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Perturbation Theory for the Helmholtz Equation

  • Fritz Gesztesy
  • Marcus Waurick
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2157)

Abstract

This chapter is of a technical nature and focuses on bounded perturbations \(\eta \in L^{\infty }(\mathbb{R}^{n})\) of the Laplacian −Δ defined on \(H^{2}(\mathbb{R}^{n})\) in \(L^{2}(\mathbb{R}^{n})\), \(n \in \mathbb{N}\), odd, and qualitative bounds for the Green’s function corresponding \((-\varDelta +\eta +\mu )^{-1}\) in terms of that of \((-\varDelta + \mathrm{Re}(\mu ))^{-1}\), for μ lying in a sector \(\varSigma _{\mu _{ 0},\vartheta }\) with apex at some μ0 > 0 and opening angle \(2\vartheta \in (0,\pi )\), along the positive real μ-axis.

References

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    C. Callias, Axial anomalies and index theorems on open spaces. Commun. Math. Phys. 62, 213–234 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
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    R. Trigub, A criterion for a characteristic function and a poly-type criterion for radial functions of several variables. Theor. Probab. Appl. 34 (4), 738–742 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Fritz Gesztesy
    • 1
  • Marcus Waurick
    • 2
  1. 1.Dept of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Institut für AnalysisTU DresdenDresdenGermany

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