scholarly journals THE QUANTUM H3 INTEGRABLE SYSTEM

2010 ◽  
Vol 25 (30) ◽  
pp. 5567-5594 ◽  
Author(s):  
MARCOS A. G. GARCÍA ◽  
ALEXANDER V. TURBINER
Keyword(s):  
Integrable System ◽  
Differential Operators ◽  
Three Dimensional ◽  
Integrable Model ◽  
Dimensional System ◽  
Property A ◽  
Algebraic Form ◽  
Infinite Dimensional ◽  
Polynomial Coefficients ◽  
Finite Dimensional

The quantum H3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H3. It is shown that the gauge-rotated H3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H3-invariants "polynomially"-isospectral to the quantum H3 model is defined.

scholarly journals SOLVABILITY OF THE F4 INTEGRABLE SYSTEM

2001 ◽  
Vol 16 (29) ◽  
pp. 4769-4801 ◽  
Author(s):  
KONSTANTIN G. BORESKOV ◽  
JUAN CARLOS LOPEZ VIEYRA ◽  
ALEXANDER V. TURBINER
Keyword(s):  
Weyl Group ◽  
Differential Operators ◽  
Coupling Constants ◽  
Dimensional Representation ◽  
Algebraic Form ◽  
Finite Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Exactly Solvable ◽  
New Variables

It is shown that the F4 rational and trigonometric integrable systems are exactly-solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of F4 root system and can be obtained by averaging over an orbit of the Weyl group. An alternative way of finding these variables exploiting a property of duality of the F4 model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational F4 model depending on two continuous and one discrete parameters is found.

scholarly journals THE QUANTUM H4 INTEGRABLE SYSTEM

2011 ◽  
Vol 26 (06) ◽  
pp. 433-447 ◽  
Author(s):  
MARCOS A. G. GARCÍA ◽  
ALEXANDER V. TURBINER
Keyword(s):  
Integrable System ◽  
Ground State ◽  
Invariant Subspaces ◽  
Harmonic Oscillators ◽  
State Function ◽  
Factorization Property ◽  
Algebraic Form ◽  
Rational Potential ◽  
Polynomial Coefficients ◽  
Finite Dimensional

The quantum H4 integrable system is a 4D system with rational potential related to the non-crystallographic root system H4 with 600-cell symmetry. It is shown that the gauge-rotated H4 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H4, is in algebraic form: it has polynomial coefficients in front of the derivatives. Any eigenfunction is a polynomial multiplied by ground-state function (factorization property). Spectra correspond to one of the anisotropic harmonic oscillators. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector α = (1, 5, 8, 12).

The Bargmann Constraint and Integrable System for a Second-Order Spectral Problem

2015 ◽  
Vol 70 (11) ◽  
pp. 913-917
Author(s):  
Wei Liu ◽  
Yafeng Liu ◽  
Shujuan Yuan
Keyword(s):  
Coordinate System ◽  
Spectral Problem ◽  
Integrable System ◽  
Evolution Equations ◽  
Lagrange Equations ◽  
Lax Pairs ◽  
Infinite Dimensional ◽  
Hamilton System ◽  
Finite Dimensional ◽  
Dimensional Soliton

AbstractIn this article, the Bargmann system related to the spectral problem (∂2+q∂+∂q+r)φ=λφ+λφx is discussed. By the Euler–Lagrange equations and the Legendre transformations, a suitable Jacobi–Ostrogradsky coordinate system is obtained. So the Lax pairs of the aforementioned spectral problem are nonlinearised. A new kind of finite-dimensional Hamilton system is generated. Moreover, the involutive solutions of the evolution equations for the infinite-dimensional soliton system are derived.

scholarly journals The initial value problem in Lagrangian drift kinetic theory

2016 ◽  
Vol 82 (3) ◽  
Author(s):  
J. W. Burby
Keyword(s):  
Phase Space ◽  
Kinetic Theory ◽  
Initial Value Problem ◽  
High Order ◽  
Dimensional System ◽  
Initial Value ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Basic Philosophy ◽  
System Phase

