Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

On the spectrum of the dynamical matrix for a class of disordered harmonic systems

Abstract: We study some aspects of the effect of mass disorder on the spectrum of the dynamical matrix of an infinite crystal in the harmonic approximation. Under suitable conditions on the masses, it is shown that the spectrum contains an absolutely continuous part and a nonempty set of isolated point eigenvalues of finite multiplicity whose number is smaller than or equal to the number of impurity atoms if the latter is finite. These conditions are satisfied only in the limiting case of zero concentration of each species of impurity. We draw some conjectures and make remarks on the spectrum under less restrictive conditions on the masses and briefly compare them with known results for random harmonic systems.

On eigenvalue accumulation for non-self-adjoint magnetic operators

Abstract: In this work, we use regularized determinant approach to study the discrete spectrum generated by relatively compact non-self-adjoint perturbations of the magnetic Schrodinger operator $(-i abla - \textbf{\textup{A}})^{2} - b$ in dimension $3$ with constant magnetic field of strength $b>0$. The situation near the Landau levels $2bq$, $q \in \mathbb{N}$, is more interesting since they play the role of thresholds of the spectrum of the free operator. First, we obtain sharp upper bounds on the number of the complex eigenvalues near the Landau levels. Under appropriate hypothesis, we then prove the presence of an infinite number of complex eigenvalues near each Landau level $2bq$, $q \in \mathbb{N}$, and the existence of sectors free of complex eigenvalues. We also prove that the eigenvalues are localized in certain sectors adjoining the Landau levels. In particular, we provide an adequate answer to the open problem from [34] about the existence of complex eigenvalues accumulating near the Landau levels. Furthermore, we prove that the Landau levels are the only possible accumulation points of the complex eigenvalues.

Extended Eigenvalues of Direct Sum of Operators

Abstract: In this paper a connection between extended eigenvalues of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. Moreover, the structure of the set of extended eigenvalues of normal compact operators has been researched.

A practical closed form solution for systems with variable eigenvalues via bilinear system theory

Abstract: This paper treats systems in state variable formulation with non-constant, parameter controlled system matrices. The synthesis of a system with controlled eigenvalues (ECS) is given. The synthesized system is a commutative bilinear system. Its solution has a closed form and is based on the solution of just one time invariant system although as many arbitrary time functions are involved as the system has independent states. The ECS is homologous to any system with a system matrix being an arbitrary, possibly time-dependent function of a single constant system matrix. All results are deduced for multiple eigenvalues of the system matrix including single eigenvalues as a special case. They are fully analogous to the solution of time invariant systems by means of the Laplace transformation.