(FUNDAMENTAL AND APPLIED MATHEMATICS)

1999, VOLUME 5, NUMBER 2, PAGES 411-416

## On singularity of solution to inverse problems of spectral analysis expressed with equations of mathematical physics

V. V. Dubrovsky
L. V. Smirnova

Abstract

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The inverse problem for the Laplacian under the Robin's boundary conditions is considered. We prove the following

Theorem. If $q$p, $p=1,2$, are real twice continuously differentiable functions on $\bar \left\{\Omega \right\}$ and there exists a subsequence $i$k of positive integers such that $|| v$ik(qp) ||L2(S) £ const |lik| b, where $v$i(qp) are orthonormal eigenfunctions of the operator $-$D +q in the case of Robin's boundary conditions with the eigenvalues l i, $i$Î N, and $0$£ b < 4-1 then there exists an infinite subsequence $i$klm of positive integers such that the conditions

l i (q1) = li (q2),   i ¹ iklm,
vi(q1)|S = vi(q2)|S,   i ¹ iklm,

imply $q$1=q2.

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