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Construction de laplaciens dont une partie finie du spectre est donnée. (Construction of Laplacians when a finite part of the spectrum is given). (French) Zbl 0636.58036
Let X be a compact connected manifold. \(F_ x\) is the set of Laplace operators associated to all \(C^{\infty}\)-metrics on X and \(S_ x\) the set of all Schrödinger operators \(H=\Delta +V\), \((\Delta \in F_ x)\) such that 0 is the smallest eigenvalue of H. Given a set F of selfadjoint positive operators the author introduces the following properties: F verifies \(A_ n\) if for every sequence \(a_ 1=0<a_ 2\leq...\leq a_ n\) there exists an \(H\in F\) having this sequence as n first eigenvalues. There is an analogous property \(B_ n\) for \(a_ 1=0<a_ 2<...<a_ n\) and \(C_ n\) if there exists \(b\geq 0\) such that \(a_ 1+b\leq...\leq a_ n+b\) is an interval of the spectrum of an \(H\in F.\)
The author proves: (1) If dim \(X\geq 3\), \(F_ x\) verifies \(A_ n\) for all n, (2) If dim X\(=2\), \(F_ x\) verifies \(B_ n\) and \(C_ n\) for all n. If \(N(X)=\sup (n|\) \(F_ x\) verifies \(A_ n)\) then if X is orientable of genus \(g\geq 3\), \([3/2+\sqrt{2g+(1/4)}]\leq N(X)\leq 4g+4,\) (3) Let \(C(X)=\max imal\) number of vertices of a complete graph imbedded in X. If X is a compact surface, \(S_ x\) verifies \(A_ n\) for \(n=C(X)\). (4) Let \(G_ d\) be the set of Laplacians coming from the Neumann problem of bounded open subsets of \(R^ d\) with piecewise \(C^ 1\) boundary. Then \(G_ d\) verifies \(A_ n\) for all n if \(d\geq 3\). \(G_ 2\) verifies \(A_ n\) iff \(n\leq 4\), \(G_ 2\) verifies \(B_ n\) for all n.
Reviewer: M.Burger

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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