Scattering resonances on truncated cones

TL;DR

This paper investigates the distribution of scattering resonances for the Laplacian on truncated Riemannian cones, providing explicit asymptotic formulas and extending previous results on non-truncated cones.

Contribution

It constructs the resolvent and scattering matrix for truncated cones and derives explicit asymptotic distribution of resonances, extending prior work on non-truncated cones.

Findings

Resonances on truncated cones are asymptotically distributed as A*r^n + o(r^n).

Laplacian on non-truncated cones has no resonances away from zero.

Explicit coefficient A in the resonance distribution formula.

Abstract

We consider the problem of finding the resonances of the Laplacian on truncated Riemannian cones. In a similar fashion to Cheeger--Taylor, we construct the resolvent and scattering matrix for the Laplacian on cones and truncated cones. Following Stefanov, we show that the resonances on the truncated cone are distributed asymptotically as Ar^n + o(r^n), where A is an explicit coefficient. We also conclude that the Laplacian on a non-truncated cone has no resonances away from zero.

Equations

$${\partial }_r^2+\frac{n-1}{r}{\partial }_r+\frac{1}{r^2}{\Delta }_h,$$

$$R(\lambda )=(\Delta +{\lambda }^2{)}^{-1}.$$

$$N(r)=\#\left\{\lambda :\lambda \text{ is a resonance and }|\lambda |\le r\right\}.$$

$$N(r)=A_n\mathrm{Vol}(Y,h)r^n+o(r^n),$$

$$L^2(C_a(Y);\mathbb{C})=\underset{j=0}{\overset{\infty }{⨁}}{L}^2((a,\infty );E_j),f(r,y)=\sum _{j=0}^{\infty }{f}_j(r){\varphi }_j(y),$$

$$R(\lambda )\left(\sum _{j=1}^{\infty }{f}_j(r){\varphi }_j(y)\right)=\underset{j=1}{\overset{\infty }{⨁}}\left(R_j(\lambda )f_j\right){\varphi }_j(y).$$

$$(R_j(\lambda )f)(r)={\int }_a^{\infty }{K}_{a,j}(r,\tilde{r})f(\tilde{r}){\tilde{r}}^{n-1}d\tilde{r}$$

$$K_{a,j}(r,\tilde{r})=\frac{\pi }{2i}{\left(\tilde{r}r\right)}^{-\frac{n-2}{2}}\{\begin{array}{ll}{H}_{{\nu }_j}^{(1)}(\lambda \tilde{r})J_{{\nu }_j}(\lambda r)-\frac{J_{{\nu }_j}(\lambda a)}{H_{{\nu }_j}^{(1)}(\lambda a)}{H}_{{\nu }_j}^{(1)}(\lambda \tilde{r})H_{{\nu }_j}^{(1)}(\lambda r)& r<\tilde{r}\\ J_{{\nu }_j}(\lambda \tilde{r})H_{{\nu }_j}^{(1)}(\lambda r)-\frac{J_{{\nu }_j}(\lambda a)}{H_{{\nu }_j}^{(1)}(\lambda a)}{H}_{{\nu }_j}^{(1)}(\lambda \tilde{r})H_{{\nu }_j}^{(1)}(\lambda r)& r>\tilde{r}\end{array}$$

$${\partial }_r^2{u}_j+\frac{n-1}{r}{\partial }_r{u}_j-\frac{{\mu }_j^2}{r^2}{u}_j+{\lambda }^2{u}_j=f_j.$$

$${\partial }_{\rho }^2\tilde{u}+\frac{n-1}{\rho }{\partial }_{\rho }\tilde{u}+\left(1-\frac{{\mu }_j^2}{{\rho }^2}\right)\tilde{u}=\frac{1}{{\lambda }^2}\tilde{f}(\rho ).$$

$$v^{\prime \prime }+\frac{1}{\rho }{v}^{\prime }+\left(1-\frac{{\nu }_j^2}{{\rho }^2}\right)v=g(\rho ),$$

$$w_1(r)=r^{-(n-2)/2}{J}_{{\nu }_j}(\lambda r),w_2(r)=r^{-(n-2)/2}{H}_{{\nu }_j}^{(1)}(\lambda r)$$

