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## Quantization due to Breaking the Commutativity of Symmetries. Wobbling Oscillator and Anharmonic Penning Trap

Russian Journal of Mathematical Physics. 2016. Vol. 23. No. 4. P. 483-489.

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).

Vologodsky V., Finkelberg M. V., Bezrukavnikov R., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 17, 2015

Sergeev A., European Mathematical Society Publishing house, 2014

This book is based on a lecture course given by the author at the Educational Center of the Steklov Mathematical Institute in 2011. It is designed for a one-semester course for undergraduate students familiar with basic differential geometry and complex and functional analysis.
The universal Teichmüller space T is the quotient of the space of quasisymmetric ...

Added: April 9, 2015

Finkelberg M. V., Rybnikov L. G., Journal of the European Mathematical Society 2012

algebra $\hat{sl}_n$. We introduce an affine, reduced, irreducible, normal quiver variety $Z$ which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on $Z$ in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) ...

Added: February 19, 2013

М.В. Карасев, Е.М. Новикова, Наноструктуры. Математическая физика и моделирование 2015 Т. 13 № 2 С. 55-92

We study the planar Penning traps in a resonance mode. The axial symmetry of the system is violated by deviation of the magnetic field from the trap axis at a small angle (the small parameter in the given model). The geometry of planar electrodes and their electric potentials are made consistent to reach a combined ...

Added: February 25, 2016

Karasev M., Novikova E., Vybornyi E., Russian Journal of Mathematical Physics 2017 Vol. 24 No. 4 P. 454-464

We introduce a notion of semiclassical bi-states. They arise from pairs of eigenstates corresponding to tunnel-splitted 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 charge dynamics. We demonstrate that under small deviation of the rectangular ...

Added: October 20, 2017

Novikova E., Russian Journal of Mathematical Physics 2021 Vol. 28 No. 3 P. 406-410

The quantum averaging method is applied to the Hamiltonian of the multifrequency resonance harmonic oscillator
perturbed by a differential operator with polynomial coefficients. The twisted product is used to transfer the averaging procedure to the space of graduated algebra of symbols. As a result, the averaged Hamiltonian is expressed in terms of generators of the quantum algebra of symmetries of the ...

Added: December 14, 2020

Novikova E., Наноструктуры. Математическая физика и моделирование 2016 Т. 15 № 2 С. 75-98

Дано описание спектральных характеристик планарной ловушки Пеннинга с кольцевой конфигурацией электродов и магнитным полем, отклоненным от аксиальной оси. Найдены соотношения между физическими параметрами, при которых наступает комбинированный частотный резонанс в гармонической (квадратичной) части гамильтониана вблизи центра ловушки. Усредненная ангармоническая часть гамильтониана представлена обыкновенным дифференциальным оператором второго порядка с полиномиальными коэффициентами, найдена асимптотика его собственных значений ...

Added: October 23, 2016

Karasev M., Novikova E., Vybornyi E., Mathematical notes 2016 Vol. 100 No. 6 P. 807-819

We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under breaking its axial symmetry (inclination of the magnetic field) and turning parameters (electric voltage, magnetic field strength and inclination angle) at double resonance. For eigenvalues of the quantum non-Lie top, under a specific variation ...

Added: October 22, 2016

Bezrukavnikov R., Finkelberg M. V., Cambridge Journal of Mathematics 2014 Vol. 2 No. 2 P. 163-190

Marc Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjecture where the symmetric group is replaced by the wreath product of S_n and Z/rZ. He has proven the original conjecture by establishing the geometric statement about the ...

Added: December 20, 2014

Novikova E., Математические заметки 2021 Т. 109 № 5 С. 747-767

For the perturbed Hamiltonian of a multifrequency resonance harmonic oscillator, a new approach for calculating the coefficients in the quantum averaging procedure is proposed. The twisted product introduced in the paper is used to transfer the averaging procedure to the space of the graduated algebra of symbols. As a result, the averaged Hamiltonian is expressed ...

Added: January 7, 2021

Takasaki K., Takebe T., Теоретическая и математическая физика (Российская Федерация) 2012 Vol. 171 No. 2 P. 683-690

We briefly review a recursive construction of hbar-dependent solutions of the Kadomtsev-Petviashvili hierarchy. We give recurrence relations for the coefficients X_n of an ħ-expansion of the operator X = X_0 + hbar X_1 + hbar^2 X_2 + ... for which the dressing operator W is expressed in the exponential form W = exp(X/hbar). The wave ...

Added: June 22, 2012

Novikova E., Наноструктуры. Математическая физика и моделирование 2017 Т. 16 № 2 С. 69-88

The scale of physical parameters determining the Penning planar trap with rectangular annular electrode is analyzed and a unifi ed relation between these parameters is obtained. This relation leads to the resonance oscillator in the leading part of the Hamiltonian for the electron in the trap. In the regime of basic hyperbolic resonance, an explicit ...

