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 …
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7 results for “Pleijel”
New proof of Zelditch's generalization using Riemannian geometry.
problem Asymptotic formula for eigenfunction sums on compact manifolds.
method Riemannian geometry methods.
result Proof of Zelditch's generalization without Fourier integral operators.
This note bounds Courant-sharp eigenvalues using geometric invariants.
problem Bounding Courant-sharp eigenvalues using geometric invariants.
method Determining an upper bound for Courant-sharp eigenvalues in terms of simple geometric invariants.
result An upper bound for Courant-sharp eigenvalues is derived.
The paper estimates the gap between eigenvalues of a clamped plate problem.
problem Estimating the gap between eigenvalues of a clamped plate problem.
method Using the asymptotic formula of Agmon and Pleijel, the paper gives an estimate for the gap between eigenvalues.
result The gap between eigenvalues is bounded by a term with a lower order $k^{ rac1n}$.
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 (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 nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …
Combines noncommutative geometry and spectral theory for new Weyl laws.
problem Developing new Weyl laws for noncommutative manifolds.
method Functional analysis, spectral theory, and Tauberian conditions.
result Generalizes and simplifies recent results on Weyl laws and integration formulas.