$$u_j(r)=\left({\int }_r^{\infty }\frac{w_2(\tilde{r})f_j(\tilde{r})}{W(w_1,w_2)(\tilde{r})}d\tilde{r}\right)w_1(r)+\left(C+{\int }_a^r\frac{w_1(\tilde{r})f_j(\tilde{r})}{W(w_1,w_2)(\tilde{r})}d\tilde{r}\right)w_2(r),$$

$$W(w_1,w_2)(r)=r^{-(n-1)}\cdot \frac{2i}{\pi }.$$

$$\left(\frac{\pi }{2i}{\int }_a^{\infty }{H}_{{\nu }_j}^{(1)}(\lambda \tilde{r}){\tilde{r}}^{\frac{n}{2}}f(\tilde{r})d\tilde{r}\right)a^{-(n-2)/2}{J}_{{\nu }_j}(\lambda a)+C{a}^{-(n-2)/2}{H}_{{\nu }_j}^{(1)}(\lambda a)=0,$$

$$C=-\frac{\pi }{2i}\frac{J_{{\nu }_j}(\lambda a)}{H_{{\nu }_j}^{(1)}(\lambda a)}{\int }_a^{\infty }{H}_{{\nu }_j}^{(1)}(\lambda \tilde{r}){\tilde{r}}^{\frac{n}{2}}f(x)dx,$$

$$\{\lambda \in \mathbb{C}:\Im \lambda >0\}$$

$$\lambda \mapsto ⟨\chi R(\lambda )\chi f,g⟩$$

$$f(r,y)=\sum _{j=0}^{\infty }{f}_j(r){\varphi }_j(y).$$

$$\Vert f{\Vert }_{L^2(C(Y))}^2={\int }_0^{\infty }\sum _{j=0}^{\infty }|f_j(r){|}^2{r}^{n-1}dr=\sum _{j=0}^{\infty }{\int }_0^{\infty }|f_j(r){|}^2{r}^{n-1}dr.$$

$$⟨\chi R(\lambda )\chi f,g⟩$$

$$+{\int }_0^{\infty }{\int }_r^{\infty }(\tilde{r}r{)}^{-\frac{n-2}{2}}\chi (r)\chi (\tilde{r})f_j(\tilde{r})g_j(r)J_{{\nu }_j}(\lambda r)H_{{\nu }_j}^{(1)}(\lambda \tilde{r}){\tilde{r}}^{n-1}{r}^{n-1}d\tilde{r}dr),$$

$$J_{\nu }(\lambda r)$$

$$H_{\nu }^{(1)}(\lambda r)$$

$$C{\int }_0^{\infty }{\int }_0^r\frac{1}{\pi {\nu }_j}\chi (r)\chi (\tilde{r})f_j(\tilde{r})g_j(r)\left[{\left(\frac{\tilde{r}}{r}\right)}^{{\nu }_j}(1+o(1))\right](\tilde{r}r{)}^{\frac{n}{2}}d\tilde{r}dr$$

$$+C{\int }_0^{\infty }{\int }_r^{\infty }\frac{1}{\pi {\nu }_j}\chi (r)\chi (\tilde{r})f_j(\tilde{r})g_j(r)\left[{\left(\frac{r}{\tilde{r}}\right)}^{{\nu }_j}(1+o(1))\right](\tilde{r}r{)}^{\frac{n}{2}}d\tilde{r}dr.$$

$$\frac{C_{\chi }}{{\nu }_j}\Vert f_j{\Vert }_{L^2}\Vert g_j{\Vert }_{L^2}.$$

$$Q(\lambda )={\tilde{\chi }}_0{R}_0(\lambda ){\chi }_0+{\tilde{\chi }}_{\infty }{R}_{\infty }(\lambda ){\chi }_{\infty },$$

