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

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 $0\leq\beta<4^{-1}$ 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 $q_1=q_2$.