Harmonic-counting measures and spectral theory of lens spaces

Abstract

In this article, associated with each lattice T ⊆ Z n the concept of a harmonic-counting measure ν T on a sphere S n−1 is introduced and it is applied to determine the asymptotic behavior of the eigenfunctions of the Laplace-Beltrami operator on a lens space. In fact, the asymptotic behavior of the cardinality of the set of independent eigenfunctions associated with the elements of T which lie in a cone is determined when T is the lattice of a lens space.

Recent results on the spectra of lens spaces

“…. , a n ) ∈ Z n satisfying that n i=1 a i s i is divisible by q. Mohades and Honari in [MH16] defined the harmonic-counting measure associated to L as the measure ν L on the Borel σ-algebra of the unit sphere S n−1 as (7. where C(U ) = {aµ : µ ∈ U, a > 0} (the cone in R n induced by U ). When U = S n−1 , (3.10) tells us that the numerator on the right hand side in the above formula equals the number of eigenvalues (counted with multiplicities) of the Laplace-Beltrami operator on L less than or equal to λ ⌊t⌋ = ⌊t⌋(⌊t⌋+2n−2).…”

“…The last section contains brief discussions on several recent articles related to the spectral theory of lens spaces. Namely, the work on the spectrum of the Dirac operator on spin lens spaces by Boldt [Bo17] and Boldt-Lauret [BL17]; the computational studies on p-isospectral lens spaces ( [GM06,La19]), the work by Bari and Hunsicker ([Ba11, BH19]) on the spectra of lens orbifolds, the article [La16] extending the one-norm method to other compact symmetric spaces of real rank one, and the harmonic counting measure introduced in [MH16].…”

A Computational Study on Lens Spaces Isospectral on Forms

“…The articles [LMR16b], [BL17], [La16a] and [La16b] follow this approach and [DD14] study in detail the examples of all-p-isospectral pairs in [LMR16a]. The articles [MH16a], [MH16] are also related to this approach.…”

The spectrum on p-forms of a lens space

“…Hence, the above mentioned characterization for 0-isospectral lens spaces can be stated as follows: L and L ′ are 0-isospectral if and only if L and L ′ are · 1 -isospectral. (See [BL17], [LMR16b], [La16], [MH17], [MH16] for related results. )…”