$$-\frac{\pi }{2}<\arg \lambda <0\text{or}\pi <\arg \lambda <\frac{3\pi }{2}.$$

$$H_{\nu }^{(1)}(z{e}^{\pi })$$

$$H_{\nu }^{(1)}(\overline{z})$$

$$z=\pm (t\coth t-t^2{)}^{1/2}+i(t^2-t\tanh t{)}^{1/2},0\le t\le t_0,$$

Full text

1. Introduction

In this note, we consider the resonances on truncated Riemannian cones and establish a Weyl-type formula for their distribution. To fix notation, we let $(Y,h)$ be a compact $(n-1)$-dimensional Riemannian manifold (with or without boundary) and let $C(Y)$ denote the cone over $Y$. In other words, $C(Y)$ is diffeomorphic to the product $(0,\infty)_{r}\times Y$ and is equipped with the incomplete Riemannian metric $g=dr^{2}+r^{2}h$. We refer the reader to the foundational work of Cheeger–Taylor [4, 5] for more details on the geometric set-up. We also introduce the truncated Riemannian cone $C_{a}(Y)$ formed by introducing a boundary at $r=a$, i.e., $C_{a}(Y)$ is diffeomorphic to $[a,\infty)_{r}\times Y$ and equipped with the same metric.

The (negative-definite) Laplacian on $C(Y)$ (or $C_{a}(Y)$ with a choice of boundary conditions) has the form

[TABLE]

where $\Delta_{h}$ denotes the Laplacian of $(Y,h)$. Its resolvent $R(\lambda)$ is given by

[TABLE]

We consider the cutoff resolvent $\chi R(\lambda)\chi$, where $\chi$ is a (fixed) smooth compactly supported function on $C(Y)$ (or $C_{a}(Y)$). One consequence of the resolvent formula of Theorem 2.1 is that the cutoff resolvent extends meromorphically to the logarithmic cover of $\mathbb{C}\setminus\{0\}$.

More precisely, we identify elements $\lambda$ of the logarithmic cover of $\mathbb{C}\setminus\{0\}$ by a magnitude $|\lambda|$ and a phase $\arg\lambda\in\mathbb{R}$. We identify the “physical half-plane” as those $\lambda$ with $\arg\lambda\in(0,\pi)$. These $\lambda$ correspond to the resolvent set $\mathbb{C}\setminus[0,\infty)$ via the map $\lambda\mapsto|\lambda|^{2}e^{2i\arg\lambda}$. The cutoff resolvent then extends to be meromorphic as a function of $\lambda$ on this logarithmic cover.

The poles of the cutoff resolvent consist of possibly finitely many $L^{2}$-eigenvalues lying in the upper half-plane (which do not appear with Dirichlet boundary conditions) and poles lying on other sheets of the cover. The latter poles are called the resonances of $\Delta$.

The main theorem of this paper counts the most physically relevant resonances for the truncated cone. In particular, we count those resonances $\lambda$ nearest to the physical half-plane, i.e., those with $\arg\lambda\in(-\frac{\pi}{2},0)$ and $\arg\lambda\in(\pi,\frac{3\pi}{2})$. The resonances on other “sheets” of the cover remain more mysterious and are related to the zeros of Hankel functions near the real axis. We consider the resonance counting function on these sheets, defined by

[TABLE]

The following theorem provides an asymptotic formula for $N(r)$.

Theorem 1.1**.**

Suppose either that the set of periodic geodesics of $(Y,h)$ has Liouville measure zero or that $Y=\mathbb{S}^{n-1}$ equipped with a constant rescaling of the standard metric. Consider the truncated cone $C_{1}(Y)$ equipped with the Dirichlet Laplacian and let $N(r)$ denote its resonance counting function on the neighboring sheets as above. We then have, as $r\to\infty$,

[TABLE]

where $A_{n}$ is an explicit constant (defined below in equation (7)) and $\operatorname{Vol}(Y,h)$ denotes the volume of the Riemannian manifold $(Y,h)$.

The constant $A_{n}\operatorname{Vol}(Y,h)$ in Theorem 1.1 is the same constant as computed by Stefanov [11] for the resonance counting function on the domain exterior to a ball in $\mathbb{R}^{n}$. When $Y=\mathbb{S}^{n-1}$ is equipped with its standard metric, the truncated cone $C_{1}(Y)$ can can be thought of as the exterior of the unit ball in Euclidean space. Theorem 1.1 recovers Stefanov’s result. (When $Y=\mathbb{S}^{n-1}$, $n$ odd, is equipped with its standard metric, the cutoff resolvent in fact continues to the complex plane; this can be seen in the resolvent formulae below.)

We also state the following theorem, which is known to the community but does not seem to be in the literature.

Theorem 1.2**.**

If $(Y,h)$ is a compact Riemannian manifold (with or without boundary) then the cone $C(Y)$ has no resonances.

