Perturbation theory for the eigenvalues of factorised symmetric matrices

AbstractWe obtain eigenvalue perturbation results for a factorised Hermitian matrix H=GJG∗ where J2=I and G has full row rank and is perturbed into G+δG, where δG is small with respect to G. This complements the earlier results on the easier case of G with full column rank. Applied to square factors G our results help to identify the so-called quasidefinite matrices as a natural class on which the relative perturbation theory for the eigensolution can be formulated in a way completely analogous to the one already known for positive definite matrices

On some properties of the Lyapunov equation for damped systems

We consider a damped linear vibrational system whose dampers depend linearly on the viscosity parameter v. We show that the trace of the corresponding Lyapunov solution can be represented as a rational function of v whose poles are the eigenvalues of a certain skew symmetric matrix. This makes it possible to derive an asymptotic expansion of the solution in the neighborhood of zero (small damping)

Block diagonalization of nearly diagonal matrices

In this paper we study the effect of block diagonalization of a nearly diagonal matrix by iterating the related Riccati equations. We show that the iteration is fast, if a matrix is diagonally dominant or scaled diagonally dominant and the block partition follows an appropriately defined spectral gap. We also show that both kinds of diagonal dominance are not destroyed after the block diagonalization

Block diagonalization of nearly diagonal matrices

In this paper we study the effect of block diagonalization of a nearly diagonal matrix by iterating the related Riccati equations. We show that the iteration is fast, if a matrix is diagonally dominant or scaled diagonally dominant and the block partition follows an appropriately defined spectral gap. We also show that both kinds of diagonal dominance are not destroyed after the block diagonalization

Spectral gap of segments of periodic waveguides

We consider a periodic strip in the plane and the associated quantum waveguide with Dirichlet boundary conditions. We analyse finite segments of the waveguide consisting of $L$ periodicity cells, equipped with periodic boundary conditions at the ``new'' boundaries. Our main result is that the distance between the first and second eigenvalue of such a finite segment behaves like $L^{-2}$.Comment: 3 page

Novel Modifications of Parallel Jacobi Algorithms

We describe two main classes of one-sided trigonometric and hyperbolic Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian matrices. These types of algorithms exhibit significant advantages over many other eigenvalue algorithms. If the matrices permit, both types of algorithms compute the eigenvalues and eigenvectors with high relative accuracy. We present novel parallelization techniques for both trigonometric and hyperbolic classes of algorithms, as well as some new ideas on how pivoting in each cycle of the algorithm can improve the speed of the parallel one-sided algorithms. These parallelization approaches are applicable to both distributed-memory and shared-memory machines. The numerical testing performed indicates that the hyperbolic algorithms may be superior to the trigonometric ones, although, in theory, the latter seem more natural.Comment: Accepted for publication in Numerical Algorithm

A GPU-based hyperbolic SVD algorithm

A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm, using a massively parallel graphics processing unit (GPU), is developed. The algorithm also serves as the final stage of solving a symmetric indefinite eigenvalue problem. Numerical testing demonstrates the gains in speed and accuracy over sequential and MPI-parallelized variants of similar Jacobi-type HSVD algorithms. Finally, possibilities of hybrid CPU--GPU parallelism are discussed.Comment: Accepted for publication in BIT Numerical Mathematic

Block-Diagonalization of Operators with Gaps, with Applications to Dirac Operators

We present new results on the block-diagonalization of Dirac operators on three-dimensional Euclidean space with unbounded potentials. Classes of admissible potentials include electromagnetic potentials with strong Coulomb singularities and more general matrix-valued potentials, even non-self-adjoint ones. For the Coulomb potential, we achieve an exact diagonalization up to nuclear charge Z=124 and prove the convergence of the Douglas-Kroll-He\ss\ approximation up to Z=62, thus improving the upper bounds Z=93 and Z=51, respectively, by H.\ Siedentop and E.\ Stockmeyer considerably. These results follow from abstract theorems on perturbations of spectral subspaces of operators with gaps, which are based on a method of H.\ Langer and C.\ Tretter and are also of independent interest

Conditional Wegner Estimate for the Standard Random Breather Potential

We prove a conditional Wegner estimate for Schr\"odinger operators with random potentials of breather type. More precisely, we reduce the proof of the Wegner estimate to a scale free unique continuation principle. The relevance of such unique continuation principles has been emphasized in previous papers, in particular in recent years. We consider the standard breather model, meaning that the single site potential is the characteristic function of a ball or a cube. While our methods work for a substantially larger class of random breather potentials, we discuss in this particular paper only the standard model in order to make the arguments and ideas easily accessible

Scattering theory for Klein-Gordon equations with non-positive energy

We study the scattering theory for charged Klein-Gordon equations: \{{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x) \epsilon^{2}(x, D_{x})\phi(t,x)=0,[2mm] \phi(0, x)= f_{0}, [2mm] \i^{-1} \p_{t}\phi(0, x)= f_{1}, {array}. where: \epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}(\p_{x_{j}} \i b_{j}(x))A^{jk}(x)(\p_{x_{k}} \i b_{k}(x))+ m^{2}(x), describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+ \bar{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x) \d x. We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein-Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case