### 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

- David Bindel and Maciej Zworski. Theory and Computation of Resonances in 1D Scattering
- MatScat on GitHub
- MatScatPy (by Sheroze Sherrifdeen and Chaitali Joshi)

### Papers

*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.

*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.

*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
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meeting external invited

#### Numerical Analysis of Resonances

Weyl at 100 Workshop (Fields Institute)

eigenbounds matscat nep pml resonance
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meeting external invited

#### Analyzing Resonances via Nonlinear Eigenvalues

ICIAM

eigenbounds matscat nep resonance
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minisymposium external invited

#### Resonances: Interpretation, Computation, and Perturbation

Cornell SCAN Seminar

eigenbounds matscat nep resonance
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seminar local

#### Resonances: Interpretation, Computation, and Perturbation

Workshop in honor of Pete Stewart at UT Austin

eigenbounds matscat nep resonance
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meeting external invited

#### Applications and Analysis of Nonlinear Eigenvalue Problems

Simon Fraser University NA Seminar

eigenbounds matscat nep resonance pml
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seminar external invited

#### Resonances and Nonlinear Eigenvalue Problems

NYCAM

eigenbounds matscat nep resonance
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meeting external

#### Numerical Analysis for Nonlinear Eigenvalue Problems

Cornell SCAN Seminar

eigenbounds matscat nep pml resonance
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seminar local

#### Bounds and Error Estimates for Resonance Problems

SIAM Annual Meeting

eigenbounds matscat nep pml resonance
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minisymposium external invited

#### Numerical Methods for Resonance Calculations

MSRI Workshop on Resonances

eigenbounds matscat nep resonance
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meeting external invited

#### Resonance Computations

NYU DOE Site Visit

eigenbounds matscat nep pml resonance
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local

#### Bounds and Error Estimates for Nonlinear Eigenvalue Problems

Berkeley Applied Math Seminar

eigenbounds matscat nep pml resonance
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seminar external invited

#### Numerical Methods for Resonance Calculations

BIRS Resonance Workshop

matscat mems nep pml resonance rf-mems
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meeting external invited

#### Eigenvaues, Resonance Poles, and Damping in MEMS

UC Berkeley LAPACK Seminar

mems pml resonance rf-mems
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seminar local