In fact, Theorem 2.1 below shows that $\lambda$ is a resonance of the truncated cone $C_{1}(Y)$ if and only if $\lambda/a$ is a resonance of the truncated cone $C_{a}(Y)$. Sending $a$ to [math] then pushes all resonances out to infinity and provides evidence for Theorem 1.2.

The proof of Theorem 1.1 has two main steps. We first separate variables and obtain an explicit resolvent formula in Theorem 2.1 to characterize the resonances as zeros of a Hankel function. In Section 3 we consider the asymptotic distribution of the zeros of each Hankel function appearing in the resolvent formula. The hypothesis on the link $(Y,h)$ is used to control the error terms when synthesizing the result. Theorem 1.2 is an immediate corollary of the resolvent formula in Theorem 2.1.

The proof of Theorem 1.1 follows an argument of Stefanov [11] very closely. Stefanov established a Weyl-type law for the distribution of resonances for the exterior of a ball in odd-dimensional Euclidean space. The main contribution of this paper is the observation that, after some natural modifications, the core of Stefanov’s argument applies to the setting of cones. Borthwick [1, 2] and Borthwick–Philipp [3] showed that a similar approach works in the asymptotically hyperbolic setting.

We further remark that we have specialized to the Dirichlet Laplacian in Theorem 1.1 only for simplicity. For Neumann or Robin boundary conditions, the resolvent formula of Theorem 2.1 has an analogous expression. The resonance counting problem then involves counting zeros of $H^{(2)^{\prime}}_{\nu}+C\nu H^{(2)}_{\nu}$, which can be handled with similar arguments.

2. Resolvent construction

In this section we write down an explicit formula (via separation of variables) for the resolvent and then show that the cut-off resolvent has a meromorphic continuation to the logarithmic cover $\Lambda$ of the complex plane. The construction is essentially contained in the work of Cheeger–Taylor [4, 5], but the resolvent is not explicitly written there.

Suppose that $\phi_{j}$ form an orthonormal family of eigenfunctions for $-\Delta_{h}$ with corresponding eigenvalues $\mu_{j}^{2}$. We decompose $L^{2}(C(Y))$ into a direct sum in terms of the eigenspaces of $-\Delta_{h}$, i.e.,

[TABLE]

where the first space is defined with respect to the volume form induced by the metric and the latter spaces can be identified (via the identification $f(r)\phi_{j}(y)\mapsto f(r)$) with the space $L^{2}((a,\infty);\mathbb{C})$ equipped with the volume form $r^{n-1}\,dr$.

For $\arg\lambda\in(0,\pi)$, the resolvent $R(\lambda)$ splits as a direct sum of operators $R_{j}(\lambda)$ acting on $L^{2}((a,\infty),E_{j})$, with measure $r^{n-1}\,dr$.

[TABLE]

In this section, we prove the following explicit formula for the $j$-th piece of the resolvent. For the cone $C(Y)$ (i.e., for $a=0$), we use the Friedrichs extension of the Laplacian to guarantee self-adjointness (though in high enough dimension the Laplacian is essentially self-adjoint):

Theorem 2.1**.**

The piece of the resolvent corresponding to the $j$-th eigenvalue has the following explicit expression on the truncated cone $C_{a}(Y)$ or the cone $C(Y)$ ($a=0$):

[TABLE]

where $K_{a,j}(r,\tilde{r})$ is given by

[TABLE]

Here $J_{\nu}$ are the standard Bessel functions of the first kind and $H^{(1)}_{\nu}$ are the Hankel functions of the first kind. The second term in both expressions should be interpreted as [math] when $a=0$.

Proof.

After separating variables, we may assume that $f=f_{j}(r)\phi_{j}(y)$. We construct the resolvent for $\Im\lambda>0$ and then meromorphically continue the expression.

Writing $u=u_{j}(r)\phi_{j}(y)$, the equation $(\Delta+\lambda^{2})u=f$ induces the following differential equation for $u_{j}$:

[TABLE]

We solve this equation by showing it is equivalent to a Bessel equation.

