October  2004, 10(4): 857-869. doi: 10.3934/dcds.2004.10.857

Eigenvalues and resonances using the Evans function

1. 

Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87111, United States

2. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States

Received  December 2002 Revised  October 2003 Published  March 2004

In this expository paper, we discuss the use of the Evans function in finding resonances, which are poles of the analytic continuation of the resolvent. We illustrate the utility of the general theory developed in [13, 14] by applying it to two physically interesting test cases: the linear Schrödinger operator and the linearization associated with the integrable nonlinear Schrödinger equation.
Citation: Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857
[1]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939

[2]

Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837

[3]

Björn Sandstede, Arnd Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 941-964. doi: 10.3934/dcds.2004.10.941

[4]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885

[5]

Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607

[6]

Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297

[7]

Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225

[8]

Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423

[9]

Carlangelo Liverani. Fredholm determinants, Anosov maps and Ruelle resonances. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1203-1215. doi: 10.3934/dcds.2005.13.1203

[10]

Frédéric Faure. Prequantum chaos: Resonances of the prequantum cat map. Journal of Modern Dynamics, 2007, 1 (2) : 255-285. doi: 10.3934/jmd.2007.1.255

[11]

T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335

[12]

Tomoki Ohsawa. Dual pairs and regularization of Kummer shapes in resonances. Journal of Geometric Mechanics, 2019, 11 (2) : 225-238. doi: 10.3934/jgm.2019012

[13]

Olga Bernardi, Matteo Dalla Riva. Analytic dependence on parameters for Evans' approximated Weak KAM solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4625-4636. doi: 10.3934/dcds.2017199

[14]

Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165

[15]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413

[16]

Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669

[17]

Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.

[18]

Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741

[19]

Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829

[20]

Motserrat Corbera, Jaume Llibre, Claudia Valls. Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2299-2337. doi: 10.3934/dcdsb.2018101

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]