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The Reconstruction of the Attracting Potential in the Sturm‐Liouville Equation through Characteristics of Negative Discrete Spectrum

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Let us consider the Sturm‐Liouville equation on the positive half‐axis with negative potential of the form q(x) = ω 2 Q(x)+Q 0(x), where functions Q and Q 0 are integrable together with derivatives of the order m + 1 and have polynomial decreasing at infinity. In the development of the Lax‐Levermore result we show that the function Q(x) + ω −2 Q 0(x) can be reconstructed with accuracy O(ω m )( only through characteristics of discrete negative spectrum of the Dirichlet problem for(*). As an application we prove that it is possible to reconstruct with prescribed accuracy a density and a compressibility of the horizontal homogeneous liquid half‐space through wavenumbers and amplitudes of surface waves excited by monochromatic source with sufficiently large but fixed frequencies ω 1 and ω 2.
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Document Type: Research Article

Affiliations: Université de Pierre et Marie Curie International Institute of Prediction Theory and Mathematical Geophysics (Moscow)

Publication date: July 1, 1996

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