Coherent states of a time-dependent forced harmonic oscillator and their Aharonov-Anandan phase

A new method to construct coherent states of a time-dependent forced harmonic oscillator was given. The close relation to the classical forced oscillator and the minimum uncertainty relation were investigated. The applied periodic force (off-resonance case), in general, will attenuate the AA phase.     

Quaternionic vector coherent states and the supersymmetric harmonic oscillator

The quaternionic vector coherent states are realized as coherent states of the supersymmetric harmonic oscillator with broken symmetry in analogy with the standard canonical coherent states of the ordinary harmonic oscillator. We study the nonclassical properties of the oscillator, such as the photon number distribution and signal-to-quantum-noise ratio in terms of these states and discuss the squeezing properties and the temporal stability of the coherent states. We obtain the orthogonal polynomials associated with the quaternionic vector coherent states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 80–98, October, 2006.     

Coherent states for a generalized oscillator with a finite-dimensional Hilbert space

A construction of oscillator-like systems connected with a given set of orthogonal polynomials and coherent states for such systems developed by the authors is extended to the case of systems with a finite-dimensional state space. As an example, we consider a generalized oscillator connected with Krawtchouk polynomials. Bibliography: 24 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 75–99.     

A map between the coulomb and the harmonic oscillator systems in a higher-dimensional space

We present an extended transformation method for mapping between higher-dimensional spaces. We report solutions of the Schrödinger equation for the Green’s function in the form of bound states and scattering states. We discuss the normalizability of the bound-state solution of a generated exactly solvable potential.     

The q-deformed harmonic oscillator, coherent states, and the uncertainty relation

For a q-deformed harmonic oscillator, we find explicit coordinate representations of the creation and annihilation operators, eigenfunctions, and coherent states (the last being defined as eigenstates of the annihilation operator). We calculate the product of the “coordinate-momentum” uncertainties in q-oscillator eigenstates and in coherent states. For the oscillator, this product is minimum in the ground state and equals 1/2, as in the standard quantum mechanics. For coherent states, the q-deformation results in a violation of the standard uncertainty relation; the product of the coordinate-and momentum-operator uncertainties is always less than 1/2. States with the minimum uncertainty, which tends to zero, correspond to the values of λ near the convergence radius of the q-exponential. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 315–322, May, 2006.     

Canonical transformations of the extended phase space and integrable systems

We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the correspondingr-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stäckel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property.     

Airy functions and transition between semiclassical and harmonic oscillator approximations for one-dimensional bound states

Theoretical and Mathematical Physics - We consider the one-dimensional Schrödinger operator with a semiclassical small parameter $$h$$ . We show that the “global” asymptotic form...     

Stability and bifurcation in a harmonic oscillator with delays

We consider a harmonic oscillator with delays. Linear stability is investigated by analyzing the associated characteristic transcendental equation. The bifurcation analysis of the equation shows that Hopf bifurcation can occur as the delay τ (taken as a parameter) crosses some critical values. The direction and stability of the Hopf bifurcation are considered by using the normal form theory due to Faria and Magalhães. An example is given to explain the results. Numerical simulations support our results.     

Geodesic transformations and harmonic spaces

We prove that a Riemannian manifold is harmonic if and only if there exists a divergence-preserving geodesic transformation with respect to each point which is not volume-preserving.     

Adelic model of harmonic oscillator

Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation.This work was partially supported by the Russian Foundation for Fundamental Research, Grant No. 93-011-147.Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 3, pp. 349–359, December, 1994.     

p-Adic and adelic harmonic oscillator with a time-dependent frequency

The classical and quantum formalism for a p-adic and adelic harmonic oscillator with a time-dependent frequency is developed, and general formulas are obtained for the main theoretical quantities. In particular, the p-adic propagator is calculated, and the existence of a simple vacuum state as well as adelic quantum dynamics is shown. A spatial discreteness and a p-adic quantum mechanical phase are noted. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 239–248, August, 2000.     

The quantum harmonic oscillator as a Zariski geometry

A structure is associated with the quantum harmonic oscillator, over a fixed algebraically closed field F$\mathbb{F}$ of characteristic 0, which is shown to be uncountably categorical. An analysis of definable sets is carried out, from which it follows that this structure is a Zariski geometry of dimension 1. It is non-classical in the sense that it is not interpretable in _ACF0${\mathit{ACF}}_0$ and in the case F=C$\mathbb{F}=\mathbb{C}$, is not a structure on a complex manifold.     

The action variable and frequency of a relativistic harmonic oscillator

We present three series representations of the frequency of a relativistic harmonic oscillator. The first two representations use two equivalent forms of the action variable. The third representation involves determining its period by direct integration. The energy dependance of the oscillator frequency is manifestly seen in all three representations. We demonstrate that all three forms yield the same expression for the frequency in the case of the weakly relativistic oscillator and have an identical nonrelativistic limit.