Changing variables to $\rho=\lambda r$ and writing $\tilde{u}(\rho)=u(\rho/\lambda)$ yields

[TABLE]

Writing $v=\rho^{(n-2)/2}\tilde{u}$, we obtain a Bessel equation for $v$:

[TABLE]

where $\nu_{j}^{2}=\mu_{j}^{2}+\left(\frac{n-2}{2}\right)^{2}$ and $g(\rho)=\frac{\rho^{(n-2)/2}}{\lambda^{2}}\tilde{f}(\rho)$.

We now proceed by the standard ODE technique of variation of parameters. One basis for the space of solutions of the homogeneous version of this Bessel equation is $\{J_{\nu_{j}}(\rho),H^{(1)}_{\nu_{j}}(\rho)\}$, where $J_{\nu}$ is the Bessel function of the first kind and $H^{(1)}_{\nu}$ is the Hankel function of the first kind. We thus may use the following basis for the space of solutions of the homogeneous equation:

[TABLE]

For $\Im\lambda>0$, $R_{j}(\lambda)f_{j}$ must lie in $L^{2}((a,\infty),r^{n-1}\,dr)$. If $f_{j}$ is compactly supported, this means that $u_{j}=R_{j}(\lambda)f_{j}$ must be a multiple of $r^{-(n-2)/2}H^{(1)}_{\nu_{j}}(\lambda r)$ near infinity. When $a>0$, $u_{j}$ must satisfy the boundary condition at $r=a$. When $a=0$, the choice of the Friedrichs extension requires that both $u_{j}$ and $u_{j}^{\prime}$ lie in the the weighted $L^{2}$ space near [math] and so $u_{j}$ must be a multiple of $r^{-(n-2)/2}J_{\nu_{j}}(\lambda r)$ near $r=0$ as any nonzero multiple of $w_{2}$ will not have this property.

We may thus write

[TABLE]

where $C$ is a yet-to-be-determined constant, the functions $w_{1}$ and $w_{2}$ are as in equation (3), and $W(w_{1},w_{2})$ is their Wronskian. The Wronskian $W$ can be easily computed in terms of the Wronskian of the Bessel and Hankel functions and seen to be

[TABLE]

We now turn our attention to the boundary condition. For $a=0$, the requirement that the solution and its derivative live in $L^{2}$ forces $C=0$, yielding the result. For $a\neq 0$, we require that $u_{j}(a)=0$, i.e.,

[TABLE]

and so we must have

[TABLE]

finishing the proof. ∎

We now claim that $\chi R(\lambda)\chi$ has a meromorphic continuation:

Lemma 2.2**.**

Given a fixed $\chi\in C^{\infty}_{c}(\mathbb{R}_{+}\times Y)$, $\chi R(\lambda)\chi$ meromorphically continues from

[TABLE]

to the logarithmic cover $\Lambda$ of the complex plane.

Proof.

We first prove the statement for the full cone; the statement for the truncated cone will follow by an appeal to the analytic Fredholm theorem.

Fix $\chi\in C^{\infty}_{c}((0,\infty))$ and regard $\chi(r)$ as a compactly supported smooth function on $C(Y)$. We let $R(\lambda)$ denote the resolvent on the non-truncated cone (i.e., $a=0$) and $K(\lambda;r,y,\widetilde{r},\widetilde{y})$ denote its integral kernel. In order to show that $\chi R(\lambda)\chi$ meromorphically continues, it suffices to show that for any $f,g\in L^{2}(C(Y))$, the function

[TABLE]

meromorphically continues to $\Lambda$.

Fix two such functions $f,g\in L^{2}(C(Y))$ and let $f_{j}(r)$ and $g_{j}(r)$ denote their coefficients in the expansion in terms of eigenfunctions of $\Delta_{h}$, i.e.,

[TABLE]

We observe that because $f$ and $g$ are square-integrable, the sum and the integral commute, i.e.,

[TABLE]

From Theorem 2.1, we may write

[TABLE]

where $J_{\nu}$ and $H_{\nu}^{(1)}$ are as above. Because each term in equation (2) meromorphically continues to the Riemann surface $\Lambda$, it suffices to show that the partial sums of the series converge locally (in $\lambda$) uniformly (in $j$).

