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

V. V. Dubrovskii ; L. V. Smirnova

Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 2, pp. 411-416.

Résumé

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 $\bar\Omega$ 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 $0\leq\beta4^{-1}$ 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 $q_1=q_2$.