Weyl Law on Asymptotically Euclidean Manifolds

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator $$Q=(1+|x|^2)(1-\varDelta )$$ Q = ( 1 + | x | 2 ) ( 1 - Δ ) on $$\mathbb {R}^d$$ R d .

Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains

This paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

Weyl asymptotics for Hanoi attractors

This paper studies the asymptotic behavior of the eigenvalue counting function of the Laplacian on some weakly self-similar fractals called Hanoi attractors. A resistance form is constructed and equipped with a suitable measure in order to obtain a Dirichlet form and its associated Laplacian. Hereby, the classical construction for p.c.f. self-similar fractals has to be modified by combining discrete and quantum graph methods.