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The Reconstruction of the Attracting Potential in the Sturm‐Liouville Equation through Characteristics of Negative Discrete Spectrum
Author(s) -
Henkin G. M.,
Novikova N. N.
Publication year - 1996
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm199697119
Subject(s) - mathematics , monochromatic color , mathematical analysis , integrable system , spectrum (functional analysis) , mathematical physics , space (punctuation) , polynomial , function (biology) , infinity , amplitude , homogeneous , physics , quantum mechanics , combinatorics , optics , linguistics , philosophy , evolutionary biology , biology
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 .