On the relation between the connected-moments expansion and the Lanczos variational scheme

Summary The recently introduced Connected-Moments Expansion (CMX) is compared to a variational Lanczos scheme for estimating ground-state energies of many-body systems. A systematic approach is given for three quantum-mechanical systems: harmonic oscillator, anharmonic oscillator and the Kondo model. A second-order analysis is given in terms of particular ratios of moments of the Hamiltonian.     

Dynamical symmetries inq-deformed quantum mechanics

The dynamical algebra of theq-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction Hamiltonian in terms of the usual canonical variables. Furthermore we construct a welldefined algebraSU _q(1,1) with consistent conjugation properties and comultiplication. We obtain non lowest weight representations of this algebra.     

Application of the algebraic approach to molecular thermodynamics and quantum deformations

An algebraic model based on Lie-algebraic techniques is applied to vibrational molecular thermodynamics. The model uses the isomorphism between the SU(2) algebra and the one-dimensional Morse oscillator. A vibrational high-temperature partition function and the related thermodynamic properties are derived in terms of the parameters of the model. The anharmonic vibrations are described as anharmonic q-bosons using a first-order expansion of a quantum deformation. It is shown, that this quantum deformation is related to the shape of the Morse potential.     

On the anharmonic oscillator in quantum mechanics and in field theory

A modified Rayleigh-Schrödinger perturbation method is used to derive explicit expansions for the eigenvalues and eigensolutions of the anharmonic oscillator. We then point out the dual relationship between the anharmonic oscillator and the Schrödinger equation for a Yukawa potential. Finally we consider an application of the method to a field-theoretic Hamiltonian, since the anharmonic oscillator plays a dominant role in many field-theoretic models.     

Algebraic methods applied to the study of energy transfer in anharmonic systems

We make use of two different methodologies to study the transition probabilities in a molecular anharmonic system in the presence of an external perturbation. For the first method, we use a series expansion of the displacement coordinate keeping up to fourth order terms; for the second method we use a deformed algebra to approximate the anharmonic Hamiltonian via a harmonic oscillator's Hamiltonian written in terms of deformed operators. We evaluate vibrational transition probabilities as a function of the collision energy and compare the results obtained with the two approaches.     

N = 4 Supersymmetric Morse Oscillator and Its Spectrum-Generating Algebra

In this paper N = 4 supersymmetry of generalized Morse oscillators in one dimension is studied. Both bound states and scattering states of its four superpartner Hamiltonians are analyzed by using unitary irreducible representations of the noncompact Lie algebra su(1,1). The spectrum-generating algebra governing the Hamiltonian of the N = 4 supersymmetric Morse oscillator is shown to be connected with the realization of Lie superalgebra osp(1,2)or B(0,1) in terms of the variables of a supersymmetric two-dimensional harmonic oscillator.     

N = 4 Supersymmetric Morse Oscillator and Its Spectrum-Generating Algebra

In this paper N = 4 supersymmetry of generalized Morse oscillators in one dimension is studied. Both bound states and scattering states of its four superpartner Hamiltonians are analyzed by using unitary irreducible representations of the noncompact Lie algebra su（1,1）. The spectrum-generating algebra governing the Hamiltonian of the N = 4 supersymmetric Morse oscillator is shown to be connected with the realization of Lie superalgebra osp（1,2） or B（0,1） in terms of the variables of a supersymmetric two-dimensional harmonic oscillator.     

SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed. PACS Numbers: 02.20.Sv, 03.65.-w, 03.65.Fd     

Secondary resonances in Penning traps. Non-lie symmetry algebras and quantum states

The Penning trap Hamiltonian (hyperbolic oscillator in a homogeneous magnetic field) is considered in the basic three-frequency resonance regime. We describe its non-Lie algebra of symmetries. By perturbing the homogeneous magnetic field, we discover that, for special directions of the perturbation, a secondary hyperbolic resonance appears in the trap. For corresponding secondary resonance algebra, we describe its non-Lie permutation relations and irreducible representations realized by ordinary differential operators. Under an additional (Ioffe) inhomogeneous perturbation of the magnetic field, we derive an effective Hamiltonian over the secondary symmetry algebra. In an irreducible representation, this Hamiltonian is a model second-order differential operator. The spectral asymptotics is derived, and an integral formula for the asymptotic eigenstates of the entire perturbed trap Hamiltonian is obtained via coherent states of the secondary symmetry algebra.     

Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation

We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the ?x ⁴-potential, which serves as a toy model for the quantum field theoretical ?⁴-theory.     

Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators

We construct a time-dependent double well potential as an exact spectral equivalent to the explicitly time-dependent negative quartic oscillator with a time-dependent mass term. Defining the unstable anharmonic oscillator Hamiltonian on a contour in the lower-half complex plane, the resulting time-dependent non-Hermitian Hamiltonian is first mapped by an exact solution of the time-dependent Dyson equation to a time-dependent Hermitian Hamiltonian defined on the real axis. When unitary transformed, scaled and Fourier transformed we obtain a time-dependent double well potential bounded from below. All transformations are carried out non-perturbatively so that all Hamiltonians in this process are spectrally exactly equivalent in the sense that they have identical instantaneous energy eigenvalue spectra.     

