# American Institute of Mathematical Sciences

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

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.
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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