288 research outputs found
On the convergence of second order spectra and multiplicity
Let A be a self-adjoint operator acting on a Hilbert space. The notion of
second order spectrum of A relative to a given finite-dimensional subspace L
has been studied recently in connection with the phenomenon of spectral
pollution in the Galerkin method. We establish in this paper a general
framework allowing us to determine how the second order spectrum encodes
precise information about the multiplicity of the isolated eigenvalues of A.
Our theoretical findings are supported by various numerical experiments on the
computation of inclusions for eigenvalues of benchmark differential operators
via finite element bases.Comment: 22 pages, 2 figures, 4 tables, research paper
Analyticity and uniform stability in the inverse spectral problem for Dirac operators
We prove that the inverse spectral mapping reconstructing the square
integrable potentials on [0,1] of Dirac operators in the AKNS form from their
spectral data (two spectra or one spectrum and the corresponding norming
constants) is analytic and uniformly stable in a certain sense.Comment: 19 page
The symmetric-Toeplitz linear system problem in parallel
[EN] Many algorithms exist that exploit the special structure of
Toeplitz matrices for solving linear systems. Nevertheless, these algorithms
are difficult to parallelize due to its lower computational cost and
the great dependency of the operations involved that produces a great
communication cost. The foundation of the parallel algorithm presented
in this paper consists of transforming the Toeplitz matrix into a another
structured matrix called Cauchy¿like. The particular properties of
Cauchy¿like matrices are exploited in order to obtain two levels of parallelism
that makes possible to highly reduce the execution time. The
experimental results were obtained in a cluster of PC¿s.Supported by Spanish MCYT and FEDER under Grant TIC 2003-08238-C02-02Alonso-Jordá, P.; Vidal Maciá, AM. (2005). The symmetric-Toeplitz linear system problem in parallel. Computational Science -- ICCS 2005,Pt 1, Proceedings. 3514:220-228. https://doi.org/10.1007/11428831_28S2202283514Sweet, D.R.: The use of linear-time systolic algorithms for the solution of toeplitz problems. k Technical Report JCU-CS-91/1, Department of Computer Science, James Cook University, Tue, 23 April 1996 15, 17, 55 GMT (1991)Evans, D.J., Oka, G.: Parallel solution of symmetric positive definite Toeplitz systems. Parallel Algorithms and Applications 12, 297–303 (1998)Gohberg, I., Koltracht, I., Averbuch, A., Shoham, B.: Timing analysis of a parallel algorithm for Toeplitz matrices on a MIMD parallel machine. Parallel Computing 17, 563–577 (1991)Gallivan, K., Thirumalai, S., Dooren, P.V.: On solving block toeplitz systems using a block schur algorithm. In: Proceedings of the 23rd International Conference on Parallel Processing, Boca Raton, FL, USA, vol. 3, pp. 274–281. CRC Press, Boca Raton (1994)Thirumalai, S.: High performance algorithms to solve Toeplitz and block Toeplitz systems. Ph.d. th., Grad. College of the U. of Illinois at Urbana–Champaign (1996)Alonso, P., Badía, J.M., Vidal, A.M.: Parallel algorithms for the solution of toeplitz systems of linear equations. In: Wyrzykowski, R., Dongarra, J., Paprzycki, M., Waśniewski, J. (eds.) PPAM 2004. LNCS, vol. 3019, pp. 969–976. Springer, Heidelberg (2004)Anderson, E., et al.: LAPACK Users’ Guide. SIAM, Philadelphia (1995)Blackford, L., et al.: ScaLAPACK Users’ Guide. SIAM, Philadelphia (1997)Alonso, P., Badía, J.M., González, A., Vidal, A.M.: Parallel design of multichannel inverse filters for audio reproduction. In: Parallel and Distributed Computing and Systems, IASTED, Marina del Rey, CA, USA, vol. II, pp. 719–724 (2003)Loan, C.V.: Computational Frameworks for the Fast Fourier Transform. SIAM Press, Philadelphia (1992)Heinig, G.: Inversion of generalized Cauchy matrices and other classes of structured matrices. Linear Algebra and Signal Proc., IMA, Math. Appl. 69, 95–114 (1994)Gohberg, I., Kailath, T., Olshevsky, V.: Fast Gaussian elimination with partial pivoting for matrices with displacement structure. Mathematics of Computation 64, 1557–1576 (1995)Alonso, P., Vidal, A.M.: An efficient and stable parallel solution for symmetric toeplitz linear systems. TR DSIC-II/2005, DSIC–Univ. Polit. Valencia (2005)Kailath, T., Sayed, A.H.: Displacement structure: Theory and applications. SIAM Review 37, 297–386 (1995
Supersymmetric Biorthogonal Quantum Systems
We discuss supersymmetric biorthogonal systems, with emphasis given to the
periodic solutions that occur at spectral singularities of PT symmetric models.
