Spectral properties of matrices associated with some directed graphs

We consider perturbations of a large Jordan matrix, either random and small in norm or of small rank. In the case of random perturbations we obtain explicit estimates which show that as the size of the matrix increases, most of the eigenvalues of the perturbed matrix converge to a certain circle with centre at the origin. In the case of finite rank perturbations we completely determine the spectral asymptotics as the size of the matrix increases.

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“…Note that bounds similar to (16) and (18) can be proved for a wide variety of other probability laws governing the coefficients of K .…”

Section: Small Random Perturbations Of Jordan Matricesmentioning

confidence: 93%

“…. , N } with re-entry law is replaced by a general directed graph; the geometry of the graph is then a vital ingredient when determining the structure of the spectrum [16]. Our results may have also connections with the analysis of para-orthogonal polynomials on the unit circle in [17].…”

Section: Perturbations Of Boundary Condition Typementioning

confidence: 94%

Perturbations of Jordan matrices

Davies

Hager

2009

Journal of Approximation Theory

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“…For more recent literature see [20]. Some of these results were rediscovered in [5,6,11], where they were used to analyze the spectra of non-self-adjoint systems of ODEs and directed finite graphs. The asymptotic distribution of the zeros of F depends heavily on the location of the extreme points of the convex hull of the set ∪ n r=1 {σ r }.…”

Section: Zeros Of Exponential Polynomials Exponential Polynomials Are...mentioning

confidence: 99%

Non-Weyl resonance asymptotics for quantum graphs

Davies

Pushnitski

2011

APDE

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We consider the resonances of a quantum graph Ᏻ that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of Ᏻ in a disc of a large radius. We call Ᏻ a Weyl graph if the coefficient in front of this leading term coincides with the volume of the compact part of Ᏻ. We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.

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“…For more recent literature see [Moreno 1973]. Some of these results were rediscovered in [Davies 2003;Davies and Incani 2010;Incani 2009], where they were used to analyze the spectra of nonselfadjoint systems of ODEs and directed finite graphs. The asymptotic distribution of the zeros of F depends heavily on the location of the extreme points of the convex hull of the set [ n r D1 f r g. We are only interested in the case in which r are distinct real numbers.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Untitled

2011

APDE

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JEAN-MARC DELORTThis paper is devoted to the construction of periodic solutions of nonlinear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash-Moser methods. Our approach avoids this, exploiting the possibility of reducing, through paradifferential conjugation, the equation under study to an equivalent form for which periodic solutions may be constructed by a classical iteration scheme. IntroductionThis paper is devoted to the existence of families of periodic solutions of Hamiltonian nonlinear Schrödinger equations on the torus ‫ޔ‬ d . Our goal is to show that such results may be proved without using Nash-Moser methods, replacing them by a technically simpler conjugation idea.We consider equations of type.!t; x; u; N u; / C f .!t; x/;where t 2 ‫,ޒ‬ x 2 ‫ޔ‬ d , F is a smooth function, vanishing at order 3 at .u; N u/ D 0, f is a smooth function on ‫ޒ‬ ‫ޔ‬ d , 2 -periodic in time, ! a frequency parameter, a real parameter and > 0 a small number. One wants to show that for small and ! in a Cantor set whose complement has small measure, the equation has time periodic solutions.Let us recall known results for that type of problems. The first periodic solutions for nonlinear wave or Schrödinger equations were constructed in [Kuksin 1993; Wayne 1990], which deal with one space dimension, with x staying in a compact interval, and imposing on the extremities of this interval convenient boundary conditions. Later on, Craig and Wayne [1993; treated the same problem for time-periodic solutions defined on ‫ޒ‬ ‫ޓ‬ 1 . Periodic solutions of nonlinear wave equations in higher space dimensions (on ‫ޒ‬ ‫ޔ‬ d , d 2) were obtained in [Bourgain 1994]. These results concern nonlinearities which are analytic. More recently, some work has been devoted to the same problem when the nonlinearity is a smooth function: Berti and Bolle [2010] have proved in this setting existence of time-periodic solutions for the nonlinear wave equation on ‫ޒ‬ ‫ޔ‬ d . We refer also to the paper of Berti, Bolle and Procesi , where the case of equations on Zoll manifolds is treated. Very recently, Berti and Procesi [2011] have studied the same problem, for wave or Schrödinger equations, on a homogeneous space. We refer also to [Craig 2000; Kuksin 2000] for more references.The proofs of all these results rely on the use of the Nash-Moser theorem, to overcome unavoidable losses of derivatives coming from the small divisors appearing when inverting the linear part of the equation. Our goal here is to show that one may construct periodic solutions of nonlinear Schrödinger equations (for large sets of frequencies), using just a standard iterative scheme instead of the quadratic scheme of the Nash-Moser method. This approach allows one to separate on the one hand the treatment of losses of derivatives coming from small divisors, and on the other hand the question of convergence of the sequence of approximations, while in a Nash-Moser scheme, both problems have to be treated at the same time....

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Spectral properties of matrices associated with some directed graphs

Abstract: We study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves.

Cited by 4 publications

References 10 publications

Perturbations of Jordan matrices

Perturbations of Jordan matrices

Non-Weyl resonance asymptotics for quantum graphs

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