scholarly journals QUANTUM LIE SYSTEMS AND INTEGRABILITY CONDITIONS

2009 ◽  
Vol 06 (08) ◽  
pp. 1235-1252 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
JAVIER DE LUCAS
Keyword(s):  
Quantum Mechanics ◽  
Differential Equations ◽  
Geometric Theory ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Integrable Quantum ◽  
Systems Of Differential Equations ◽  
Integrability Conditions ◽  
Mechanical Models ◽  
Lie Systems

The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analyzed from a geometric perspective. In this paper we use both developments to obtain a geometric theory of integrability in Quantum Mechanics and we use it to provide a series of non-trivial integrable quantum mechanical models and to recover some known results from our unifying point of view.

scholarly journals QUANTUM SINGULARITIES IN STATIC SPACETIMES

2011 ◽  
Vol 20 (05) ◽  
pp. 729-743 ◽  
Author(s):  
JOÃO PAULO M. PITELLI ◽  
PATRICIO S. LETELIER
Keyword(s):  
Quantum Mechanics ◽  
Wave Operator ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Mathematical Framework ◽  
Physical Content ◽  
Dirac Equations ◽  
Quantum Particles ◽  
Klein Gordon ◽  
Quantum Singularities

We review the mathematical framework necessary to understand the physical content of quantum singularities in static spacetimes. We present many examples of classical singular spacetimes and study their singularities by using wave packets satisfying Klein–Gordon and Dirac equations. We show that in many cases the classical singularities are excluded when tested by quantum particles but unfortunately there are other cases where the singularities remain from the quantum mechanical point of view. When it is possible we also find, for spacetimes where quantum mechanics does not exclude the singularities, the boundary conditions necessary to turn the spatial portion of the wave operator to be self-adjoint and emphasize their importance to the interpretation of quantum singularities.

scholarly journals MASS DEFORMATIONS OF SUPER YANG–MILLS THEORIES IN D = 2 + 1, AND SUPER-MEMBRANES: A NOTE

2009 ◽  
Vol 24 (03) ◽  
pp. 193-211 ◽  
Author(s):  
ABHISHEK AGARWAL
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theories ◽  
Mass Term ◽  
Quantum Mechanical ◽  
Explicit Formulas ◽  
Yang Mills ◽  
Mechanical Models ◽  
Mass Terms ◽  
Matter Fields ◽  
Matrix Quantum

Mass deformations of supersymmetric Yang–Mills theories in three spacetime dimensions are considered. The gluons of the theories are made massive by the inclusion of a nonlocal gauge and Poincaré invariant mass term due to Alexanian and Nair, while the matter fields are given standard Gaussian mass-terms. It is shown that the dimensional reduction of such mass-deformed gauge theories defined on R3 or R × T2 produces matrix quantum mechanics with massive spectra. In particular, all known massive matrix quantum mechanical models obtained by the deformations of dimensional reductions of minimal super Yang–Mills theories in diverse dimensions are shown also to arise from the dimensional reductions of appropriate massive Yang–Mills theories in three spacetime dimensions. Explicit formulas for the gauge theory actions are provided.

scholarly journals Nonlinear singularly perturbed systems of differential equations: A survey

1995 ◽  
Vol 1 (4) ◽  
pp. 275-301
Author(s):  
Angela Slavova
Keyword(s):  
Differential Equations ◽  
Singular Perturbations ◽  
Point Of View ◽  
Singularly Perturbed ◽  
Theory Theory ◽  
Singularly Perturbed Systems ◽  
Constructive Methods ◽  
Systems Of Differential Equations ◽  
Perturbed Systems

In this paper a survey of the most effective methods in singular perturbations is presented. Many applied problems can be modeled by nonlinear singularly perturbed systems, as, for example, problems in kinetics, biochemistry, semiconductors theory, theory of electrical chains, economics, and so on. In this survey we consider averaging and constructive methods that are very useful from the point of view of their numerical and computer realizations.

scholarly journals APPLICATIONS OF LIE SYSTEMS IN DISSIPATIVE MILNE–PINNEY EQUATIONS

2009 ◽  
Vol 06 (04) ◽  
pp. 683-699 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
JAVIER DE LUCAS
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Geometric Approach ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Particular Solutions ◽  
Systems Of Differential Equations ◽  
Lie Systems ◽  
Linear Differential

We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express the general solution of a dissipative Milne–Pinney equation in terms of particular solutions of a system of second-order linear differential equations and a set of constants.

Locality or Non-Locality in Quantum Mechanics: Hidden Variables without "Spooky Action-at-a-Distance"

2001 ◽  
Vol 56 (1-2) ◽  
pp. 5-15
Author(s):  
Yakir Aharonov ◽  
Alonso Botero ◽  
Marian Scully
Keyword(s):  
Quantum Mechanics ◽  
Hidden Variables ◽  
Point Of View ◽  
Quantum Mechanical ◽  
The Third ◽  
Pacs 03.65 ◽  
Action At A Distance ◽  
Non Locality

Abstract The folklore notion of the "Non-Locality of Quantum Mechanics" is examined from the point of view of hidden-variables theories according to Belinfante's classification in his Survey of Hidden Variables Theories. It is here shown that in the case of EPR, there exist hidden variables theories that successfully reproduce quantum-mechanical predictions, but which are explicitly local. Since such theories do not fall into Belinfante's classification, we propose an expanded classification which includes similar theories, which we term as theories of the "third" kind. Causal implications of such theories are explored. -Pacs: 03.65.Bz

Stability of Linear Systems With Multitask Right-Hand Member

2018 ◽  
pp. 74-112 ◽  
Author(s):  
G. V. Alferov ◽  
G. G. Ivanov ◽  
P. A. Efimova ◽  
A. S. Sharlay
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Computational Models ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Point Of View ◽  
Control Laws ◽  
Systems Of Differential Equations ◽  
The Stability

To study the dynamics of mechanical systems and to define the construction parameters and control laws, it is necessary to have computational models accurately describing properties of real mechanisms. From a mathematical point of view, the computational models of mechanical systems are actually the systems of differential equations. These models can contain equations that also describe non-mechanical phenomena. In this chapter, the problems of stability and asymptotic stability conditions for the motion of mechanical systems with holonomic and non-holonomic constraints are under consideration. Stability analysis for the systems of differential equations is given in term of the second Lyapunov's method. With the use of the set-theoretic approach, the necessary and sufficient conditions for stability and asymptotic stability of zero solution of the considered system are formulated. The proposed approaches can be used to study the stability of the motion for robot manipulators, transport, space, and socio-economic systems.

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