FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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

{\bfseries Theorem.} If $q_p$, $p=1,2$, are real twice continuously differentiable functions on $\bar{\Omega}$ and there exists a subsequence $i_k$ of positive integers such that $\|v_{i_k}(q_p)\|_{L_2(S)}\leq \mathrm{const} |\lambda_{i_k}|^{\beta}$, where $v_i(q_p)$ are orthonormal eigenfunctions of the operator $-\Delta+q$ in the case of Robin's boundary conditions with the eigenvalues $\lambda_i$, $i\in\mathbb N$, and $0\leq\beta<4^{-1}$ then there exists an infinite subsequence $i_{k_{l_m}}$ of positive integers such that the conditions
$$
\lambda_i(q_1)=\lambda_i(q_2),\ \ i\neq i_{k_{l_m}},\quad
v_i(q_1)|_S=v_i(q_2)|_S,\ \ i\neq i_{k_{l_m}},
$$
 imply $q_1=q_2$.

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Last modified: July 6, 1999