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

View as HTML
View as gif image
View as LaTeX source

```
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 $q_1=q_2$.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/99/992/99204t.htm

Last modified: July 6, 1999