Scattering and resonances

Linear eigenproblems for standing waves; nonlinear eigenproblems for almost-standing waves.

Summary

Most work on numerical analysis of eigenvalue problem involves either self-adjoint problems or problems that can be strongly approximated by something finite dimensional (i.e. operators with compact resolvent). Many interesting problems from scattering theory and from the stability theory of traveling waves fall into neither of these categories. For example, resonance poles in scattering theory give information about the asymptotic dynamics of trapped waves leaking from a bounded subdomain in the same way that eigenvalues give information about asymptotic steady-state vibrations on finite domains. There are several ways to define resonances, but for computation it is often convenient to describe them as solutions to nonlinear eigenvalue problems. We work on fast algorithms to find solutions to such nonlinear eigenproblems and analysis to describe the error in what we compute – and to tell us what solutions our computations might miss.

Links

Papers

D. Bindel and A. Hood, “Localization Theorems for Nonlinear Eigenvalues,” SIAM Review, vol. 57, no. 4, pp. 585–607, Dec. 2015.
SIGEST feature article. 
@article{2015-sirev,
  author = {Bindel, David and Hood, Amanda},
  title = {Localization Theorems for Nonlinear Eigenvalues},
  journal = {SIAM Review},
  publisher = {SIAM},
  volume = {57},
  number = {4},
  pages = {585--607},
  month = dec,
  year = {2015},
  notable = {SIGEST feature article.},
  doi = {10.1137/15M1026511}
}

Abstract:

Let $T : \Omega \rightarrow {\Bbb C}^{n\times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset {\Bbb C}$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.

D. Bindel and A. Hood, “Localization Theorems for Nonlinear Eigenvalues,” SIAM Journal on Matrix Analysis, vol. 34, no. 4, pp. 1728–1749, 2013.
2015 SIAG/LA award (best journal paper in applied LA in three years). 
@article{2013-simax,
  author = {Bindel, David and Hood, Amanda},
  title = {Localization Theorems for Nonlinear Eigenvalues},
  journal = {SIAM Journal on Matrix Analysis},
  volume = {34},
  number = {4},
  pages = {1728--1749},
  year = {2013},
  doi = {10.1137/130913651},
  arxiv = {http://arxiv.org/abs/1303.4668},
  notable = {2015 SIAG/LA award (best journal paper in applied LA in three years).}
}

Abstract:

Let $T : \Omega \rightarrow {\Bbb C}^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $\Omega \subset {\Bbb C}$. A point $\lambda \in \Omega$ is an eigenvalue if the matrix $T(\lambda)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin’s theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.

D. Bindel and M. Zworski, “Symmetry of Bound and Antibound States in the Semiclassical Limit,” Letters in Math Physics, vol. 81, no. 2, pp. 107–117, Aug. 2007. 
@article{2007-symmetry,
  author = {Bindel, David and Zworski, Maciej},
  title = {Symmetry of Bound and Antibound States in the Semiclassical Limit},
  journal = {Letters in Math Physics},
  volume = {81},
  number = {2},
  pages = {107--117},
  month = aug,
  year = {2007},
  doi = {10.1007/s11005-007-0178-7}
}

Abstract:

Motivated by a recent numerical observation we show that in one dimensional scattering a barrier separating the interaction region from infinity implies approximate symmetry of bound and antibound states. We also outline the numerical procedure used for an efficient computation of one dimensional resonances.

Talks

Some perturbation theorems for nonlinear eigenvalue problems

Workshop on Dissipative Spectral Theory, Cardiff
eigenbounds matscat nep pml resonance • meeting external invited

Numerical Analysis of Resonances

Weyl at 100 Workshop (Fields Institute)
eigenbounds matscat nep pml resonance • meeting external invited

Analyzing Resonances via Nonlinear Eigenvalues

ICIAM
eigenbounds matscat nep resonance • minisymposium external invited

Resonances: Interpretation, Computation, and Perturbation

Cornell SCAN Seminar
eigenbounds matscat nep resonance • seminar local

Resonances: Interpretation, Computation, and Perturbation

Workshop in honor of Pete Stewart at UT Austin
eigenbounds matscat nep resonance • meeting external invited

Applications and Analysis of Nonlinear Eigenvalue Problems

Simon Fraser University NA Seminar
eigenbounds matscat nep resonance pml • seminar external invited

Resonances and Nonlinear Eigenvalue Problems

NYCAM
eigenbounds matscat nep resonance • meeting external

Numerical Analysis for Nonlinear Eigenvalue Problems

Cornell SCAN Seminar
eigenbounds matscat nep pml resonance • seminar local

Bounds and Error Estimates for Resonance Problems

SIAM Annual Meeting
eigenbounds matscat nep pml resonance • minisymposium external invited

Numerical Methods for Resonance Calculations

MSRI Workshop on Resonances
eigenbounds matscat nep resonance • meeting external invited

Resonance Computations

NYU DOE Site Visit
eigenbounds matscat nep pml resonance • local

Bounds and Error Estimates for Nonlinear Eigenvalue Problems

Berkeley Applied Math Seminar
eigenbounds matscat nep pml resonance • seminar external invited

Numerical Methods for Resonance Calculations

BIRS Resonance Workshop
matscat mems nep pml resonance rf-mems • meeting external invited

Eigenvaues, Resonance Poles, and Damping in MEMS

UC Berkeley LAPACK Seminar
mems pml resonance rf-mems • seminar local