By the asymptotic expansions of Bessel functions for large order, we know [6, 10.19] that, locally in $\lambda\in\Lambda$, and for $r\in\operatorname{supp}\chi$,

[TABLE]

as $\nu\to\infty$ through the positive reals. In particular, for $j$ large enough, each term in equation (2) can be bounded by

[TABLE]

Observe that in the first integral, $\widetilde{r}/r$ is bounded by $1$, while $r/\widetilde{r}$ is bounded by $1$ in the second.

Because $\chi$ is compactly supported, we may therefore bound each term (for $j$ large enough) by

[TABLE]

This sequence is absolutely summable, so the partial sums of the series in equation (2) converge locally uniformly. This establishes that the cut-off resolvent on the full cone ($a=0$) meromorphically extends to the logarithmic cover $\Lambda$ of the complex plane.

We now proceed to the case of the truncated cone ($a>0$). We proceed by an appeal to the analytic Fredholm theorem.

Fix $\chi_{0},\chi_{\infty}\in C^{\infty}((a,\infty))$ so that $\chi_{0}(r)$ is supported near $r=a$, $\chi_{\infty}(r)$ is identically zero near $r=a$, and $\chi_{0}+\chi_{\infty}=1$. We let $R_{\infty}(\lambda)$ denote the resolvent on the non-truncated cone and $R_{0}(\lambda)$ denote the resolvent on a compact manifold with boundary into which the support of $\chi_{0}$ embeds isometrically. We define the parametrix

[TABLE]

where $\tilde{\chi}$ have similar support properties and are identically $1$ on the support of their counterparts. Applying $\Delta+\lambda^{2}$ yields a remainder of the form $I+\sum[\Delta,\tilde{\chi}_{i}]R_{i}(\lambda)\chi_{i}$. Both terms are compact and the operator is invertible for large $\Im\lambda$ by Neumann series, so applying $R_{a}(\lambda)$ to both sides and inverting the remainder shows that it has a meromorphic continuation. ∎

3. Proof of Theorem 1.1

By the formula for the resolvent in Theorem 2.1, the resonances of $R_{a}(\lambda)$ correspond to those $\lambda$ for which $H_{\nu_{j}}^{(1)}(\lambda a)=0$ for some $j$. For simplicity we will discuss only the case $a=1$ as the other cases can be found by rescaling. As mentioned in the introduction, we consider only those resonances nearest to the upper half-plane, i.e., those with

[TABLE]

Because $\nu_{j}$ is real, we may relate the zeros of $H_{\nu_{j}}^{(1)}(\lambda)$ in the region given by equation (5) to zeros of $H^{(2)}_{\nu_{j}}(\lambda)$ in the quadrant $0<\arg\lambda<\frac{\pi}{2}$ via analytic continuation formulae. Indeed, it is well-known [6, 10.11.5, 10.11.9] that

[TABLE]

The first of these equations identifies zeros of $H^{(1)}_{\nu}$ in $\pi<\arg\lambda<\frac{3\pi}{2}$ to zeros of $H^{(2)}_{\nu}$ in the first quadrant; the second equation does the same for zeros of $H^{(1)}_{\nu}$ with $-\frac{\pi}{2}<\arg\lambda<0$. In particular, each zero of $H^{(2)}_{\nu}$ with $0\leq\arg\lambda\leq\pi/2$ corresponds to exactly two resonances.

For large enough $\nu$, the zeros of the Hankel function $H^{(2)}_{\nu}$ in the first quadrant lie near the boundary of (a scaling of) an “eye-like” domain $K\subset\mathbb{C}$. The domain $K$ is symmetric about the real axis and is bounded by the following curve and its conjugate:

[TABLE]

where $t_{0}$ is the positive root of $t=\coth t$. We refer to the piece of the boundary of $K$ lying in the upper half-plane by $\partial K_{+}$.

The constant $A_{n}$ given above is given by the following:

[TABLE]

where $B_{n-1}$ is the $(n-1)$-dimensional unit ball. Observe that, up to a factor of the volume of the unit sphere (which is replaced by the volume of $Y$ in the theorem statement), the constant $A_{n}$ is the same constant computed by Stefanov [11].

