For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.

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.

We introduce a notion of semiclassical bi-states. They arise from pairs of eigenstates corresponding to tunnel-split eigenlevels and generate 2-level subsystems in a given quantum system. As an example, we consider the planar Penning trap with rectangular electrodes assuming the 3: (−1) resonance regime of the charge dynamics. We demonstrate that, under a small deviation of the rectangular shape of electrodes from the square shape (symmetry breaking), there appear instanton pseudoparticles, semiclassical bi-states, and 2-level subsystems in such a quantum trap.

Using geometrical and algebraic ideas, we study tunnel eigenvalue asymptotics and tunnel bilocalization of eigenstates for certain class of operators (quantum Hamiltonians) including the case of Penning traps, well known in physical literature. For general hyperbolic traps with geometric asymmetry, we study resonance regimes which produce hyperbolic type algebras of integrals of motion. Such algebras have polynomial (non-Lie) commutation relations with creation-annihilation structure. Over this algebra, the trap asymmetry (higher-order anharmonic terms near the equilibrium) determines a pendulum-like Hamiltonian in action-angle coordinates. The symmetry breaking term generates a tunneling pseudoparticle (closed instanton). We study the instanton action and the corresponding spectral splitting.

The properties of flux-cored wires of the rutile and metal-powder type for welding and surfacing in industry and constructional engineering are examined. It is shown that flux-cored wires have obvious advantages in comparison with solid section wires both in the welding and technological properties of the welded joints. They are recommended for wide use.

The Schrödinger operator for a spinless charge inside a layer with parabolic confinement profile and homogeneous magnetic field is considered. The Lorentz (cyclotron) and the confinement frequencies are assumed to be equal to each other. After inclination of the layer normal from the magnetic field direction there appears a pseudospin su(2)-field removing the resonance degeneracy of Landau levels. Under deviations of the layer surface from the plane shape, a longitudinal geometric current is created. In circulations around surface warping, there is a nontrivial quantum phase transition generated by an element of the π1-homotopy group and a hidden degree of freedom (spectral degeneracy) associated with a “charge” of geometric poles on the layer. The quantization rule contains an additional parity index related to the algebraic number of geometric poles and the Landau level number. The resonance pseudospin phase-shift represents an example of general Aharonov–Bohm type topologic phenomena in quantum (semiclassical or adiabatic) systems with delta-function singularities in symplectic structure.

A geometric construction of the `ala Planck action integral (quantization rule) determining adiabatic terms for fast-slow systems is considered. We demonstrate that in the first (after zero) adiabatic approximation order, this geometric rule is represented by a deformed fast symplectic 2-form. The deformation is controlled by the noncommutativity of the slow adiabatic parameters. In the case of one fast degree of freedom, the deformed symplectic form incorporates the contraction of the slow Poisson tensor with the adiabatic curvature.The same deformed fast symplectic structure is used to represent the improved adiabatic invariant in a geometric form.

We discuss two examples of classical mechanical systems which can become quantum either because of degeneracy of an integral of motion or because of tuning parameters at resonance. In both examples, the commutativity of the symmetry algebra is breaking, and noncommutative symmetries arise. Over the new noncommutative algebra, the system can reveal its quantum behavior including the tunneling effect. The important role is played by the creation-annihilation regime for the perturbation or anharmonism. Activation of this regime sometimes needs in an additional resonance deformation (Cartan subalgebra breaking).

We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under the breaking of its axial symmetry (inclination of the magnetic field) and tuning parameters (electric voltage, magnetic field strength and inclination angle) at double resonance. For eigenvalues of the quantum non-Lie top, under a specific variation of the voltage on the trap electrode, there exists an avoided crossing effect and a corresponding effect of bilocalization of quantum states on pairs of closed trajectories belonging to common energy levels. This quantum tunneling happens on the symplectic leaves of the symmetry algebra, and hence it generates a tunneling of quantum states of the electron between the 3D-tori in the whole 6D-phase space. We present a geometric formula for the leading term of asymptotics of the tunnel energy-splitting in terms of symplectic area of membranes bounded by invariantly defined instantons.

For slow-fast Hamiltonian systems with one fast degree of freedom, we describe the construction of the complete adiabatic invariant and the complete adiabatic term at once in all asymptotic orders by using the small parameter "dynamics" and parallel translations in the phase space.

