arXiv research

We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results …

New proof of Zelditch's generalization using Riemannian geometry.

Let $Ω\subset \mathbb R^d\,, d\geq 2$, be a bounded open set, and denote by $λ\_j(Ω), j\geq 1$, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues $λ\_j(Ω)$, for which there exists an associated eigenf…

The paper estimates the gap between eigenvalues of a clamped plate problem.

We consider the heat kernel (and the zeta function) associated with Laplace type operators acting on a general irreducible rank 1 locally symmetric space X. The set of Minakshisundaram- Pleijel coefficients {A_k(X)}_{k=0}^{\infty} in the short-time asymptotic expansion of the heat kernel is calculated explicitly.

We address the question of determining the eigenvalues $λ\_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …

Combines noncommutative geometry and spectral theory for new Weyl laws.