Bi-Orbital States in Hyperbolic Traps

We consider a charge in a general electromagnetic trap near a hyperbolic stationary point. The two-dimensional trap Hamiltonian is the sum of a hyperbolic harmonic part and higher order anharmonic corrections. We suppose that two frequencies of the harmonic part are under the resonance 1 : (?1). In this case, anharmonic terms define the dynamics and an effective Hamiltonian on the space of motion constants of the ideal harmonic operator. We show that if the anharmonic part is symmetric, then the effective Hamiltonian has unstable equilibriums and separatrix, which define distinct classically allowed regions in the space of motion constants of the ideal trap. The corresponding stationary states of the trapped charge can form bi-orbital states, i.e., a state localized on two distinct classical trajectories. We obtain semiclassical asymptotics of the energy splitting corresponding to the charge tunneling between these two trajectories in the phase space and express it in terms of complex instantons.     

Nonuniversal quantities from dual renormalization group transformations

Using a simplified version of the renormalization group (RG) transformation of Dyson's hierarchical model, we show that one can calculate all the nonuniversal quantities entering into the scaling laws by combining an expansion about the high-temperature fixed point with a dual expansion about the critical point. The magnetic susceptibility is expressed in terms of two dual quantities transforming covariantly under an RG transformation and has a smooth behavior in the high-temperature limit. Using the analogy with Hamiltonian mechanics, the simplified example discussed here is similar to the anharmonic oscillator, while more realistic examples can be thought of as coupled oscillators, allowing resonance phenomena.     

On dynamical symmetry algebra for aq-extension of the harmonic oscillator

Dynamical symmetry algebra for aq-analogue of the linear harmonic oscillator in quantum mechanics is explicitly constructed in terms ofq-difference raising and lowering operators, which factorize governing Hamiltonian for this model.     

Algebraic approach to thermodynamic properties of diatomic molecules

A simple model extending Lie algebraic techniques is applied to the analysis of thermodynamic vibrational properties of diatomic molecules. Local anharmonic effects are described by means of a Morse-like potential and the corresponding anharmonic bosons are associated with the SU(2) algebra. The total number of anharmonic bosons, fixed by the potential shape, is determined for a large number of diatomic molecules. A vibrational high-temperature partition function and the related thermodynamic functions are derived and studied in terms of the parameters of the model. The idea of a critical temperature is introduced in relation to the specific heat. A physical interpretation in terms of a quantum deformation associated with the model is given.     

Unified Method for Dynamical Groups of Some Anharmonic Potentials

Realizations of the creation and annihilation operators for some important anharmonic potentials, such as the Morse potential, the modified Pöschl–Teller potential (MPT), the pseudoharmonic oscillator, and infinitely deep square-well potential, are presented by a factorization method. It is shown that the operators for the Morse potential and the MPT potential satisfy the commutation relations of an SU(2) algebra, but those of the pseudoharmonic oscillator and the infinitely deep square-well potential constitute an SU(1, 1) algebra. The matrix elements of some related operators are analytically obtained. The harmonic limits of the SU(2) operators for the Morse and MPT potentials are studied as the Weyl algebra.     

Anyons,deformed oscillator algebras and projectors

We initiate an algebraic approach to the many-anyon problem based on deformed oscillator algebras. The formalism utilizes a generalization of the deformed Heisenberg algebras underlying the operator solution of the Calogero problem. We define a many-body Hamiltonian and an angular momentum operator which are relevant for a linearized analysis in the statistical parameter ν. There exists a unique ground state and, in spite of the presence of defect lines, the anyonic weight lattices are completely connected by the application of the oscillators of the algebra. This is achieved by supplementing the oscillator algebra with a certain projector algebra.     

A model for a stochastic multiphoton absorption process of an anharmonic oscillator

A model of multiphoton absorption of i.r. laser radiation by an anharmonic, molecular, vibrational mode is discussed. The multiphoton absorption is described as a stochastic process for which the reservoir, including rotation and other vibrational modes, modulates the frequency of the active oscillator stochastically. The treatment of the nonlinear oscillator is based on quantization of a classical, driven anharmonic oscillator.     

Energy features of an adiabatically loaded anharmonic oscillator

An excited anharmonic oscillator is considered under conditions of adiabatic (i.e., slow, as compared to the oscillation period) loading with an external force tending to a constant value at long times. The energy characteristics of the adiabatically loaded anharmonic oscillator, such as the instantaneous energy of the oscillator, the maximum kinetic (oscillation) energy, and the kinetic and potential energies averaged over the period, are analytically calculated as a function of the steady-state force. The analytical results are confirmed by the data of numerical calculations. It is established that the external force gives rise to a redistribution of the average kinetic and potential components of the initial energy of the anharmonic oscillator and that the transferred energy portions at a small external force considerably exceed the average work done by the external force.