For these periodic solutions, the dual functions are associated polynomials
that obey inhomogeneous equations. We construct in detail some explicit
examples for the supersymmetric pairs of potentials V_{+/-}(z) = -U(z)^2 +/-
z(d/(dz))U(z) where U(z) = \sum_{k>0}u_{k}z^{k}. In particular, we consider the
cases generated by U(z) = z and z/(1-z). We also briefly consider the effects
of magnetic vector potentials on the partition functions of these systems.Comment: Changes are made to conform to the published version. In particular,
some errors are corrected on pp 12-1
A unified approach to Darboux transformations
We analyze a certain class of integral equations related to Marchenko
equations and Gel'fand-Levitan equations associated with various systems of
ordinary differential operators. When the integral operator is perturbed by a
finite-rank perturbation, we explicitly evaluate the change in the solution. We
show how this result provides a unified approach to Darboux transformations
associated with various systems of ordinary differential operators. We
illustrate our theory by deriving the Darboux transformation for the
Zakharov-Shabat system and show how the potential and wave function change when
a discrete eigenvalue is added to the spectrum.Comment: final version that will appear in Inverse Problem
Global Bifurcation of Rotating Vortex Patches
We rigorously construct continuous curves of rotating vortex patch solutions to the two-dimensional Euler equations. The curves are large in that, as the parameter tends to infinity, the minimum along the interface of the angular fluid velocity in the rotating frame becomes arbitrarily small. This is consistent with the conjectured existence [30, 38] of singular limiting patches with 90 corners at which the relative fluid velocity vanishes. For solutions close to the disk, we prove that there are “cat's-eyes”-type structures in the flow, and provide numerical evidence that these structures persist along the entire solution curves and are related to the formation of corners. We also show, for any rotating vortex patch, that the boundary is analytic as soon as it is sufficiently regular.</p
On Approximation of the Eigenvalues of Perturbed Periodic Schrodinger Operators
This paper addresses the problem of computing the eigenvalues lying in the
gaps of the essential spectrum of a periodic Schrodinger operator perturbed by
a fast decreasing potential. We use a recently developed technique, the so
called quadratic projection method, in order to achieve convergence free from
spectral pollution. We describe the theoretical foundations of the method in
detail, and illustrate its effectiveness by several examples.Comment: 17 pages, 2 tables and 2 figure
Smilansky's model of irreversible quantum graphs, I: the absolutely continuous spectrum
In the model suggested by Smilansky one studies an operator describing the
interaction between a quantum graph and a system of one-dimensional
oscillators attached at several different points in the graph. The present
paper is the first one in which the case is investigated. For the sake of
simplicity we consider K=2, but our argument is of a general character. In this
first of two papers on the problem, we describe the absolutely continuous
spectrum. Our approach is based upon scattering theory
Biorthogonal Quantum Systems
Models of PT symmetric quantum mechanics provide examples of biorthogonal
quantum systems. The latter incorporporate all the structure of PT symmetric
models, and allow for generalizations, especially in situations where the PT
construction of the dual space fails. The formalism is illustrated by a few
exact results for models of the form H=(p+\nu)^2+\sum_{k>0}\mu_{k}exp(ikx). In
some non-trivial cases, equivalent hermitian theories are obtained and shown to
be very simple: They are just free (chiral) particles. Field theory extensions
are briefly considered.Comment: 34 pages, 5 eps figures; references added and other changes made to
conform to journal versio
Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE Revisited
We derive expansions of the resolvent
Rn(x;y;t)=(Qn(x;t)Pn(y;t)-Qn(y;t)Pn(x;t))/(x-y) of the Hermite kernel Kn at the
edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the
finite n expansion of Qn(x;t) and Pn(x;t). Using these large n expansions, we
give another proof of the derivation of an Edgeworth type theorem for the
largest eigenvalue distribution function of GUEn. We conclude with a brief
discussion on the derivation of the probability distribution function of the
corresponding largest eigenvalue in the Gaussian Orthogonal Ensemble (GOEn) and
Gaussian Symplectic Ensembles (GSEn)
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