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Spectral analysis of the Epstein operator

Published online by Cambridge University Press:  14 November 2011

J. C. Guillot  and
C. H. Wilcox
J. C. Guillot
Affiliation:
Département de Mathématiques, Université de Paris Nord, 93430 Villetaneuse, France
C. H. Wilcox
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, U.S.A.
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Synopsis

The Epstein operator is defined by

where x = (x1, …, xn) ∈ Rn, y ∈ R,

and H, K, L, M are real constants such that c2(y) > 0. The operator arises in the study of acoustic wave propagation in plane-stratified fluids with sound speed c(y) at depth y. In this paper it is shown that A defines a selfadjoint operator in the Hilbert space ℋ = L2(Rn + 1c−2(y) dx dy) where dx = dx1 … dxn. The spectral family of A is constructed, the spectrum is shown to be continuous and an eigenfunction expansion for A is given in terms of families of improper eigenfunctions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 1-2 , 1978 , pp. 85 - 98
Copyright
Copyright © Royal Society of Edinburgh 1978

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