Asymptotiques spectrales et géométrie des nombres

In this thesis, we study the spectrum of the Laplacian and of other related operators. When defined on either a closed compact Riemannian manifold, or a manifold with boundary and self-adjoint boundary conditions, the Laplacian Δ has a real and discrete spectrum   𝜆1(M) ≤ 𝜆2≤ (M)… ↗∞  accumulating only at ∞ The numbers 𝜆j (M) are those for which there is a non-trivial solution to the equation  Δ𝜑 + 𝜆𝜑 = 0.  We are more specifically interested in the asymptotic behaviour of the counting function  N(𝜆; M) ≔ #{ 𝜆j (M) ˂ 𝜆}.  Hermann Weyl has shown in 1911 [80] what is now known as Weyl’s law,  R(𝜆; M) ≔ N(𝜆; M) − ω^d/〖(2π)〗_d  Vol(M)𝜆d/2   as 𝜆 ⟶ ∞  where ωd is the volume of the unit ball in dimension d. We want to determine the size of  R(𝜆; M) ≔ N(𝜆; M) − ω^d/〖(2π)〗_d  Vol(M)𝜆d/2. In the context at hand, we have translated this problem in terms of the geometry of numbers, the study of the interaction between lattice points, e.g. ℤd and convex sets. In the first chapter, we make a precise description of the problems studied and how they can be linked to the geometry of numbers. Furthermore, we describe in more detail themain techniques that we have used.  The second chapter, titled On a generalised Gauss circle problem and integrated density of states [54], has been written in collaboration with Leonid Parnovski. There, we study the spectrum of the Laplacian on a product of a flat torus and Euclidean space. In this case, the spectrum is not discrete. However, we study the integrated density of states, which takes the role of the eigenvalue counting function and also satisfies Weyl’s law. We obtain upper and lower bounds on R(𝜆) in this context, which depend on the relative dimensions of the flat torus and Euclidean space. When the dimension of the torus is strictly smaller than that of the Euclidean space the upper and lower bound share the same polynomial order. We also obtain an asymptotic expansion up to constant order for the integrated density of states of a magnetic Schrödinger operator with constant potential.  The third chapter, titled The Steklov spectrum of cuboids [26] has been written together with Alexandre Girouard, Iosif Polterovich and Alessandro Savo. We study the Steklov spectrum, i.e.the spectrum of the Dirichlet-to-Neumann operator on cuboids of any dimension. Eigenvalue asymptotics for this operator had not been very much studied on domains whose boundaries are not smooth and cuboids provide a first example of such domains. The spectrum is discrete and only accumulates at infinity, and we obtain a two-term Weyl’s law for the Steklov spectrum. We also obtain an isoperimetric inequality for the first non-trivial eigenvalue. Finally, we prove that some sequence of eigenfunctions concentrates along edges, which are subsets of measure zero, a phenomenon named scarring.  In the last chapter, titled Eigenvalue optimisation on flat tori and lattice points in anisotropically expanding domains [53], we turn our attention to the spectrum of the Laplacian on flat tori. We obtain bounds on R(𝜆) depending on the injectivity radius. We then use those bounds to obtain that any sequence of flat tori 𝕋K maximising the kth eigenvalue of the Laplacian must degenerate when dimension is inferior or equal to 10. To do so, we have stated the problem at hand in terms of counting points of ℤd inside anisotropically expanding domains, generalising results of Yuri Kordyukov and Andrei Yakovlev [49].

Keyphrases: Spectral geometry, Spectral asymptotics, Geometry of numbers, Density of states, Steklov problem, Weyl's law