We use below two different parametrizations of the piece of $\partial K_{+}$ lying the in the quadrant $0\leq\arg z\leq\pi/2$. The first parametrization is by the argument of $z$, i.e., by the map

[TABLE]

For the second parametrization, we introduce the function $\rho$, defined by

[TABLE]

where (following Stefanov [11, Section 4] and Olver [10, Chapter 10]) the branches of the functions above are chosen so that $\zeta$ is real when $z$ is. Another characterization is that the principal branches are chosen when $0<z<1$ and continuity is demanded elsewhere.

The boundary $\partial K$ is the vanishing set of $\Re\rho$. This yields a parametrization of the part of $\partial K_{+}$ lying in $0\leq\arg z\leq\pi/2$:

[TABLE]

The transition between the two parametrizations is given by

[TABLE]

The function $\zeta$ defined in equation (8) is the solution of the ODE

[TABLE]

that is infinitely differentiable on the positive real axis (including at $z=1$). As is implicit in equation (8), it can be analytically continued to the complex plane with a branch cut along the negative real axis.

Because the resonances correspond to zeros of $H^{(2)}_{\nu_{j}}$, we must also consider the asymptotic distribution of the $\nu_{j}$. In what follows, we consider only the case when the periodic geodesics of $(Y,h)$ have measure zero.111When $(Y,h)$ is a sphere, the analysis is simplified slightly. In that case, one replaces the use of the Weyl formula with explicit formulae for the eigenvalues $\mu_{j}^{2}$ and their multiplicities. The eigenvalues $\mu_{j}^{2}$ of $\Delta_{h}$ obey Weyl’s law:

[TABLE]

Here $\operatorname{Vol}(B_{n-1})$ denotes the volume of the unit ball in $\mathbb{R}^{n-1}$ and $\operatorname{Vol}(Y,h)$ is the volume of $Y$ equipped with the metric $h$. In general, $R(\mu)=O(\mu^{n-2})$, but if we now impose the dynamical hypothesis (that the set of periodic geodesics of $(Y,h)$ has Liouville measure zero), then a theorem of Duistermaat–Guillemin [7] (in the boundaryless case) and Ivrii [8, 9] (in the boundary case) shows that

[TABLE]

The non-periodicity assumption then allows us to count eigenvalues on intervals of length one:

[TABLE]

As $\nu_{j}^{2}=\mu_{j}^{2}+(n-2)^{2}/4$, the same counting formula holds for $\nu_{j}$, i.e.,

[TABLE]

We now turn our attention to the zeros of the Hankel function $H_{\nu}^{(2)}(z)$ with $\arg z\in[0,\pi/2]$. An argument from Watson [12, pages 511–513] is easily adapted to give a precise count of the number of zeros of $H^{(2)}_{\nu}$ in this sector. Indeed, that argument shows that the number of zeros is given by the closest integer to $\nu/2-1/4$ (when $\nu-1/2$ is an integer, there is a zero on the imaginary axis and so rounds up).

As $\nu\to\infty$ through positive real values, we have an asymptotic expansion [6, 10.20.6] relating the Hankel function to the Airy function

[TABLE]

Here $A_{k}$ and $B_{k}$ are real and infinitely differentiable for $\zeta\in\mathbb{R}$. This expansion is uniform in $|\arg z|\leq\pi-\delta$ for fixed $\delta>0$. In particular, for large enough $\nu$, the zeros of the Hankel function are well-approximated by zeros of the Airy function and we may identify each zero $h_{\nu,k}$ of the Hankel function $H^{(2)}_{\nu}$ with a zero of the Airy function $\operatorname{Ai}(-z)$.

Let $a_{k}$ denote the $k$-th zero of the Airy function $\operatorname{Ai}(-z)$; all $a_{k}$ are positive and

[TABLE]

We now define $\lambda_{\nu,k}$ and $\widetilde{\lambda}_{\nu,k}$ via the Airy zeros and their leading approximations:

[TABLE]

where $k=1,\dots,\lfloor\nu/2+1/4\rfloor$. By the Hankel expansion (10), $|h_{\nu,k}-\lambda_{\nu,k}|\leq C/\nu$ for large enough $\nu$ while $|h_{\nu,k}-\widetilde{\lambda}_{\nu,k}|\leq C/\nu$ for large enough $\nu$ and $k$. As we have identified $\lfloor\nu/2+1/4\rfloor$ approximate zeros, we can conclude that these account for all $h_{\nu,k}$.

