Showing papers on "Ladder operator published in 2000"
Book•
21 Sep 2000
TL;DR: In this paper, the authors present a model of a classical supersymmetric model of superpotential potential in a classical poisson-bracket and a classical Supersymmetric Hamiltonian.
Abstract: GENERAL REMARKS ON SUPERSYMMETRY Background BASIC PRINCIPLES OF SUPERSYMMETRIC QUANTUM MECHANICS SUSY and the Oscillator Problem Superpotential and Setting Up a Supersymmetric Hamiltonian Physical Interpretation of Hs Properties of the Partner Hamiltonians Applications Superspace Formalism Other Schemes of SUSY SUPERSYMMETRIC CLASSICAL MECHANICS Classical Poisson Bracket, Its Generalizations Some Algebraic Properties of the Generalized Poisson Bracket A Classical Supersymmetric Model SUSY Breaking, Witten Index and Index Condition SUSY Breaking Witten Index Finite Temperature SUSY Regulated Witten Index Index Condition q-Deformation and Index Condition Parabosons Deformed Parabose States and Index Condition Witten's Index and Higher-Derivative SUSY Explicit SUSY Breaking and Singular Superpotentials FACTORIZATION METHOD, SHAPE INVARIANCE AND GENERATION OF SOLVABLE PROBLEMS Preliminary Remarks Factorization Method of Infeld and Hull Shape Invariance Condition Self-Similar Potentials A Note on the Generalized Quantum Condition Non-Uniqueness of the Factorizability Phase Equivalent Potentials Generation of Exactly Solvable Potentials in SUSYQM Conditionally Solvable Potentials and SUSY RADIAL PROBLEMS AND SPIN-ORBIT COUPLING SUSY and the Radial Problems Radial Problems Using Ladder Operator Techniques in SUSYQM Isotropic Oscillator and Spin-Orbit Coupling SUSY in D- Dimensions SUPERSYMMETRY IN NONLINEAR SYSTEMS The KdV Equation Conservation Laws in Nonlinear Systems Lax Equations SUSY and Conservation Laws in the KdV - MKdV Systems Darboux's Method SUSY and Conservation Laws in the KdV-SG Systems Supersymmetric KdV Conclusion PARASUPERSYMMETRY Introduction Models of PSUSYQM PSUSY of Arbitrary Order p Truncated Oscillator and PSUSYQM Multidimensional Parasuperalgebras APPENDIX Note: Each chapter also contains a References section
250 citations
TL;DR: In this paper, the authors define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures.
Abstract: We define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures. We find such structures by means of the associative version of Nijenhuis tensors. Explicit examples, e.g. for the harmonic oscillator, are given.
124 citations
TL;DR: In this article, exact upper and lower bounds on the spectrum of the square of the Hermitian Wilson Dirac operator were derived for domain-wall fermions in odd dimensions.
Abstract: New exact upper and lower bounds are derived on the spectrum of the square of the Hermitian Wilson Dirac operator It is hoped that the derivations and the results will be of help in the search for ways to reduce the cost of simulations using the overlap Dirac operator The bounds also apply to the Wilson Dirac operator in odd dimensions and are therefore relevant to domain-wall fermions as well (c) 2000 The American Physical Society
110 citations
96 citations
TL;DR: In this article, the Jacobi identity for intertwining operator algebras is introduced and the main properties of genus-zero conformal field theories, including vertex operator algebra, modules, intertwining operators, Verlinde algebra, and fusing and braiding matrices, are incorporated into this identity.
Abstract: We prove a generalized rationality property and a new identity that we call the ''Jacobi identity'' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in [H4] and [H6], the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.
67 citations
TL;DR: In this article, the authors constructed vertex operator algebras associated with self-orthogonal ternary codes and showed that the vertex operator algebra V contains a subalgebra T 2 10 5 4 4 4, 0 [ L, 3 and 5 5 4 2.
62 citations
TL;DR: In this article, the Schrodinger operator on graphs is studied and the spectral statistics of a unitary operator which represents the quantum evolution of a quantum map on the graph are derived.
