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

Publisher： John Wiley & Sons Inc

E-ISSN： 1467-9590|97|1|19-52

ISSN： 0022-2526

Source： STUDIES IN APPLIED MATHEMATICS, Vol.97, Iss.1, 1996-07, pp. : 19-52

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Abstract

Let us consider the Sturm‐Liouville equationon the positive half‐axis with negative potential of the form q(x) = ω2Q(x)+Q0(x), where functions Q and Q0 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−2Q0(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 ω1 and ω2.