Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals

Beyn, Wolf-Juergen; Latushkin, Yuri; Rottmann-Matthes, Jens

doi:10.48550/arxiv.1210.3952

Cited by 3 publications

(19 citation statements)

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“…A common analytical tool to verify assumption (4.11) in applications is to study the zeroes of the so-called Evans function, see [36], [55]. For numerical purposes however, we prefer to solve boundary eigenvalue problems subject to finite boundary conditions and to employ a contour method, see Section 5 and [5].…”

Section: 3mentioning

confidence: 99%

“…Of interest here is another source of non-linearity: the use of projection boundary conditions when solving linear eigenvalue problems for operators such as (4.6) on a bounded interval J = [x − , x + ]. In the following we summarise two of the major results from [5] on this problem. Contour methods have been developed over the last years ([1], [4], [33]) and have become rather popular since no a-priori knowledge about the location of eigenvalues is assumed.…”

Section: Non-linear Eigenvalue Problemsmentioning

confidence: 99%

“…We mention that Theorem 5.1 generalises to eigenvalues of arbitrary geometric and algebraic multiplicity. With the proper definition of generalised eigenvectors of (5.1) it turns out that the Jordan normal form of the matrix E L in (5.6) inherits the exact multiplicity structure of the non-linear eigenvalue problem, see [5,Theorem 2.8]. For the overall algorithm one approximates the integrals in (5.3) by a quadrature rule (for analytical contours Γ the trapezoidal sum is sufficient since it leads to exponential convergence [4]) and solves linear systems L(λ)u k = v k , k = 1, .…”

Section: Non-linear Eigenvalue Problemsmentioning

confidence: 99%

“…. , κ, see [4], [5,Section 2.2]. There is even an extension of the contour method to cases where the nondegeneracy condition (5.4) is violated.…”

Section: Non-linear Eigenvalue Problemsmentioning

confidence: 99%

“…The evaluation of the matrix E(λ) from (5.2) requires to solve inhomogeneous equations Such λ-dependent boundary matrices P ± , Q ± ∈ C(Ω, R 2m,m ) occur with the so-called projection boundary conditions ( [3]) and lead to fast convergence towards the solution of (5.8) as x ± → ±∞. The matrices are determined in such a way (see [5,Section 4]) that…”

Section: Non-linear Eigenvalue Problemsmentioning

confidence: 99%

See 4 more Smart Citations

Computation and stability of waves in equivariant evolution equations

Beyn

Otten

2019

Spectral Structures and Topological Methods in Mathematics

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Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant evolution equation. In numerical computations the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain degree by the project: linear stability implies non-linear (asymptotic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of non-linear eigenvalue problems via a contour method.

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Section: 3mentioning

confidence: 99%

Section: Non-linear Eigenvalue Problemsmentioning

confidence: 99%

Section: Non-linear Eigenvalue Problemsmentioning

confidence: 99%

“…. , κ, see [4], [5,Section 2.2]. There is even an extension of the contour method to cases where the nondegeneracy condition (5.4) is violated.…”

Section: Non-linear Eigenvalue Problemsmentioning

confidence: 99%

Section: Non-linear Eigenvalue Problemsmentioning

confidence: 99%

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Computation and stability of waves in equivariant evolution equations

Beyn

Otten

2019

Spectral Structures and Topological Methods in Mathematics

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Energy scales and black hole pseudospectra: the structural role of the scalar product

Gasperin

Jaramillo

2022

Class. Quantum Grav.

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A pseudospectrum analysis has recently provided evidence of a potential generic instability of black hole (BH) quasinormal mode (QNM) overtones under high-frequency perturbations. Such instability analysis depends on the assessment of the size of perturbations. The latter is encoded in the scalar product and its choice is not unique. Here, we address the impact of the scalar product choice, advocating for founding it on the physical energy scales of the problem. The article is organized in three parts: basics, applications and heuristic proposals. In the first part, we revisit the energy scalar product used in the hyperboloidal approach to QNMs, extending previous effective analyses and placing them on solid spacetime basis. The second part focuses on systematic applications of the scalar product in the QNM problem: i) we demonstrate that the QNM instability is not an artifact of previous spectral numerical schemes, by implementing a finite elements calculation from a weak formulation; ii) using Keldysh’s asymptotic expansion of the resolvent, we provide QNM resonant expansions for the gravitational waveform, with explicit expressions of the expansion coefficients; iii) we propose the notion of 'epsilon-dual QNM expansions' to exploit BH QNM instability in BH spectroscopy, complementarily exploiting both non-perturbed and perturbed QNMs, the former informing on large scales and the latter probing small scales. The third part enlarges the conceptual scope of BH QNM instability proposing: a) spiked perturbations are more efficient in triggering BH QNM instabilities than smooth ones, b) a general picture of the BH QNM instability problem is given, supporting the conjecture (built on Burnett’s conjecture on the spacetime high-frequency limit) that Nollert-Price branches converge universally to logarithmic Regge branches in the high-frequency limit and c) aiming at a fully geometric description of QNMs, BMS states are hinted as possible asymptotic/boundary degrees of freedom for an inverse scattering problem.

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Energy scales and black hole pseudospectra: the structural role of the scalar product

Gasperin,

Jaramillo

2021

Preprint

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Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals

Cited by 3 publications

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Computation and stability of waves in equivariant evolution equations

Computation and stability of waves in equivariant evolution equations

Energy scales and black hole pseudospectra: the structural role of the scalar product

Energy scales and black hole pseudospectra: the structural role of the scalar product

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