Abstract: We consider the Schrodinger operator on graphs and study the spectral statistics of a unitary operator which represents the quantum evolution, or a quantum map on the graph. This operator is the quantum analogue of the classical evolution operator of the corresponding classical dynamics on the same graph. We derive a trace formula, which expresses the spectral density of the quantum operator in terms of periodic orbits on the graph, and show that one can reduce the computation of the two-point spectral correlation function to a well defined combinatorial problem. We illustrate this approach by considering an ensemble of simple graphs. We prove by a direct computation that the two-point correlation function coincides with the circular unitary ensemble expression for 2 × 2 matrices. We derive the same result using the periodic orbit approach in its combinatorial guise. This involves the use of advanced combinatorial techniques which we explain.
32 citations
TL;DR: In this paper, the vertex operator algebra MD was constructed using a code D in Z2 × Z2 and all the irreducible modules of MD were computed in Z 2 × Z 2.
31 citations
Dissertation•
01 Jan 2000
29 citations
TL;DR: In this article, the authors considered the spectral shift function where H 0 is a Schrodinger operator with a variable Riemannian metric and an electromagnetic field and V is a perturbation by a multiplication operator.
Abstract: We consider the spectral shift function where H 0 is a Schrodinger operator with a variable Riemannian metric and an electromagnetic field and V is a perturbation by a multiplication operator. We prove the Weyl type asymptotic formula for in the large coupling constant limit
28 citations
TL;DR: In this article, an integral representation for the effective Dirac operator with the anti-periodic boundary condition in the fifth direction was derived and exponential bounds were obtained for gauge fields with small lattice field strength.
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 links these systems.
TL;DR: In this paper, a regularized quantum counterpart of the free scalar wave equation is shown to admit a self-adjoint extension and is not a positive operator, unlike its classical counterpart.
Abstract: Every (one-polarization) cylindrical wave solution of vacuum general relativity is completely determined by a corresponding axisymmetric solution to the free scalar wave equation on an auxiliary (2 + 1)-dimensional flat spacetime. The physical metric at radius R is determined by the energy, (R ), of the scalar field in a box (in the flat spacetime) of radius R . In a recent work, among other important results, Ashtekar and Pierri have introduced a strategy to study the quantum geometry in this system, through a regularized quantum counterpart of (R ). We show that this regularized object is a densely defined symmetric operator, thereby correcting an error in their proof of this result. We argue that it admits a self-adjoint extension and show that the operator, unlike its classical counterpart, is not positive.
TL;DR: In this article, a new formulation of the 3D Boltzmann non linear operator, without assuming Grad's angular cuto hypothesis, was proposed for intermolecular laws behaving as 1=r s,w ith s> 2.
Abstract: We propose a new formulation of the 3D Boltzmann non linear operator, without assuming Grad's angular cuto hypothesis, and for intermolecular laws behaving as 1=r s ,w ith s> 2. It involves natural pseudo dierential operators, under a form which is analogous to the Landau operator. It may be used in the study of the associated equations, and more precisely in the non homogeneous framework.
TL;DR: In this paper, the OPE formula in non-commutative field theory has been tested in the context of the mixing of UV and IR effects, and it has been shown that the product of two fields in general cannot be described by a series expansion of single local operator insertions.
Abstract: Motivated by the mixing of UV and IR effects, we test the OPE formula in non-commutative field theory. First we look at the renormalization of local composite operators, identifying some of their characteristic IR/UV singularities. Then we find that the product of two fields in general cannot be described by a series expansion of single local operator insertions.
TL;DR: In this article, the first step towards a function theory for Rarita-Schwinger equation on general spin manifolds is taken. But the main result contained in the paper is a complete classification of polynomial solutions of Rarit-Schwinging equation on R n.
Abstract: There is a certain family of conformally invariant first order elliptic systems which include the Dirac operator as its first member, and the Rarita-Schwinger operator, as the second simplest operator in the row. Its basic properties on general spin manifolds are described there. The aim of the paper is to do first step towards a function theory for Rarita-Schwinger equation. The main result contained in the paper is a complete classification of polynomial solutions of Rarita-Schwinger equation on R n . Relations with Clifford analysis and representation theory are discussed.
CERN1
TL;DR: In this article, the authors describe the numerical treatment of Neuberger's lattice Dirac operator as implemented in a practical application and discuss the improvements they have found to accelerate the numerical computations and give an estimate of the expense when using this operator in practice.
Abstract: We describe in some detail our numerical treatment of Neuberger’s lattice Dirac operator as implemented in a practical application. We discuss the improvements we have found to accelerate the numerical computations and give an estimate of the expense when using this operator in practice.