Existing high-order variational drift kinetic theories contain unphysical rapidly varying modes that are not seen at low orders. These unphysical modes, which may be rapidly oscillating, damped or growing, are ushered in by a failure of conventional high-order drift kinetic theory to preserve the structure of its parent model’s initial value problem. In short, the (infinite dimensional) system phase space is unphysically enlarged in conventional high-order variational drift kinetic theory. I present an alternative, ‘renormalized’ variational approach to drift kinetic theory that manifestly respects the parent model’s initial value problem. The basic philosophy underlying this alternate approach is that high-order drift kinetic theory ought to be derived by truncating the all-orders system phase-space Lagrangian instead of the usual ‘field$+$particle’ Lagrangian. For the sake of clarity, this story is told first through the lens of a finite-dimensional toy model of high-order variational drift kinetics; the analogous full-on drift kinetic story is discussed subsequently. The renormalized drift kinetic system, while variational and just as formally accurate as conventional formulations, does not support the troublesome rapidly varying modes.

Center Manifold of Fractional Dynamical System

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Dynamical System ◽  
Center Manifold ◽  
High Dimensional ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Reduction Methods ◽  
Fractional Dynamical System ◽  
Lower Dimensional

Dimension reduction of dynamical system is a significant issue for technical applications, as regards both finite dimensional system and infinite dimensional systems emerging from either science or engineering. Center manifold method is one of the main reduction methods for ordinary differential systems (ODSs). Does there exists a similar method for fractional ODSs (FODSs)? In other words, does there exists a method for reducing the high-dimensional FODS into a lower-dimensional FODS? In this study, we establish a local fractional center manifold for a finite dimensional FODS. Several examples are given to illustrate the theoretical analysis.

scholarly journals Adapting the range of validity for the Carleman linearization

2016 ◽  
Vol 14 ◽  
pp. 51-54 ◽  
Author(s):  
Harry Weber ◽  
Wolfgang Mathis
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Nonlinear Differential Equation ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Liouville Problem ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Differential

Abstract. In this contribution, the limitations of the Carleman linearization approach are presented and discussed. The Carleman linearization transforms an ordinary nonlinear differential equation into an infinite system of linear differential equations. In order to transform the nonlinear differential equation, orthogonal polynomials which represent solutions of a Sturm–Liouville problem are used as basis. The determination of the time derivate of this basis yields an infinite dimensional linear system that depends on the considered nonlinear differential equation. The infinite linear system has the same properties as the nonlinear differential equation such as limit cycles or chaotic behavior. In general, the infinite dimensional linear system cannot be solved. Therefore, the infinite dimensional linear system has to be approximated by a finite dimensional linear system. Due to limitation of dimension the solution of the finite dimensional linear system does not represent the global behavior of the nonlinear differential equation. In fact, the accuracy of the approximation depends on the considered nonlinear system and the initial value. The idea of this contribution is to adapt the range of validity for the Carleman linearization in order to increase the accuracy of the approximation for different ranges of initial values. Instead of truncating the infinite dimensional system after a certain order a Taylor series approach is used to approximate the behavior of the nonlinear differential equation about different equilibrium points. Thus, the adapted finite linear system describes the local behavior of the solution of the nonlinear differential equation.

Active Control of Flexible Structures Subject to Distributed and Seismic Disturbances

1993 ◽  
Vol 115 (4) ◽  
pp. 649-657 ◽  
Author(s):  
Akira Ohsumi ◽  
Yuichi Sawada
Keyword(s):  
Active Control ◽  
Flexible Structures ◽  
Distributed Parameter ◽  
Dimensional System ◽  
Random Disturbance ◽  
Modal Expansion ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Real Earthquake ◽  
The Mathematical Model

