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Sur l’unicité d’un problème inverse spectral bidimensionnel. (On the uniqueness of an inverse spectral bidimensional problem). (French) Zbl 0679.35083
We show that the spectral characteristics $$(\lambda_ n,\| \phi_ j(.,.,\lambda_ n)\|)_{n\geq 0}$$; $$j=1,...,m_ n$$, are sufficient to ensure the uniqueness of the associated inverse spectral bidimensional problem: $$\Delta \phi +(\lambda -q)\phi =0,$$ in (0,1)$$\times (0,1)$$, and $$q(x,y)=q(x).$$
Theorem. Let $$B(q,j_ 1,j_ 2,j_ 3,j_ 4)$$ be the problem: $\Delta \phi +(\lambda -q)\phi =0\quad in\quad \Omega,$ $\partial_ n\phi +j_ 1\phi =0\quad on\quad]0,1[\times \{0\},\quad \partial_ n\phi +j_ 2\phi =0\quad on\quad]0,1[\times \{1\},$ $\partial_ n\phi +j_ 3\phi =0\quad on\quad \{0\}\times]0,1[,\quad \partial_ n\phi +j_ 4\phi =0\quad on\quad \{1\}\times]0,1[.$ Let p and q in $$C^ 0(0,1):\int^{1}_{0}p(t)dt=\int^{1}_{0}q(t)dt.$$
Let $$(\lambda_ n,\| \phi_ j(.,.,\lambda_ n)\|)_{n\geq 0}$$, $$(\mu_ n,\| \psi_ j(.,.,\mu_ n)\|)_{n\geq 0}$$; $$j=1,...,m_ n$$ be the spectral characteristics of the problems $$B(p,j_ 1,j_ 2,j_ 3,j_ 4)$$ and $$B(q,j_ 1,j_ 2,j_ 3,j_ 4)$$, respectively.
If $$\lambda_ n=\mu_ n$$ and $$\| \phi_ j(.,.,\lambda_ n)\| =\| \psi_ j(.,.,\mu_ n)\|$$ for all $$n\geq 0$$ then $$p=q$$ in (0,1).