TL;DR: The algebraic structure and the relationships between the eigenspaces of the Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are investigated through the Cherednik operators.
Abstract: The algebraic structure and the relationships between the eigenspaces of the Calogero-Sutherland model (CSM) and the Sutherland model (SM) on a circle are investigated through the Cherednik operators. We find an exact connection between the simultaneous nonsymmetric eigenfunctions of the ${A}_{N\ensuremath{-}1}$ Cherednik operators, from which the eigenfunctions of the CSM and SM are constructed, and the monomials. This construction allows us to simultaneously diagonalize both CSM and SM (after gauging away the Hamiltonians by suitable measures) and also enables us to write down a harmonic oscillator algebra involving the Cherednik operators, which yields the raising and lowering operators for both of these models. The connections of the CSM with free oscillators and the SM with free particles on a circle are established in a novel way. We also point out the subtle differences between the excitations of the CSM and the SM.
TL;DR: Inverse scattering for real-valued short range potentials on Rn is studied in this article, where it is shown that the scattering matrix at fixed energy is the pull-back of a pseudo-differential operator and that the symbol of the operator determines the asymptotics of the potential.
Abstract: Inverse scattering for real-valued short range potentials on Rn is studied. It is shown that the scattering matrix at fixed energy is the pull-back of a pseudo-differential operator and that the symbol of the operator determines the asymptotics of the potential. This is done by an explicit construction of the Poisson operator for the scattering problem as an oscillatory integral
TL;DR: In this article, it was shown that for the screened Coulomb potential and isotropic harmonic oscillator, there exists an infinite number of closed orbits for suitable angular momentum values, and that the energy (but not angular momentum) raising and lowering operators can be constructed from a factorization of the radial Schrodinger equation.
Abstract: It is shown that for the screened Coulomb potential and isotropic harmonic oscillator, there exists an infinite number of closed orbits for suitable angular momentum values. At the aphelion (perihelion) points of classical orbits, an extended Runge-Lenz vector for the screened Coulomb potential and an extended quadrupole tensor for the screened isotropic harmonic oscillator are still conserved. For the screened two-dimensional (2D) Coulomb potential and isotropic harmonic oscillator, the dynamical symmetries SO3 and SU(2) are still preserved at the aphelion (perihelion) points of classical orbits, respectively. For the screened 3D Coulomb potential, the dynamical symmetry SO4 is also preserved at the aphelion (perihelion) points of classical orbits. But for the screened 3D isotropic harmonic oscillator, the dynamical symmetry SU(2) is only preserved at the aphelion (perihelion) points of classical orbits in the eigencoordinate system. For the screened Coulomb potential and isotropic harmonic oscillator, only the energy (but not angular momentum) raising and lowering operators can be constructed from a factorization of the radial Schrodinger equation.
TL;DR: In this paper, the Brillouin-Wigner theory and the Rayleigh-Schrodinger theory are formulated using projection operator techniques and the determination of the perturbed eigenvalue can be decoupled from that of perturbed state.
Abstract: The time-independent perturbation theory in quantum mechanics is formulated using projection operator techniques. The determination of the perturbed eigenvalue can be decoupled from that of the perturbed eigenstate. Both the Brillouin–Wigner theory and the Rayleigh–Schrodinger theory come out straightforwardly. Both degenerate and nondegenerate cases can be treated in a unified way for arbitrarily high order perturbations.
TL;DR: In this paper, a general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems, where the time-evolutional propagator of a multi-vel spin system is decomposed completely into a product of a series of elementary propagators.
Abstract: A general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems The time-evolutional propagator of a multilevel spin system is decomposed completely into a product of a series of elementary propagators Then the unitary time evolution of the system can be determined exactly through the decomposed propagator This decomposition may be simplified with the help of the properties of the finite dimensional Liouville operator space and of its three operator subspaces, and the operator algebra structure of spin Hamiltonian of the system The Liouville operator space contains the even-order multiple-quantum, the zero-quantum, and the longitudinal magnetization and spin order operator subspace, and moreover, each former subspace contains its following subspaces The propagator can be decomposed readily and completely for a spin system whose Hamiltonian is a member of the longitudinal magnetization and spin order operator subspace If the Hamiltonian of a spin system is a zero-quantum operator this decomposition may be implemented by making a zero-quantum unitary transformation on the Hamiltonian to convert it into the diagonalized Hamiltonian, while if the Hamiltonian is an even-order multiple-quantum operator the decomposition may be carried out by diagonalizing the Hamiltonian with an even-order multiple-quantum unitary transformation When the Hamiltonian is a member of the Liouville operator space but not any element of its three subspaces the decomposition may be achieved first by making an odd-order multiple-quantum and then an even-order multiple-quantum unitary transformation to convert it into the diagonalized Hamiltonian Parameter equations to determine the unknown parameters in the decomposed propagator are derived for the general case and approaches to solve the equations are proposed
01 Jan 2000
TL;DR: In this paper, it is shown that the standard SU(1,1) and SU(2) coherent states are the unique states which minimize the second order characteristic inequality for the three generators.
