Factorization of second-order matrix differential operators and a matrix Darboux transformation

2003 ◽  
Vol 81 (8) ◽  
pp. 977-988
Author(s):  
Paul Bracken
Keyword(s):  
General Solution ◽  
Differential Operators ◽  
First Order ◽  
A Value ◽  
Order Matrix ◽  
Matrix Differential Operators ◽  
Potential Matrix ◽  
Matrix Differential ◽  
Matrix Operators ◽  
First Order Differential Equations

It is shown that a class of matrix Schrödinger operators can be factored into a product of two first-order matrix operators. The equations that relate the elements in these first-order operators to the elements of the potential matrix of the Schrödinger operator are obtained. They are found to be coupled first-order differential equations in the variables of the first-order matrix operators. Finally, an example of a factorization of a matrix operator is obtained, and a general solution associated to a value of the spectral parameter is given. PACS Nos.: 02.30.Mq, 12.39.Pn

scholarly journals gln+1 ALGEBRA OF MATRIX DIFFERENTIAL OPERATORS AND MATRIX QUASI-EXACTLY-SOLVABLE PROBLEMS

2013 ◽  
Vol 53 (5) ◽  
pp. 462-469
Author(s):  
Yuri F. Smirnov ◽  
Alexander Turbiner
Keyword(s):  
Differential Operators ◽  
First Order ◽  
Matrix Generalization ◽  
Matrix Coefficients ◽  
Matrix Differential Operators ◽  
Exactly Solvable ◽  
Matrix Differential ◽  
Solvable Problems

The generators of the algebra <em>gl<sub>n+1</sub></em> in the form of differential operators of the first order acting on <strong>R</strong><sup><em>n</em></sup> with matrix coefficients are explicitly written. The algebraic Hamiltonians for matrix generalization of 3−body Calogero and Sutherland models are presented.

SIMILARITY OF OPERATORS ON TENSOR PRODUCTS OF SPACES AND MATRIX DIFFERENTIAL OPERATORS

2018 ◽  
Vol 106 (1) ◽  
pp. 19-30
Author(s):  
MICHAEL GIL’
Keyword(s):  
Differential Operators ◽  
Normal Operator ◽  
Variable Coefficients ◽  
Closed Convex Hull ◽  
Constant Matrix ◽  
Norm Estimates ◽  
Matrix Coefficients ◽  
Matrix Differential Operators ◽  
Constant Coefficients ◽  
Matrix Differential

Let ${\mathcal{H}}=\mathbb{C}^{n}\otimes {\mathcal{E}}$ be the tensor product of a Euclidean space $\mathbb{C}^{n}$ and a separable Hilbert space ${\mathcal{E}}$. Our main object is the operator $G=I_{n}\otimes S+A\otimes I_{{\mathcal{E}}}$, where $S$ is a normal operator in ${\mathcal{E}}$, $A$ is an $n\times n$ matrix, and $I_{n},I_{{\mathcal{E}}}$ are the unit operators in $\mathbb{C}^{n}$ and ${\mathcal{E}}$, respectively. Numerous differential operators with constant matrix coefficients are examples of operator $G$. In the present paper we show that $G$ is similar to an operator $M=I_{n}\otimes S+\hat{D}\times I_{{\mathcal{E}}}$ where $\hat{D}$ is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of $G$, nonregular on the closed convex hull $\operatorname{co}(G)$ of the spectrum of $G$. The functions $G^{-\unicode[STIX]{x1D6FC}}\;(\unicode[STIX]{x1D6FC}>0)$ and $(\ln G)^{-1}$ are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of $G$ regular on $\operatorname{co}(G)$. Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.

scholarly journals Vibration analysis of rotating rings with complex support stiffnesses

2018 ◽  
Vol 237 ◽  
pp. 01010
Author(s):  
Fuchun Yang ◽  
Yue Zhang ◽  
Hailong Li
Keyword(s):  
Differential Operators ◽  
Natural Frequencies ◽  
Frequency Separation ◽  
Vibration Modes ◽  
Governing Equations ◽  
Nodal Diameter ◽  
Vibration Properties ◽  
Matrix Differential Operators ◽  
Matrix Differential ◽  
Space Matrix

Vibration characteristics of rotating rings with complex support stiffnesses are studied. The complex stiffnesses of the rotating ring include discrete stiffnesses and partially distributed stiffnesses. The governing equations are established by Hamilton’s principle. The governing equations are cast in matrix differential operators and discretized using Galerkin’s method. The eigenvalue problem is dealt with state space matrix and the natural frequencies and vibration modes are obtained. The properties of natural frequencies and vibration modes of rotating rings are studied. The results illustrate that frequency separation and frequency veering happen with the increase of rotation speed. The vibration modes are not dominated by only one nodal diameter while dominated by several nodal diameters because the discrete and partially distributed stiffnesses disrupt the axisymmetry of rotating rings. The influences of several parameters to vibration properties of rotating rings are also investigated.

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