The purpose of this paper is to present a method of active control for suppressing the vibration of a mechanically flexible cantilever beam which is subject to a distributed random disturbance and also a seismic input at the clamped end. First, the mathematical model of the flexible structure is established by a stochastic partial differential equation which describes the Euler-Bernoulli type distributed parameter system with internal viscous damping and subject to the seismic and distributed random inputs. Second, the distributed parameter model, which is considered as an infinite-dimensional system, is reduced to a finite-dimensional one by using the modal expansion, and split into the controlled part and the uncontrolled (residual) one. The principal approach is to regard the observation spillover due to uncontrolled part as a colored observation noise and construct an estimator, and then we construct the optimal control system. Finally, simulation studies are presented by using a real earthquake accelerogram data.

scholarly journals Entropic Uncertainty Relations via Direct-Sum Majorization Relation for Generalized Measurements

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 270 ◽  
Author(s):  
Kyunghyun Baek ◽  
Hyunchul Nha ◽  
Wonmin Son
Keyword(s):  
Three Dimensional ◽  
Uncertainty Relations ◽  
Dimensional System ◽  
Direct Sum ◽  
Entropic Uncertainty Relations ◽  
Finite Dimensional ◽  
Uncertainty Bound ◽  
Three Dimensional System ◽  
Unsharp Observables ◽  
Positive Operator Valued Measure

We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches (Friendland, S.; Gheorghiu, V.; Gour, G. Phys. Rev. Lett. 2013, 111, 230401; Rastegin, A.E.; Życzkowski, K. J. Phys. A, 2016, 49, 355301), particularly by extending the direct-sum majorization relation first introduced in (Rudnicki, Ł.; Puchała, Z.; Życzkowski, K. Phys. Rev. A 2014, 89, 052115). We illustrate the usefulness of our uncertainty relations by considering a pair of qubit observables in a two-dimensional system and randomly chosen unsharp observables in a three-dimensional system. We also demonstrate that our bound tends to be stronger than the generalized Maassen–Uffink bound with an increase in the unsharpness effect. Furthermore, we extend our approach to the case of multiple POVM measurements, thus making it possible to establish entropic uncertainty relations involving more than two observables.

scholarly journals SOLVABILITY OF THE G2 INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (22) ◽  
pp. 3885-3903 ◽  
Author(s):  
MARCOS ROSENBAUM ◽  
ALEXANDER TURBINER ◽  
ANTONIO CAPELLA
Keyword(s):  
Integrable System ◽  
Configuration Space ◽  
Symmetric Functions ◽  
Differential Operators ◽  
Coupling Constants ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
Exactly Solvable ◽  
Three Body

It is shown that the three-body trigonometric G2 integrable system is exactly solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the Hamiltonian can be expressed as a quadratic polynomial in the generators of some Lie algebra of differential operators in a finite-dimensional representation. Four infinite families of eigenstates, represented by polynomials, and the corresponding eigenvalues are described explicitly.

scholarly journals Global Stabilization of Nonlinear Finite Dimensional System with Dynamic Controller Governed by 1 − d Heat Equation with Neumann Interconnection

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 227
Author(s):  
Mohsen Dlala ◽  
Abdallah Benabdallah
Keyword(s):  
State Feedback ◽  
Linear Part ◽  
Coupled System ◽  
High Gain ◽  
Dimensional System ◽  
Nonlinear Part ◽  
Infinite Dimensional ◽  
Dynamic Controller ◽  
Finite Dimensional ◽  
Linear Growth Condition

This paper deals with the stabilization of a class of uncertain nonlinear ordinary differential equations (ODEs) with a dynamic controller governed by a linear 1−d heat partial differential equation (PDE). The control operates at one boundary of the domain of the heat controller, while at the other end of the boundary, a Neumann term is injected into the ODE plant. We achieve the desired global exponential stabilization goal by using a recent infinite-dimensional backstepping design for coupled PDE-ODE systems combined with a high-gain state feedback and domination approach. The stabilization result of the coupled system is established under two main restrictions: the first restriction concerns the particular classical form of our ODE, which contains, in addition to a controllable linear part, a second uncertain nonlinear part verifying a lower triangular linear growth condition. The second restriction concerns the length of the domain of the PDE which is restricted.

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