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
Full Text:

References:

 [1] V. ARNOLD , Modes and Quasi-Modes (Journal of Functional Analysis and its applications, vol. 6, 1972 , p. 94-101). MR 45 #6331 | Zbl 0251.70012 · Zbl 0251.70012 [2] C. ANNÉ , Écrasement d’anses et spectre du laplacien (Ann. scient. Éc. Norm. Sup. 4e série (20, 1987 , p. 271 à 280). Numdam | Zbl 0634.58035 · Zbl 0634.58035 [3] G. BESSON , Sur la multiplicité de la première valeur propre des surfaces riemanniennes (Ann. Institut Fourier, vol. 30, 1980 , p. 109-128). Numdam | MR 81h:58059 | Zbl 0417.30033 · Zbl 0417.30033 [4] M. BURGER , Estimations des petites valeurs propres du laplacien d’un revêtement de variétés riemanniennes compactes (C. R. Acad. Sci. Paris, t. 302, série I, 1986 , p. 191-194). MR 87f:58164 | Zbl 0585.53035 · Zbl 0585.53035 [5] B. COLBOIS et Y. COLIN DE VERDIÈRE , Multiplicité de la première valeur propre positive du laplacien d’une surface à courbure constante [Comment. Math. Helv., 1987 (à paraître)]. [6] CHAVEL et FELDMANN , Spectra of Manifolds with Small Handles (Comment. math. Helv., vol. 56, 1981 , p. 83-102). MR 82j:58110 | Zbl 0473.53037 · Zbl 0473.53037 [7] S. CHENG , Eigenfunctions and Nodal Sets (Commentarii Math. Helv., vol. 51, 1979 , p. 43-55). MR 53 #1661 | Zbl 0334.35022 · Zbl 0334.35022 [8] B. COLBOIS , Petites valeurs propres du laplacien sur une surface de Riemann compacte et graphes (C. R. Acad. Sci. Paris, t. 301, série I, 1985 , p. 927-930). MR 88b:58139 | Zbl 0582.53034 · Zbl 0582.53034 [9] Y. COLIN DE VERDIÈRE , Spectres de variétés riemanniennes et spectres de graphes [Proc. ICM, Berkeley, 1986 (à paraître)]. Zbl 0693.58034 · Zbl 0693.58034 [10] Y. COLIN DE VERDIÈRE , Sur la multiplicité de la première valeur propre non nulle du laplacien (Commentarii Math. Hel., vol. 61, 1986 , p. 254-270). MR 88b:58140 | Zbl 0607.53028 · Zbl 0607.53028 [11] Y. COLIN DE VERDIÈRE , Sur une hypothèse de transversalité d’Arnold [Comment. Math. Helv., 1987 (à paraître)]. Zbl 0672.58046 · Zbl 0672.58046 [12] Y. COLIN DE VERDIÈRE , Sur un nouvel invariant des graphes finis et un critère de planarité (Prépublications de l’Institut Fourier, 71, 1987 , p. 1-15). [13] J. DODZIUK , Difference Equations, Isoperimetric Inequalities and Transience of Certain Random Walks (Trans. Amer. Math. Soc., vol. 284, 1984 , p. 787-794). MR 85m:58185 | Zbl 0512.39001 · Zbl 0512.39001 [14] B. HELFFER et J. SJÖSTRAND , Puits multiples en semi-classique, I (Comm. P.D.E., vol. 1984, p. 337-408) ; II (Ann. I.H.P. (Phys. théorique), vol. 42, 1985 , p. 127-212 ; III (Math. Nachrichten, 127, 1985 , p. 263-313) ; IV (Comm. in P.D.E., vol. 10, 1985 , p. 245-340) ; V, VI (à paraître). · Zbl 0546.35053 [15] M. KAC , Can One Hear the Shape of a Drum ? (Amer. Math. Monthly, vol. 73, 1966 , p. 1-23). MR 34 #1121 | Zbl 0139.05603 · Zbl 0139.05603 [16] L. PAYNE , G. POLYA et H. WEINBERGER , On the Ratio of Consecutive Eigenvalues (J. of Math. and Phys., vol. 35, 1956 , p. 289-298). MR 18,905c | Zbl 0073.08203 · Zbl 0073.08203 [17] M. PROTTER , Can One Hear the Shape of a Drum ? , Revisited, Preprint, Berkeley, 1985 . · Zbl 0645.35074 [18] T. PIGNATORO et D. SULLIVAN , Ground State and Lowest Eigenvalue of the Laplacian for Non-Compact Hyperbolic Surfaces (Comm. Math. Phys., vol. 104, 1986 , p. 529-535). Article | Zbl 0602.58046 · Zbl 0602.58046 [19] G. RINGEL , Map Color Theorem , Springer, 1974 . MR 50 #1955 | Zbl 0287.05102 · Zbl 0287.05102 [20] RAUCH et TAYLOR , Potential and Scattering Theory in Widely Perturbed Domains (Journal of Functional Analysis, vol. 18, 1975 , p. 27-59). MR 51 #13476 | Zbl 0293.35056 · Zbl 0293.35056
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