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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), 67-75.  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), 1715-1736.  Google Scholar

[3]

W. J. Beyn, S. Selle and V. Th\"ummler, Freezing multipulses and multifronts, SIAM J. Appl. Dyn. Syst., 7 (2008), 577-608.  Google Scholar

[4]

D. Edmunds and W. Evans, "Spectral Theory and Differential Operators," in "Oxford Mathematical Monographs, Oxford Science Publications". The Clarendon Press, Oxford University Press, 1987.  Google Scholar

[5]

T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Google Scholar

[6]

R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.  Google Scholar

[7]

B. Sandstede, Stability of multiple-pulse solutions, Trans. Amer. Math. Soc., 350 (1998), 429-472.  Google Scholar

[8]

A. Scheel and J. Wright, Colliding dissipative pulses--the shooting manifold, J. Differential Equations, 245 (2008), 59-79.  Google Scholar

[9]

S. Zelik and A. Mielke, "Multi-pulse Evolution and Space-time Chaos in Dissipative Systems," Mem. Amer. Math. Soc., vol. 198, 2009.  Google Scholar

[10]

J. Alexander and C. Jones, Existence and stability of asymptotically oscillatory double pulses, J. Reine Angew. Math., 446 (1994), 49-79.  Google Scholar

[11]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," in Lecture Notes in Mathematics, vol. 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[12]

J. Wright, Separating dissipative pulses: the exit manifold, J. Dynam. Differential Equations, 21 (2009), 315-328.  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), 67-75.  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), 1715-1736.  Google Scholar

[3]

W. J. Beyn, S. Selle and V. Th\"ummler, Freezing multipulses and multifronts, SIAM J. Appl. Dyn. Syst., 7 (2008), 577-608.  Google Scholar

[4]

D. Edmunds and W. Evans, "Spectral Theory and Differential Operators," in "Oxford Mathematical Monographs, Oxford Science Publications". The Clarendon Press, Oxford University Press, 1987.  Google Scholar

[5]

T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Google Scholar

[6]

R. Pego and M. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys., 164 (1994), 305-349.  Google Scholar

[7]

B. Sandstede, Stability of multiple-pulse solutions, Trans. Amer. Math. Soc., 350 (1998), 429-472.  Google Scholar

[8]

A. Scheel and J. Wright, Colliding dissipative pulses--the shooting manifold, J. Differential Equations, 245 (2008), 59-79.  Google Scholar

[9]

S. Zelik and A. Mielke, "Multi-pulse Evolution and Space-time Chaos in Dissipative Systems," Mem. Amer. Math. Soc., vol. 198, 2009.  Google Scholar

[10]

J. Alexander and C. Jones, Existence and stability of asymptotically oscillatory double pulses, J. Reine Angew. Math., 446 (1994), 49-79.  Google Scholar

[11]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," in Lecture Notes in Mathematics, vol. 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[12]

J. Wright, Separating dissipative pulses: the exit manifold, J. Dynam. Differential Equations, 21 (2009), 315-328.  Google Scholar

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