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

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 .