TAILIEUCHUNG - Đề tài " Inverse spectral problems and closed exponential systems "

Consider the inverse eigenvalue problem of the Schr¨dinger operator deo fined on a finite interval. We give optimal and almost optimal conditions for a set of eigenvalues to determine the Schr¨dinger operator. These conditions are o simple closedness properties of the exponential system corresponding to the known eigenvalues. The statements contain nearly all former results of this topic. We give also conditions for recovering the Weyl-Titchmarsh m-function from its values m(λn ). | Annals of Mathematics Inverse spectral problems and closed exponential systems By Mikl os Horv rath Annals of Mathematics 162 2005 885 918 Inverse spectral problems and closed exponential systems By Miklos Horvath Abstract Consider the inverse eigenvalue problem of the Schrodinger operator defined on a finite interval. We give optimal and almost optimal conditions for a set of eigenvalues to determine the Schrodinger operator. These conditions are simple closedness properties of the exponential system corresponding to the known eigenvalues. The statements contain nearly all former results of this topic. We give also conditions for recovering the Weyl-Titchmarsh m-function from its values m Xn . 1. Introduction Consider the Schroodinger operator Ly -y q x y over the segment 0 n with a potential q E L1 0 n real-valued. The eigenvalue problem Ly Xy on 0 n y 0 cos a y 0 sin a 0 y n cos 3 y n sin 3 0 defines a sequence of eigenvalues Xo X1 Xn . Xn E R Xn I they form together the spectrum ơ q a 3 . In the inverse eigenvalue problems we aim to recover the potential q from a given set of eigenvalues not necessarily taken from the same spectrum . The first result of this type is given in Research supported by the Hungarian NSF Grants OTKA T 32374 and T 37491. 886 MIKLOS HORVATH Theorem a Ambarzumian 1 . Let q E C 0 n and consider the Neumann eigenvalue problem y 0 y n 0 . a 3 n 2 . If the eigenvalues are An n2 n 0 then q 0. Later it was observed by G. Borg that the knowledge of the first eigenvalue Ao 0 plays a crucial role here he also found the general rule that in most cases two spectra are needed to recover the potential Theorem B Borg 5 . Let q E L1 0 n ơ1 ơ q 0 3 ơ2 ơ q a2 3 sina2 0 and - ơ2 if sin 3 0 ơ2 ơ2 Ao if sin 3 0. Then ơ1 u ỡ2 determines the potential . and no proper subset has the same property. Here determination means that there is no other potential q E L1 0 n with ơ1 ơ ỡ2 ơ . There is a related extension Theorem C Levinson .

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