We study the dynamics of a charge in the planar Penning trap in which the direction of the magnetic field does not coincide with the trap axis. Under a certain combined resonance condition on the deviation angle and magnitudes of magnetic and electric fields, the trajectories of a charge are near-periodic.We describe the reduction to a top-like system with one degree of freedom on the space with quadratic Poisson brackets and study the stability of the equilibrium points of this system.

We discuss the general opportunity to create (asymptotically) a completely integrable system from the original perturbed system by inserting additional perturbing terms. After such an artificial insertion, there appears an opportunity to make the secondary averaging and secondary reduction of the original system. Thus, in this way, the 3 D -system becomes 1-dimensional. We demonstrate this approach by the example of a resonance Penning trap.

For slow--fast quantum systems, we compute first corrections to the quantum action and to the effective slow Hamiltonian.

СОБСТВЕННЫЕ СОСТОЯНИЯ КВАНТОВОЙ НАНОЛОВУШКИ ПЕННИНГА-ИОФФЕ В РЕЗОНАНСНОМ РЕЖИМЕОбсуждается выбор физических параметров квантовой наноловушки Пеннинга с возмущающим неоднородным магнитным полем Иоффе, а также роль резонансных частотных режимов.Приведена

The equilibrium structure of the urotropine molecule is characterized by means of gas electron diffraction (GED) with the involvement of quantum chemistry and vibrational spectroscopy. A structural analysis of the GED data is performed based on the parameters of the intramolecular potential function using of the program complex SYMM/DISP/ELDIFF/LARGE. The quadratic and cubic force constants of the urotropine molecule were obtained earlier on the basis of calculations at the MP2(full)/cc-pVTZ level and assuming molecular symmetry T (d) . The values of the equilibrium geometric parameters r (e) of the urotropine molecule are found. The experimental structural parameters are in good agreement with those calculated at the MP2(full)/cc-pVTZ level.

We discuss the choice of physical parameters of a quantum Penning nanotrap under the action of a perturbing inhomogeneous Ioffe magnetic field and also the role of frequency resonance modes. We present a general scheme for constructing the asymptotic behavior of the eigenstates by the generalized geometric quantization method and obtain the reproducing measure in the integral representation of eigenfunctions.

For quantum as well classical slow-fast systems, we develop a general method which allows one to compute the adiabatic invariant (approximate integral of motion), its symmetries, the adiabatic guiding center coordinates and the effective scalar Hamiltonian in all orders of a small parameter. The scheme does not exploit eigenvectors or diagonalization, but is based on the ideas of isospectral deformation and zero-curvature equations, where the role of "time" is played by the adiabatic (quantization) parameter. The algorithm includes the construction of the zero-curvature adiabatic connection and its splitting generated by averaging up to an arbitrary order in the small parameter.

We consider the one-dimensional Schrödinger operator in the semiclassical regime assuming that its double-well potential is the sum of a finite "physically given" well and a square shape probing well whose width or depth can be varied (tuned). We study the dynamics of initial state localized in the physical well. It is shown that if the probing well is not too close to the physical one and if its parameters are specially tuned, then the tunnel catch effect appears, i.e. the initial state starts tunneling oscillations between the physical and probing wells. The asymptotic formula for the probability of finding the state in the probing well is obtained. We show that the observation of the tunnel catch effect can be used to determine the energy level of the initial state, and we obtain the corresponding asymptotic formula for the initial state energy. We also calculate the leading term of the tunneling splitting of energy levels in this double well potential.

Обсуждаются физические параметры квантовой наноловушки Пеннинга. В случае резонанса 3:(-1) между поперечными частотами ловушки дано описание воспроизводящей меры на симплектических листах, отвечающих неприводимым представлениям нелиевской алгебры симметрий с кубическими коммутационными соотношениями. Неоднородность магнитного поля и негармоническая часть электрического потенциала ловушки после двойного квантового усреднения порождают эффективный гамильтониан, который в неприводимом представлении становится обыкновенным дифференциальным оператором второго порядка. Получена интегральная формула для асимптотических собственных состояний возмущенной 3:(-1) резонансной ловушки Пеннинга через собственные функции этого оператора, когерентные состояния и воспроизводящую меру.