×

On an inverse problem for the Laplace operator with continuous potential. (Russian) Zbl 0714.35087

Let \(\Pi =\{0\leq x\leq a\), \(0\leq y\leq b\}\), \(\Pi_{1/4}=\{0\leq x\leq a/2\), \(0\leq y\leq b/2\},\)
L: the boundary value problem \(-\Delta u=\lambda u\) with \(u|_{\partial \Pi}=0,\)
E(\(\lambda\)): the resolution of the identity with respect to L, \(\beta >0\), and \(T=\int \lambda^{\beta}dE(\lambda).\)
The orthogonal eigenfunctions \(v_{m,n}=(2/\sqrt{ab})\sin (\pi mx/y)\sin (\pi ny/b)\) of the operator T correspond to the eigenvalues \(\lambda_{m,n}=(\pi^ 2m^ 2/a^ 2+ \pi^ 2n^ 2/b^ 2)^{\beta}.\)
Let \(\sigma\) (T) be the spectrum of T, and \(d_{m,n}=\rho (\lambda_{m,n},\sigma (T)\setminus \lambda_{m,n})\). The index t in \(\lambda_ t\) is \(t=\#\{\lambda_{m,n}\); \(\lambda_{m,n}\leq \lambda_ t\}\). Let \(\{\lambda_{t_ k}\}^ N_{k=1}\), \(N\leq \infty\), be the sequence of the simple eigenvalues of T. Next let P be the multiplication operator of the function p(x,y)\(\in C({\bar \Pi})\) satisfying \(p(a- x,y)=p(x,y)=p(x,b-y)\), and \[ (1)\quad \iint_{\Pi}p(x,y)\cos (2\pi mx/a)dx dy=\iint_{\Pi}p(x,y)\cos (2\pi ny/b)dx dy=0. \] \(\{\mu_ t,u_ t(x,y)\}\) is the eigenvalue and eigenfunction of the operator \(T+P\), where the index t is given by the increasing order of Re\(\mu_ t\). First of all the authors prove the inequality \(| \mu_{t_ k}- \lambda_{t_ k}| \leq const\| p\|_{\infty}\). Next they prove the
Theorem: If \(\sum^{N}_{k=1}d^{-1}_{t_ k}<\infty\), there exists a closed cube \(u(0,\epsilon)=\{p\); \(\| p\|_{\infty}\leq \epsilon \}\) in C(\({\bar \Pi}{}_{1/4})\) such that for any sequence \(\{\xi_{t_ k}\}^ N_{k=1}\), satisfying the condition \(\sum^{N}_{k=1}| \xi_{t_ k}-\lambda_{t_ k}| \leq c(a,b,\epsilon)\epsilon\), there exists one and only one potential p(x,y)\(\in u(0,\epsilon)\) satisfying the above condition (1) and the following properties:
a) \(\mu_{t_ k}=\xi_{t_ k},\)
b) \(\iint_{\Pi}p(x,y)\cos (2\pi mx/a)\cdot \cos (2\pi ny/b)dx dy=0\), if \(\lambda_{m,n}\not\in \{\lambda_{t_ k}\}^ N_{k=1},\)
c) The Fourier series of p(x,y) converges uniformly and absolutely in \(\Pi_{1/4}.\)
One of their examples is the following: If \(a^ 2/b^ 2\) is an algebraic number with order \(n\geq 2\) and \(3\geq \beta >2\), then the potential p(x,y) is given by cosine polynomial.
Reviewer: H.Yamagata

MSC:

35R30 Inverse problems for PDEs
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
PDFBibTeX XMLCite