arXiv subjects

Jonathan Huntley

Publications and source records attributed to Jonathan Huntley.

4 recordsLinked to original sources

arXiv2003

Spectral theory of automorphic forms and analysis of invariant operators on $SL_3({\cal{Z}}$ with applications

We study a variety of problems in the spectral theory of automorphic forms using entirely analytic techniques such as Selberg trace formula, asymptotics of Whittaker functions and behavior of heat kernels. Error terms for Weyl's law and an analog of Selberg's eigenvalue conjecture for $SL_3({\bf Z})$ is given. We prove the following: Let $\cal H$ be the homogeneous space associated to the group $PGL_3(\bf R)$. Let $X = Γ{\backslash SL_3({\bf Z}})$ and consider the first non-trivial eigenvalue $λ_1$ of the Laplacian on $L^2(X)$. Using geometric considerations, we prove the inequality $λ_1 > 3pi^2/10> 2.96088.$ Since the continuous spectrum is represented by the band $[1,\infty)$, our bound on $λ_{1}$ can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space. Brief comment on relevance of automorphic forms to applications in high energy physics is given.

hep-th↗

arXiv1998

On an Analog of Selberg's Eigenvalue Conjecture for SL_3(Z)

Let H be the homogeneous space associated to the group PGL_3(R). Let X=Γ/H where Γ=SL_3(Z) and consider the first non-trivial eigenvalue λ_1 of the Laplacian on L^2(X). Using geometric considerations, we prove the inequality λ_1<pi^2/10. Since the continuous spectrum is represented by the band [1,\infty), our bound on λ_1 can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

math.SP↗

arXiv1998

Weyl's Law with Error Estimate

Let X=Sl(3,Z)\Sl(3,R)/SO(3,R). Let N(lambda) denote the dimension of the space of cusp forms with Laplace eigenvalue less than lambda. We prove that N(lambda)=C lambda^(5/2)+O(lambda^2) where C is the appropriate constant establishing Weyl's law with a good error term for the noncompact space X. The proof uses the Selberg trace formula in a form that is modified from the work of Wallace and also draws on results of Stade and Wallace and techniques of Huntley and Tepper. We also, in the course of the proof, give an upper bound on the number of cusp forms that can violate the Ramanujan conjecture.

hep-th↗

arXiv1994

On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces

We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T) = \sum\limits_{\lambda_n \leq T}(T-\lambda_n)^w $$ where $\{\lambda_n\}$ is the set of eigenvalues of the Laplacian which acts on the space of smooth functions on $M$. If $M$ is non-compact, then the weighted counting function is defined via the inverse Laplace transform. Now let $M_{\ell}$ denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface $M_{0}$. As an example of our results, we have the following theorem: There is an explicitly defined function $G_{\ell,w}(T)$ which depends solely on $\ell$, $w$, and $T$ such that for $w > 3/2$ and $T>0$, we have $$N_{M_{\ell},w}(T) = G_{\ell,w}(T) +N_{M_{0},w}(T) +o(1)$$ for $\ell \to 0$. We also consider the setting when $w < 3/2$, and we obtain a new proof of the continuity of small eigenvalues on degenerating hyperbolic Riemann surfaces of finite volume.

math.DG↗