2,346 research outputs found

    An inverse Sturm-Liouville problem with a fractional derivative

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    In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order α∈(1,2)\alpha\in(1,2) of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of Computational Physic

    Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials

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    We discuss inverse spectral theory for singular differential operators on arbitrary intervals (a,b)⊆R(a,b) \subseteq \mathbb{R} associated with rather general differential expressions of the type τf=1r(−(p[f′+sf])′+sp[f′+sf]+qf),\tau f = \frac{1}{r} \left(- \big(p[f' + s f]\big)' + s p[f' + s f] + qf\right), where the coefficients pp, qq, rr, ss are Lebesgue measurable on (a,b)(a,b) with p−1p^{-1}, qq, rr, s∈Lloc1((a,b);dx)s \in L^1_{\text{loc}}((a,b); dx) and real-valued with p≠0p\not=0 and r>0r>0 a.e.\ on (a,b)(a,b). In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in this paper include those implied by the spectral measure, by two-spectra and three-spectra, as well as local Borg-Marchenko-type inverse spectral results. The special cases of Schr\"odinger operators with distributional potentials and Sturm--Liouville operators in impedance form are isolated, in particular.Comment: 29 page

    Inverse eigenvalue problem for discrete three-diagonal Sturm-Liouville operator and the continuum limit

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    In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation of the Schroedinger operator on a bounded interval as a finite three-diagonal symmetric Jacobi matrix. This derivation is more correct in comparison with previous works which used only single-diagonal matrix. It is demonstrated that inverse problem procedure is nothing else than well known Gram-Schmidt orthonormalization in Euclidean space for special vectors numbered by the space coordinate index. All the results of usual inverse problem with continuous coordinate are reobtained by employing a limiting procedure, including the Goursat problem -- equation in partial derivatives for the solutions of the inversion integral equation.Comment: 19 pages There were made some additions (and reformulations) to the text making the derivation of the results more precise and understandabl
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