Pseudo-laplaciens II
Annales de l'Institut Fourier, Tome 33 (1983) no. 2, pp. 87-113.

Dans cet article, nous étudions une famille d’opérateurs auto-adjoints Δa dérivés du laplacien sur une surface de Riemann d’aire finie et ayant au voisinage de l’infini la structure d’un cylindre [b,+[×R/Z muni d’une métrique à courbure constante -1. Après avoir étudié la théorie spectrale de tels opérateurs, nous donnons, comme application, un théorème prévoyant l’absence générique de valeurs propres immergées dans le spectre continu du laplacien de ces surfaces. Nous montrons enfin comment ceci permet de conjecturer une relation entre les zéros de la fonction zêta de Riemann et les valeurs propres d’un certain opérateur auto-adjoint ; cette dernière partie est basée sur des travaux récents de D. Hejhal.

In this paper we study a family of self-adjoint operators Δa which are related to the Laplace operator on some Riemann surface with finite area and one hyperbolic cusp at infinity. We study first the spectral theory of these operators; then we give an application: we prove that, generically, the Laplace operator on that kind of surface has no eigenvalue imbedded in the continuum. Finally we give a conjecture relating the zeros of the Riemann zeta function with some eigenvalues of a self-adjoint operator : that is related with some recent work by D. Hejhal.

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Colin De Verdière, Yves. Pseudo-laplaciens II. Annales de l'Institut Fourier, Tome 33 (1983) no. 2, pp. 87-113. doi : 10.5802/aif.917. http://www.numdam.org/articles/10.5802/aif.917/

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