AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

Let A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

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

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.

Vibration analysis of rotating rings with complex support stiffnesses

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.

Sharp eigenvalue enclosures for the perturbed angular Kerr–Newman Dirac operator

We examine a certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients. The strategy relies on computing the second-order spectrum relative to subspaces of continuous piecewise linear functions. For smooth perturbations of the angular Kerr–Newman Dirac operator, explicit rates of convergence linked to regularity of the eigenfunctions are established. Numerical tests which validate and sharpen by several orders of magnitude the existing benchmarks are also included.