Abstract: The three ways of generalization of canonical coherent states are briefly reviewed and compared with the emphasis laid on the (minimum) uncertainty way. The characteristic uncertainty relations, which include the Schroedinger and Robertson inequalities, are extended to the case of several states. It is shown that the standard SU(1,1) and SU(2) coherent states are the unique states which minimize the second order characteristic inequality for the three generators. A set of states which minimize the Schroedinger inequality for the Hermitian components of the su_q(1,1) ladder operator is also constructed. It is noted that the characteristic uncertainty relations can be written in the alternative complementary form.
TL;DR: In this paper, the authors study the nonclassical properties and algebraic characteristics of the negative binomial states introduced by Barnett recently and find that they are excited geometric states, which are essentially Perelomov'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 Perelomov'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. d 32.80.Pj Optical cooling of atoms; trapping
TL;DR: In this paper, the exact reflection and transmission coefficients for supersymmetric shape-invariant potentials barriers are calculated by an analytical continuation of the asymptotic wavefunctions 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 wavefunctions obtained via the introduction of new generalized ladder operators. The general form of the wavefunction is obtained by the use of the F (- ,+ )-matrix formalism of Froman and Froman which is related to the evolution of asymptotic wavefunction coefficients.
TL;DR: In this article, a superposition of the negative binomial states is introduced and the sub-Poissonian statistics and squeezing properties of the superposition states are studied in detail.
Abstract: The states formed by a superposition of the negative binomial states are introduced. The sub-Poissonian statistics and squeezing properties of the superposition states are studied in detail. Moreover, we obtain the ladder operator formalism of the superposition state and find that this state is a type of two-photon nonlinear coherent state. A scheme for the generation of the superposition state is also discussed.
TL;DR: In this paper, it was shown that the radial momentum operator is unitarily equivalent to the momentum operator on L 2, which is not self-adjoint and has no selfadjoint extensions.
Abstract: The non self-adjointness of the radial momentum operator has been noted before by several authors, but the various proofs are incorrect. We give a rigorous proof that the $n$-dimensional radial momentum operator is not self- adjoint and has no self-adjoint extensions. The main idea of the proof is to show that this operator is unitarily equivalent to the momentum operator on $L^{2}[(0,\infty),dr]$ which is not self-adjoint and has no self-adjoint extensions.
TL;DR: The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained in this paper, where the state bears the generally deformed oscillator algebraic structure.
Abstract: The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained. The state bears the generally deformed oscillator algebraic structure. It is found that the Perelomov's coherent state is a su(1, 1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su(1, 1) displacement operator in terms of the hypergeometric functions and the expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1, 1) optical systems are discussed.
TL;DR: By virtue of the technique of integration within an ordered product of operators, this article constructed the normally ordered operator fiedholm equation, and used it to derive some new operator formulas.
Abstract: By virtue of the technique of integration within an ordered product of operators we construct the normally ordered operator fiedholm equation. We use it to derive some new operator formulas. For Weyl correspondence, operator fiedholm equation can also be constructed. Some applications of the operator Fkedholm equation are given.
Posted Content•
TL;DR: In this article, a non-Hermitian regularization of the SUSY-type isospectrality and degeneracy of harmonic oscillators with a centrifugal spike is presented.
Abstract: Harmonic oscillators with a centrifugal spike are analysed, via a non-Hermitian regularization, within a complexified SUSY quantum mechanics The formalism enables us to construct the factorized creation and annihilation operators We show how the real though, generically, non-equidistant spectrum complies with the current SUSY-type isospectrality and degeneracy in an unusual way