American Institute of Mathematical Sciences

Advanced
January  2013, 18(1): 273-281. doi: 10.3934/dcdsb.2013.18.273

On the spectrum of the superposition of separated potentials.

J. Douglas Wright 1,

1. 

Drexel University, Department of Mathematics, 3141 Chestnut Ave, Philadelphia, PA 19104, United States

Received  April 2012 Revised  July 2012 Published  September 2012

Suppose that $V(x)$ is an exponentially localized potential and $L$ is a constant coefficient differential operator. A method for computing the spectrum of $L+V(x-x_1) + ... + V(x-x_N)$ given that one knows the spectrum of $L+V(x)$ is described. The method is functional theoretic in nature and does not rely heavily on any special structure of $L$ or $V$ apart from the exponential localization. The result is aimed at applications involving the existence and stability of multi-pulses in partial differential equations.
Keywords: Multipulses, long distance interactions, perturbation theory for linear operators., eigenvalues.
Mathematics Subject Classification: Primary: 47A55, 35P05; Secondary: 47A4.
Citation: J. Douglas Wright. On the spectrum of the superposition of separated potentials.. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 273-281. doi: 10.3934/dcdsb.2013.18.273
References:
[1]

H. Ammari and S. Moskow, Asymptotic expansions for eigenvalues in the presence of small inhomogeneities,, Math. Methods Appl. Sci., 26 (2003), 67. Google Scholar

[2]

H. Ammari, H. Kang and H. Lee, Asymptotic expansions for eigenvalues of the Lamsystem in the presence of small inclusions,, Comm. Partial Differential Equations, 32 (2007), 1715. Google Scholar

[3]

W. J. Beyn, S. Selle and V. Th\"ummler, Freezing multipulses and multifronts,, SIAM J. Appl. Dyn. Syst., 7 (2008), 577. Google Scholar

[4]

D. Edmunds and W. Evans, "Spectral Theory and Differential Operators,", in, (1987). Google Scholar

[5]

T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, (1980). Google Scholar

[6]

R. Pego and M. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305. Google Scholar

[7]

B. Sandstede, Stability of multiple-pulse solutions,, Trans. Amer. Math. Soc., 350 (1998), 429. Google Scholar

[8]

A. Scheel and J. Wright, Colliding dissipative pulses--the shooting manifold,, J. Differential Equations, 245 (2008), 59. Google Scholar

[9]

S. Zelik and A. Mielke, "Multi-pulse Evolution and Space-time Chaos in Dissipative Systems,", Mem. Amer. Math. Soc., (2009). Google Scholar

[10]

J. Alexander and C. Jones, Existence and stability of asymptotically oscillatory double pulses,, J. Reine Angew. Math., 446 (1994), 49. Google Scholar

[11]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", in Lecture Notes in Mathematics, (1981). Google Scholar

[12]

J. Wright, Separating dissipative pulses: the exit manifold,, J. Dynam. Differential Equations, 21 (2009), 315. Google Scholar

show all references

References:
[1]

H. Ammari and S. Moskow, Asymptotic expansions for eigenvalues in the presence of small inhomogeneities,, Math. Methods Appl. Sci., 26 (2003), 67. Google Scholar

[2]

H. Ammari, H. Kang and H. Lee, Asymptotic expansions for eigenvalues of the Lamsystem in the presence of small inclusions,, Comm. Partial Differential Equations, 32 (2007), 1715. Google Scholar

[3]

W. J. Beyn, S. Selle and V. Th\"ummler, Freezing multipulses and multifronts,, SIAM J. Appl. Dyn. Syst., 7 (2008), 577. Google Scholar

[4]

D. Edmunds and W. Evans, "Spectral Theory and Differential Operators,", in, (1987). Google Scholar

[5]

T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, (1980). Google Scholar

[6]

R. Pego and M. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305. Google Scholar

[7]

B. Sandstede, Stability of multiple-pulse solutions,, Trans. Amer. Math. Soc., 350 (1998), 429. Google Scholar

[8]

A. Scheel and J. Wright, Colliding dissipative pulses--the shooting manifold,, J. Differential Equations, 245 (2008), 59. Google Scholar

[9]

S. Zelik and A. Mielke, "Multi-pulse Evolution and Space-time Chaos in Dissipative Systems,", Mem. Amer. Math. Soc., (2009). Google Scholar

[10]

J. Alexander and C. Jones, Existence and stability of asymptotically oscillatory double pulses,, J. Reine Angew. Math., 446 (1994), 49. Google Scholar

[11]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", in Lecture Notes in Mathematics, (1981). Google Scholar

[12]

J. Wright, Separating dissipative pulses: the exit manifold,, J. Dynam. Differential Equations, 21 (2009), 315. Google Scholar

[1]

Yaiza Canzani, A. Rod Gover, Dmitry Jakobson, Raphaël Ponge. Nullspaces of conformally invariant operators. Applications to $\boldsymbol{Q_k}$-curvature. Electronic Research Announcements, 2013, 20: 43-50. doi: 10.3934/era.2013.20.43
[2]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[3]

François Hamel, Emmanuel Russ, Nikolai Nadirashvili. Comparisons of eigenvalues of second order elliptic operators. Conference Publications, 2007, 2007 (Special) : 477-486. doi: 10.3934/proc.2007.2007.477
[4]

Miklós Horváth, Márton Kiss. A bound for ratios of eigenvalues of Schrodinger operators on the real line. Conference Publications, 2005, 2005 (Special) : 403-409. doi: 10.3934/proc.2005.2005.403
[5]

John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019
[6]

Fioralba Cakoni, Shari Moskow, Scott Rome. The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions. Inverse Problems & Imaging, 2015, 9 (3) : 725-748. doi: 10.3934/ipi.2015.9.725
[7]

Joyce R. McLaughlin and Arturo Portnoy. Perturbation expansions for eigenvalues and eigenvectors for a rectangular membrane subject to a restorative force. Electronic Research Announcements, 1997, 3: 72-77.

[8]

John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations & Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001
[9]

Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233
[10]

Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012
[11]

Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299
[12]

Wenxian Shen, Xiaoxia Xie. On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1665-1696. doi: 10.3934/dcds.2015.35.1665
[13]

Roberto Alicandro, Giuliano Lazzaroni, Mariapia Palombaro. Derivation of a rod theory from lattice systems with interactions beyond nearest neighbours. Networks & Heterogeneous Media, 2018, 13 (1) : 1-26. doi: 10.3934/nhm.2018001
[14]

Raz Kupferman, Asaf Shachar. On strain measures and the geodesic distance to $SO_n$ in the general linear group. Journal of Geometric Mechanics, 2016, 8 (4) : 437-460. doi: 10.3934/jgm.2016015
[15]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623
[16]

Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019041
[17]

Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457
[18]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
[19]

Igor Chueshov, Irena Lasiecka, Justin Webster. Flow-plate interactions: Well-posedness and long-time behavior. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 925-965. doi: 10.3934/dcdss.2014.7.925
[20]

Ian Johnson, Evelyn Sander, Thomas Wanner. Branch interactions and long-term dynamics for the diblock copolymer model in one dimension. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3671-3705. doi: 10.3934/dcds.2013.33.3671

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences