On the spectrum of the superposition of separated potentials.

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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.

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

Received April 2012 Revised July 2012 Published September 2012

Abstract
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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.
[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.
[3]	W. J. Beyn, S. Selle and V. Th\"ummler, Freezing multipulses and multifronts,, SIAM J. Appl. Dyn. Syst., 7 (2008), 577.
[4]	D. Edmunds and W. Evans, "Spectral Theory and Differential Operators,", in, (1987).
[5]	T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, (1980).
[6]	R. Pego and M. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305.
[7]	B. Sandstede, Stability of multiple-pulse solutions,, Trans. Amer. Math. Soc., 350 (1998), 429.
[8]	A. Scheel and J. Wright, Colliding dissipative pulses--the shooting manifold,, J. Differential Equations, 245 (2008), 59.
[9]	S. Zelik and A. Mielke, "Multi-pulse Evolution and Space-time Chaos in Dissipative Systems,", Mem. Amer. Math. Soc., (2009).
[10]	J. Alexander and C. Jones, Existence and stability of asymptotically oscillatory double pulses,, J. Reine Angew. Math., 446 (1994), 49.
[11]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", in Lecture Notes in Mathematics, (1981).
[12]	J. Wright, Separating dissipative pulses: the exit manifold,, J. Dynam. Differential Equations, 21 (2009), 315.

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References:

[1]	H. Ammari and S. Moskow, Asymptotic expansions for eigenvalues in the presence of small inhomogeneities,, Math. Methods Appl. Sci., 26 (2003), 67.
[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.
[3]	W. J. Beyn, S. Selle and V. Th\"ummler, Freezing multipulses and multifronts,, SIAM J. Appl. Dyn. Syst., 7 (2008), 577.
[4]	D. Edmunds and W. Evans, "Spectral Theory and Differential Operators,", in, (1987).
[5]	T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, (1980).
[6]	R. Pego and M. Weinstein, Asymptotic stability of solitary waves,, Comm. Math. Phys., 164 (1994), 305.
[7]	B. Sandstede, Stability of multiple-pulse solutions,, Trans. Amer. Math. Soc., 350 (1998), 429.
[8]	A. Scheel and J. Wright, Colliding dissipative pulses--the shooting manifold,, J. Differential Equations, 245 (2008), 59.
[9]	S. Zelik and A. Mielke, "Multi-pulse Evolution and Space-time Chaos in Dissipative Systems,", Mem. Amer. Math. Soc., (2009).
[10]	J. Alexander and C. Jones, Existence and stability of asymptotically oscillatory double pulses,, J. Reine Angew. Math., 446 (1994), 49.
[11]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", in Lecture Notes in Mathematics, (1981).
[12]	J. Wright, Separating dissipative pulses: the exit manifold,, J. Dynam. Differential Equations, 21 (2009), 315.

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