2,064 research outputs found
Generalization and variations of Pellet's theorem for matrix polynomials
We derive a generalized matrix version of Pellet's theorem, itself based on a
generalized Rouch\'{e} theorem for matrix-valued functions, to generate upper,
lower, and internal bounds on the eigenvalues of matrix polynomials. Variations
of the theorem are suggested to try and overcome situations where Pellet's
theorem cannot be applied.Comment: 20 page
On the eigenvalues and eigenvectors of nonsymmetric saddle point matrices preconditioned by block triangular matrices
Block lower triangular and block upper triangular matrices are popular preconditioners for nonsymmetric saddle point matrices. In this note we show that a block lower triangular preconditioner gives the same spectrum as a block upper triangular preconditioner and that the eigenvectors of the two preconditioned systems are related
Generalized Householder Transformations for the Complex Symmetric Eigenvalue Problem
We present an intuitive and scalable algorithm for the diagonalization of
complex symmetric matrices, which arise from the projection of
pseudo--Hermitian and complex scaled Hamiltonians onto a suitable basis set of
"trial" states. The algorithm diagonalizes complex and symmetric
(non--Hermitian) matrices and is easily implemented in modern computer
languages. It is based on generalized Householder transformations and relies on
iterative similarity transformations T -> T' = Q^T T Q, where Q is a complex
and orthogonal, but not unitary, matrix, i.e, Q^T equals Q^(-1) but Q^+ is
different from Q^(-1). We present numerical reference data to support the
scalability of the algorithm. We construct the generalized Householder
transformations from the notion that the conserved scalar product of
eigenstates Psi_n and Psi_m of a pseudo-Hermitian quantum mechanical
Hamiltonian can be reformulated in terms of the generalized indefinite inner
product [integral of the product Psi_n(x,t) Psi_m(x,t) over dx], where the
integrand is locally defined, and complex conjugation is avoided. A few example
calculations are described which illustrate the physical origin of the ideas
used in the construction of the algorithm.Comment: 14 pages; RevTeX; font mismatch in Eqs. (3) and (15) is eliminate
Inflation with a graceful exit in a random landscape
We develop a stochastic description of small-field inflationary histories
with a graceful exit in a random potential whose Hessian is a Gaussian random
matrix as a model of the unstructured part of the string landscape. The
dynamical evolution in such a random potential from a small-field inflation
region towards a viable late-time de Sitter (dS) minimum maps to the dynamics
of Dyson Brownian motion describing the relaxation of non-equilibrium
eigenvalue spectra in random matrix theory. We analytically compute the
relaxation probability in a saddle point approximation of the partition
function of the eigenvalue distribution of the Wigner ensemble describing the
mass matrices of the critical points. When applied to small-field inflation in
the landscape, this leads to an exponentially strong bias against small-field
ranges and an upper bound on the number of light fields
participating during inflation from the non-observation of negative spatial
curvature.Comment: Published versio
Topological objects in QCD
Topological excitations are prominent candidates for explaining
nonperturbative effects in QCD like confinement. In these lectures, I cover
both formal treatments and applications of topological objects. The typical
phenomena like BPS bounds, topology, the semiclassical approximation and chiral
fermions are introduced by virtue of kinks. Then I proceed in higher dimensions
with magnetic monopoles and instantons and special emphasis on calorons.
Analytical aspects are discussed and an overview over models based on these
objects as well as lattice results is given.Comment: 28 pages, 17 figures; Lectures given at 45th Internationale
Universitaetswochen fuer Theoretische Physik (International University School
of Theoretical Physics): Conceptual and Numerical Challenges in Femto- and
Peta-Scale Physics, Schladming, Styria, Austria, 24 Feb - 3 Mar 200
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