Added: January 30, 2018

Karasev M., Novikova E., Vybornyi E., Mathematical notes 2017 Vol. 102 No. 5-6 P. 776-786

In the model of Penning trap with a geometric asymmetry we study a resonance regime which produces a hyperbolic type algebra of integrals of motion. The algebra has qubic (non-Lie) commutation relations with creation-anihilation structure. The anharmonic part of the trap potential determines a top-like Hamiltonian over this algebra. The symmetry breaking term generates a ...

Added: October 20, 2017

Finkelberg M. V., Rybnikov L. G., Algebraic Geometry 2014 Vol. 1 No. 2 P. 166-180

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on ...

Added: October 25, 2013

Popov A. M., Lebedeva I. V., Knizhnik A. A. et al., Computational Materials Science 2014 Vol. 92 P. 84-91

The interaction and tunneling conductance between oppositely located ends of coaxial carbon nanotubes are studied by the example of two (11, 11) nanotubes with open ends terminated by hydrogen atoms. The Green function formalism is applied to determine the tunneling current through the nanotube ends as a function of the distance between the ends, relative orientation ...

Added: October 17, 2016

Karasev M., Vybornyi E., Journal of Mathematical Physics 2016

We consider the one-dimensional Schrodinger 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 an initial state localized in the physical well. It is shown that ...

Added: October 23, 2015

Losev Ivan, Compositio Mathematica 2017 Vol. 153 No. 12 P. 2445-2481

In this paper we study categories O over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden, Licata, Proudfoot and Webster. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories O. We ...

Added: October 15, 2017

Lasukov V. V., Abdrashitova M.O., Russian Physics Journal 2020 Vol. 63 No. 4 P. 631-648

A quantum solution of the classical electrodynamics equations has been found. It is shown that all information on the multiparticle process of creation of scalar pairs of particles by a nonstationary self-acting electric field is contained in solutions of the d’Alembert single-particle equation. The existence of a quantum solution of the d’Alembert equation is determined ...

Added: September 7, 2020

О.В. Благодырева, М.В. Карасев, Е.М. Новикова, Наноструктуры. Математическая физика и моделирование 2013 Т. 9 № 1 С. 5-18

We discuss physical parameters of quantum Penning nanotraps. In the case of 3:(-1) resonance between transverse frequencies of the trap we describe the reproducing measure on symplectic leaves corresponding to irreducible representations of non-Lie symmetry algebra with qubic commutation relations. Nonhomogeneity of the magnetic field and anharmonicity of the electric potential of the trap, after ...

Added: November 20, 2013

Arseyev P., Mantsevich V. N., Maslova N. S., JETP Letters 2012 Vol. 95 No. 10 P. 589-594

The possibility of non-adiabatic electron pumping in the system of three coupled quantum dots attached to the leads is discussed. We have found out that periodical changing of energy level position in the middle quantum dot results in non zero mean tunneling current appeared due to non-adiabatic non-equilibrium processes. The same principle can be used ...

Added: October 28, 2014

Pereskokov A., Липская А. В., Вестник Московского энергетического института 2010 № 6 С. 99-109

Рассмотрены радиально-симметричные решения уравнения типа Хартри, содержащего как кулоновский потенциал, так и интегральную нелинейность с потенциалом взаимодействия Юкавы. В квазиклассическом приближении выведены и исследованы уравнения для самосогласованного потенциала. Выписано правило квантования типа Бора-Зоммерфельда. Найдены асимптотические собственные значения и собственные функции. ...

Added: December 16, 2012

Vybornyi E., Theoretical and Mathematical Physics 2014 Vol. 178 No. 1 P. 93-114

We consider the one-dimensional stationary Schr¨odinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the well-known formulas to the ...

Added: December 23, 2013

Avetisov V. A., Gorsky A., Nechaev S. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2016 Vol. 93 No. 1 P. 012302-1-012302-7

We consider an equilibrium ensemble of large Erdos-Renyi topological random networks with fixed vertex ˝ degree and two types of vertices, black and white, prepared randomly with the bond connection probability p. The network energy is a sum of all unicolor triples (either black or white), weighted with chemical potential of triples μ. Minimizing the ...

Added: March 14, 2016

Vybornyi E., Наноструктуры. Математическая физика и моделирование 2015 Т. 12 № 1 С. 5-84

We consider the problem of constructing semiclassical asymptotic expansions of discrete spectrum and the corresponding stationary states of one-dimensional Schrödinger operator in the case of resonance tunneling. We consider two basic models: tunneling in an asymmetric double-well potential on a line and momentum tunneling of a particle in a potential field on a circle. For ...

Added: February 12, 2016