scholarly journals Application of a Rank-One Perturbation to Pendulum Systems

2020 ◽  
Vol 12 (5) ◽  
pp. 47
Author(s):  
Traor´e. G. Y. Arouna ◽  
M. Dosso ◽  
J.-C. Koua Brou
Keyword(s):  
Perturbation Theory ◽  
Strong Stability ◽  
Double Pendulum ◽  
Simple Pendulum ◽  
Rank One Perturbation ◽  
Stability And Instability ◽  
Rank One

From a perturbation theory proposed by Mehl, et al., a study of the rank-one perturbation of the problems governed by pendulum systems is presented. Thus, a study of motion of the simple pendulum, double and triple pendulums with oscillating support, not coupled as coupled by a spring and double pendulum with fixed support is proposed. Finally (strong) stability and instability zones are calculated for each studied system.

scholarly journals Reconstruction of Differential Operators with Frozen Argument

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 24
Author(s):  
Oles Dobosevych ◽  
Rostyslav Hryniv
Keyword(s):  
Perturbation Theory ◽  
Spectral Properties ◽  
General Framework ◽  
Differential Operators ◽  
Inverse Spectral Problem ◽  
Complete Characterization ◽  
Rank One Perturbation ◽  
Clear Explanation ◽  
Rank One

We study spectral properties of a wide class of differential operators with frozen arguments by putting them into a general framework of rank-one perturbation theory. In particular, we give a complete characterization of possible eigenvalues for these operators and solve the inverse spectral problem of reconstructing the perturbation from the resulting spectrum. This approach provides a unified treatment of several recent studies and gives a clear explanation and interpretation of the obtained results.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Analysis of the Strong and Weak Monotonic External Stability of the Resonance in a Perturbed Dynamical System

2021 ◽  
Vol 16 ◽  
pp. 180-191
Author(s):  
Vladislav V. Lyubimov
Keyword(s):  
Dynamical System ◽  
Sufficient Conditions ◽  
Strong Stability ◽  
New Classification ◽  
External Stability ◽  
Fast Phase ◽  
Stability And Instability ◽  
Long Time ◽  
Independent Variable

A perturbed dynamical system involving two ordinary differential equations is under review. Whereupon, the differential equation for determining the fast phase contains the ratio of the two frequencies. When these frequencies coincide for a long time, a resonance is implemented in this system. The aim of this paper is to obtain the conditions of monotonic external stability and instability of this resonance. The sufficient conditions for the external stability and instability of the resonance defined in this paper assume that the signs of the analyzed derivatives remain unchanged in the non-resonant section of the change in the independent variable. This paper gives a new classification of the phenomenon of external stability of resonance, which includes weak, linear, and strong stability. It should be noted that the conditions of monotonic external stability and instability of the resonance presented in this paper can be used in various scientific and technological problems, in which resonances are observed. Particularly, the concluding part of the paper considers the application of the results obtained within the framework of the problem of the perturbed motion of a rigid body relative to a fixed point.

SINGULARLY FINITE RANK NONSYMMETRIC PERTURBATIONS ${\mathcal H}_{-2}$-CLASS OF A SELF-ADJOINT OPERATOR

2021 ◽  
Vol 9 (1) ◽  
pp. 140-151
Author(s):  
O. Dyuzhenkova ◽  
M. Dudkin
Keyword(s):  
Finite Rank ◽  
Scalar Product ◽  
Separable Hilbert Space ◽  
Point Spectrum ◽  
Adjoint Operator ◽  
Formal Expression ◽  
Rank One Perturbation ◽  
Rank One ◽  
Negative Space ◽  
First Time

The singular nonsymmetric rank one perturbation of a self-adjoint operator from classes ${\mathcal H}_{-1}$ and ${\mathcal H}_{-2}$ was considered for the first time in works by Dudkin M.E. and Vdovenko T.I. \cite{k8,k9}. In the mentioned papers, some properties of the point spectrum are described, which occur during such perturbations. This paper proposes generalizations of the results presented in \cite{k8,k9} and \cite{k2} in the case of nonsymmetric class ${\mathcal H}_{-2}$ perturbations of finite rank. That is, the formal expression of the following is considered \begin{equation*} \tilde A=A+\sum \limits_{j=1}^{n}\alpha_j\langle\cdot,\omega_j\rangle\delta_j, \end{equation*} where $A$ is an unperturbed self-adjoint operator on a separable Hilbert space ${\mathcal H}$, $\alpha_j\in{\mathbb C}$, $\omega_j$, $\delta_j$, $j=1,2, ..., n<\infty$ are vectors from the negative space ${\mathcal H}_{-2}$ constructed by the operator $A$, $\langle\cdot,\cdot\rangle$ is the dual scalar product between positive and negative spaces.

Quantum instability in the kicked rotator with rank-one perturbation

1990 ◽  
Vol 151 (6-7) ◽  
pp. 289-294 ◽  
Author(s):  
B. Milek ◽  
P. Seba
Keyword(s):  
Rank One Perturbation ◽  
Rank One
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