 2 begins by converting function theoretic interpolation problems into problems concerning the properties of operators on a Hilbert space. Throughout Chapter 2 we are concerned with finding a particular class of polynomials. Although the exact form of the polynomials is unknown to us, we are aware of various properties they must possess. We use these properties to help us define a suitable class of polynomials. In Chapter 3 we define a class of polynomials based on the results of Chapter 2. We present
TL;DR: In this paper, a class of eigenfunctions of an analytic difference operator generalizing the special Lam? operator was studied, with particular attention to quantum-mechanical aspects.
Abstract: We study a class of eigenfunctions of an analytic difference operator generalizing the special Lam? operator , paying particular attention to quantum-mechanical aspects. We show that in a suitable scaling limit the pertinent eigenfunctions lead to the eigenfunctions of the operator in a finite volume. We establish various orthogonality and non-orthogonality results by direct calculations, generalize the `one-gap picture' associated with the above Lam? operator, and obtain duality properties for the hyperbolic, trigonometric and rational specializations.
TL;DR: In this paper, a vanishing theorem for certain isotypical components of the kernel of the S1-equivariant Dirac operator with coefficients in an admissible Clifford module was proved.
Abstract: We prove a vanishing theorem for certain isotypical components of the kernel of the S1-equivariant Dirac operator with coefficients in an admissible Clifford module. The method is based on changing the metric by a conformal (generally unbounded) factor and studying the effect of this change on the Dirac operator and its kernel. In the cases relevant to S1-actions we find that the kernel of the new operator is naturally isomorphic to the kernel of the original operator.
TL;DR: In this article, the von Neumann-Liouville time evolution equation is represented in a discrete quantum phase space and the mapped Liouville operator and corresponding Wigner function are explicitly written for the problem of a magnetic moment interacting with a magnetic field and the precessing solution is found.
Abstract: The von Neumann-Liouville time evolution equation is represented in a discrete quantum phase space. The mapped Liouville operator and the corresponding Wigner function are explicitly written for the problem of a magnetic moment interacting with a magnetic field and the precessing solution is found. The propagator is also discussed and a time interval operator, associated to a unitary operator which shifts the energy levels in the Zeeman spectrum, is introduced. This operator is associated to the particular dynamical process and is not the continuous parameter describing the time evolution. The pair of unitary operators which shifts the time and energy is shown to obey the Weyl–Schwinger algebra.
TL;DR: In this paper, the spectrum of a streaming operator with diffuse reflection boundary condition arising in transport theory is observed in L 1 space and it is shown that the spectrum is completely determined by these integral operators and consists of countable isolated eigenvalues, each of which has finite algebraic multiplicity.
TL;DR: The absolute continuity of Dirac operator spectrum in R2 with the scalar potential V and the vector potential A=(A1, A2) being periodic functions with a common period lattice was shown in this paper.
Abstract: We prove the absolute continuity of the Dirac operator spectrum inR2 with the scalar potential V and the vector potential A=(A1, A2) being periodic functions (with a common period lattice) such that V, Aj≠Llocq(R2), q>2.
TL;DR: In this article, the exact reflection and transmission coefficients for supersymmetric shape-invariant potentials barriers are calculated by an analytical continuation of the asymptotic wave functions obtained via the introduction of new generalized ladder operators.
Abstract: Exact reflection and transmission coefficients for supersymmetric shape-invariant potentials barriers are calculated by an analytical continuation of the asymptotic wave functions obtained via the introduction of new generalized ladder operators. The general form of the wave function is obtained by the use of the F-matrix formalism of Froman and Froman which is related to the evolution of asymptotic wave function coefficients.
TL;DR: In this article, the code vertex operator algebras under coordinates change were studied and the results showed that code vertex operators are invariant to the coordinates change of the vertex operator.
Abstract: (1999). Code vertex operator algebras under coordinates change. Communications in Algebra: Vol. 27, No. 9, pp. 4587-4605.
TL;DR: In this article, the authors studied the nonclassical properties and algebraic characteristics of the negative binomial states introduced by Barnett recently and found that they are excited geometric states, which are essentially Peremolov's SU(1, 1) coherent states.
Abstract: We study the nonclassical properties and algebraic characteristics of the negative binomial states introduced by Barnett recently. The ladder operator formalism and displacement operator formalism of the negative binomial states are found and the algebra involved turns out to be the SU(1,1) Lie algebra via the generalized Holstein-Primarkoff realization. These states are essentially Peremolov's SU(1,1) coherent states. We reveal their connection with the geometric states and find that they are excited geometric states. As intermediate states, they interpolate between the number states and geometric states. We also point out that they can be recognized as the nonlinear coherent states. Their nonclassical properties, such as sub-Poissonian distribution and squeezing effect are discussed. The quasiprobability distributions in phase space, namely the Q and Wigner functions, are studied in detail. We also propose two methods of generation of the negative binomial states.