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## Inserted perturbations generating asymptotical integrability

Mathematical notes. 2014. Vol. 96. No. 6. P. 965-970.

We discuss the general opportunity to create (asymptotically) a comletely 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, by this way, the $3D$-system becomes $1$-dimensional. We demonstrate this approach by the example of a resonance Penning trap.

Publication based on the results of:

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., 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

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

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

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

Derbyshev A. E., Povolotsky A. M., Priezzhev V. B., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2015 Vol. 91 P. 022125

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. (2012) P05014] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as ...

Added: February 19, 2015

Sechin I., Zotov A., Physics Letters B 2018 Vol. 781 P. 1-7

In this paper we discuss R -matrix-valued Lax pairs for sl N Calogero-Moser model and their relation to integrable quantum long-range spin chains of the Haldane-Shastry-Inozemtsev type. First, we construct the R -matrix-valued Lax pairs for the third flow of the classical Calogero-Moser model. Then we notice that the scalar parts (in the auxiliary space) of the M -matrices ...

Added: September 18, 2018

Marshakov A., Семенякин Н. С., Journal of High Energy Physics 2019 Vol. 100 No. 10 P. 1-52

We discuss relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It is shown that glN XXZ-type spin chain on M sites is isomorphic to a cluster integrable system with N × M rectangular Newton polygon and N × M fundamental domain of a ...

Added: October 21, 2019

Marshakov A., International Journal of Modern Physics A 2013 Vol. 28 No. 3-4 P. 1340007

We propose an explicit construction for the integrable models on Poisson submanifolds of the Lie groups. The integrals of motion are computed in cluster variables via the Lax map. This generalized construction for the co-extended loop groups allows to formulate, in general terms, some new classes of integrable models. ...

Added: March 28, 2013

Feigin B. L., Jimbo M., Mukhin E., Communications in Mathematical Physics 2019 No. 367 P. 455-481

On a Fock space constructed from mn free bosons and lattice Z mn , we give a level n action of the quantum toroidal algebra E m associated to gl m , together with a level m action of the quantum toroidal algebra E n associated to gl n . We prove that the E ...

Added: December 10, 2019

A. Zabrodin, A. Zotov, Nuclear Physics B 2018 Vol. 927 P. 550-565

We discuss a self-dual form or the Backlund transformations for the continuous (in time variable) glN Ruijsenaars-Schneider model. It is based on the first order equations in N+M complex variables which include N positions of particles and M dual variables. The latter satisfy equations of motion of the glM Ruijsenaars-Schneider model. In the elliptic case ...

Added: February 15, 2018

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

М.В. Карасев, Е.М. Новикова, Теоретическая и математическая физика 2014 Т. 179 № 3 С. 406-425

The choice of physical parameters of the quantum Penning nanotrop with a perturbing in homogeneous magnetic Ioffe field is discussed, as well as the role of the resonance frequency modes. The general scheme for constructing the asymptotics of eigenstates by themethods of generalized geometric quantization is presented. The reproducing measure in theintegral representation of eigenfunctions ...

Added: March 8, 2014

Zabrodin A., Zotov A., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 20 P. 1-12

We discuss the correspondence between the Knizhnik–Zamolodchikov equations associated with GL(N) and the n-particle quantum Calogero model in the case when n is not necessarily equal to N. This can be viewed as a natural 'quantization' of the quantum-classical correspondence between quantum Gaudin and classical Calogero models. ...

Added: June 7, 2017

Marshakov A., Journal of Geometry and Physics 2012 Vol. 003 P. 16-36

We discuss the Poisson structures on Lie groups and propose an explicit construction of the integrable models on their appropriate Poisson submanifolds. The integrals of motion for the SL(N)-series are computed in cluster variables via the Lax map. This construction, when generalised to the co-extended loop groups, gives rise not only to alternative descriptions of relativistic Toda systems, but allows ...

Added: February 11, 2013

Duval C., Shevchishin V., Valent G., Journal of Geometry and Physics 2015 Vol. 87 P. 461-481

We obtain, in local coordinates, the explicit form of the two-dimensional, superintegrable
systems of Matveev and Shevchishin involving linear and cubic integrals. This enables us
to determine for which values of the parameters these systems are indeed globally defined
on S^2. ...

Added: March 23, 2015

Buryak A., Dubrovin B., Guere J. et al., International Mathematics Research Notices 2020 Vol. 2020 No. 24 P. 10381-10446

In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus 1 quantum correction and, as an application, compute completely the quantization ...

Added: April 21, 2020

Povolotsky A. M., Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 46 P. 465205

The conditions of the integrability of general zero range chipping models with factorized steady states, which were proposed in Evans et al (2004 J. Phys. A: Math. Gen. 37 L275), are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle ...

Added: November 14, 2013

Marshall I., International Mathematics Research Notices 2015 Vol. 18 P. 8925-8958

A Poisson structure is defined on the space {\mathcal {W}} of twisted polygons in {\mathbb {R}}^{\nu }. Poisson reductions with respect to two Poisson group actions on {\mathcal {W}} are described. The \nu =2 and \nu =3 cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice ...

Added: November 28, 2014

Marshakov A., Миронов А. Д., Морозов А. Ю., Journal of Geometry and Physics 2011 Vol. 61 P. 1203-1222

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block ...

Added: February 28, 2013

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2019 No. 074003 P. 1-22

We establish the exact laws of large numbers for two time additive quantities in the raise and peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the ...

Added: October 8, 2019

Васильев М., Zabrodin A., Zotov A., Nuclear Physics B - Proceedings Supplements 2020 Vol. 952 No. 114931 P. 1-20

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all ...

Added: August 20, 2020