On Weyl’s Asymptotics and Remainder Term for the Orthogonal and Unitary Groups

Abstract

We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by Avakumovic (Math Z 65:327–344, [1]) and Hörmander (Acta Math 121:193–218, [15]) and show that for the scale of orthogonal and unitary groups $\mathbf{SO}(N)$, $\mathbf{SU}(N)$, $\mathbf{U}(N)$ and $\mathbf{Spin}(N)$ it is not sharp. While for negative sectional curvature improvements are possible and known, cf. e.g., Duistermaat and Guillemin (Invent Math 29:39–79, [7]), here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for $\mathbf{U}(N)$]. Furthermore here the improvements are sharp and quantitative relating to the dimension and rank of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum.

Asymptotics of Weighted Integrals Involving Bessel Functions

In this appendix we present the proof of an estimate used earlier in the paper. This is concerned with the asymptotics of Bessel functions and their weighted integrals.

Proposition 6.1

Let $\alpha \ge 2$ and $\beta >-1/2$. Then there exist constants $M>0$ and $c>0$ such that for any $z \ge M$ we have

$$\begin{aligned} \int _0^z t^{\alpha +\beta } J_{\beta }(t) \, dt \le c z^{\alpha +\beta -\frac{1}{2}}. \end{aligned}$$

(90)

Proof

Using the identity established in Lemma 6.1 below we can write

$$\begin{aligned} \int _{0}^{z} t^{\alpha +\beta } J_{\beta }(t) \, dt&= z^{\alpha +\beta }J_{\beta +1}(z) - (\alpha -1) \int _0^z t^{\alpha +\beta -1}J_{\beta +1}(t) \, dt. \end{aligned}$$

Next in virtue of the asymptotic decay of Bessel function at infinity (see, e.g., Stein and Weiss [30]) it follows that there exists $M>0$ such that $J_{\beta +1}(z) \le {cz^{-\frac{1}{2}}}$ for $z>M$. Hence we can write

$$\begin{aligned} \int _{0}^{z} t^{\alpha +\beta } J_{\beta }(t) \, dt \le c_1 z^{\alpha +\beta -\frac{1}{2}} + c_2 \int _M^z t^{\alpha +\beta -\frac{3}{2}} \, dt \end{aligned}$$

from which the conclusion follows at once. $\square$

Lemma 6.1

Let $\alpha \ge 2$ and $\beta >-1/2$ with $z \in {\mathbb R}$.Then the following identity holds

$$\begin{aligned} \int _{0}^{z} t^{\alpha +\beta } J_{\beta }(t) \, dt = z^{\alpha +\beta }J_{\beta +1}(z) - (\alpha -1) \int _0^z t^{\alpha +\beta -1}J_{\beta +1}(t) \, dt \end{aligned}$$

(91)

Proof

Starting from the following weighted integral identity for Bessel functions (cf., e.g., Grafakos [11] Appendix B.3)

$$\begin{aligned} \int _0^z J_{\beta -1}(t) t^\beta \, dt = z^{\beta } J_{\beta }(z) \end{aligned}$$

we can write

$$\begin{aligned} \int _0^z t^{\alpha +\beta -1}J_{\beta +1} (t) dt&= \int _0^z t^{\alpha -2} \int _0^t s^{\beta +1}J_{\beta }(s) \, ds dt = \int _0^z s^{\beta +1}J_{\beta }(s) \int _s^z t^{\alpha -2} \, dt ds. \end{aligned}$$

Now integrating the term on the right gives

$$\begin{aligned} \int _0^z t^{\alpha +\beta -1}J_{\beta +1} (t) \, dt&= \frac{1}{\alpha -1}\int _0^z s^{\beta +1}J_{\beta }(s) [z^{\alpha -1}-s^{\alpha -1}] \, ds \\&= \frac{1}{\alpha -1}z^{\alpha +\beta }J_{\beta +1}(z) - \frac{1}{\alpha -1}\int _0^z s^{\beta +\alpha }J_{\beta }(s) \, ds \end{aligned}$$

and so the conclusion follows by simple manipulation of the above. $\square$