We now divide our attention into those zeros with small argument and those with large argument. We introduce the auxiliary counting function

[TABLE]

We first address those with small argument. Fix $\epsilon>0$ and consider those zeros with $|z|<r$ and $\arg z\in[0,\epsilon]$. We need count those $\lambda_{\nu,k}$ with $\arg\lambda_{\nu,k}\in[0,\epsilon]$ and $|\lambda_{\nu,k}|\leq r$. As $|\lambda_{\nu,k}|$ is comparable to $\nu$, we can overcount these zeros by counting all $\lambda_{\nu,k}$ with argument in $[0,\epsilon]$ and $\nu\leq Cr$.

Because $|\rho|\leq C\epsilon^{3/2}$ for those $\lambda_{\nu,k}$ with $\arg\lambda_{\nu,k}\in[0,\epsilon]$, we must only count those $a_{k}$ with $a_{k}\leq C\nu^{2/3}\epsilon$. The leading order asymptotic [6, 9.9.6] for the zeros of the Airy function shows that this number is $O(\nu\epsilon^{3/2})$.

We now count those resonances with argument in $[0,\epsilon]$. Putting together the asymptotic for $\nu_{j}$ in equation (3) with the previous two paragraphs, we have (with $m(\nu_{j})$ denoting the multiplicity of $\nu_{j}$)

[TABLE]

We now consider those resonances with argument in $[\epsilon,\pi/2]$. For large enough $\nu$, the approximations $\widetilde{\lambda}_{\nu,k}$ are valid for these resonances. We count those approximate resonances with $\nu_{j}\in[\rho,\rho+1)$ and $\arg\lambda_{\nu,k}\in[\theta,\theta+\Delta\theta]$. We start by introducing, for fixed $\nu$, the number $\Delta k_{\nu}$ of $\widetilde{\lambda}_{\nu,k}$ with argument lying in $[\theta,\theta+\Delta\theta]$. Observe that the definition of $\widetilde{\lambda}_{\nu,k}$ relates $\Delta k_{\nu}$ with $\Delta t$ by

[TABLE]

where $\Delta t$ denotes the change in $t$ corresponding to $\Delta\theta$ in the parametrizations above. Note that $\Delta t$ is independent of the choice of $\nu$. We can then write

[TABLE]

By the definition of the approximate zeros $\widetilde{\lambda}_{\nu,k}$, we can estimate their size $|\widetilde{\lambda}_{\nu,k}|$ in terms of $|z(\theta)|$, provided that $\arg\widetilde{\lambda}_{\nu,k}\in[\theta,\theta+\Delta\theta]$, yielding

[TABLE]

In particular, if $\nu_{j}|z(\theta)|\geq r$ but $|\lambda_{\nu,k}|\leq r$, then $\nu_{j}\in\left[\frac{r}{|z(\theta)|}(1-c\Delta\theta),\frac{r}{|z(\theta)|}\right]$. We may thus rewrite our counting function as follows:

[TABLE]

By our improved Weyl’s law (3), the second term is $O(r^{n-2})$.

We now focus our attention on the first term (here $\lfloor\cdot\rfloor$ denotes the “floor” function):

[TABLE]

Again by Weyl’s law, we observe that the second term is $O(r^{n-2})$. By relating $\Delta t$ and $\Delta k_{\nu}$ we can rewrite the first term:

[TABLE]

By Weyl’s law (3), the second term is $O(r^{n-1})$, so we again consider the first term.

As $\Delta t$ is independent of $\nu_{j}$, we may use Weyl’s law as well on the first term:

[TABLE]

We finally introduce a Riemann sum in $t$ to understand this main term:

[TABLE]

Here the prefactor of $2$ disappeared because the first integral parametrizes only half of $\partial K_{+}$. It reappears in the statement of Theorem 1.1 because each zero here corresponds to two resonances (one on each sheet). We further observe that the constant $A_{n}\operatorname{Vol}(Y,h)$ agrees with the leading term found in the Euclidean case found by Stefanov [11].

Sending $\epsilon$ to [math] establishes the theorem for the approximate zeros $\lambda_{\nu,k}$. Because each $\lambda_{\nu,k}$ is in a $C/\nu$ neighborhood of a zero $h_{\nu,k}$, this finishes the proof of the theorem.

Acknowledgments

Part of this research formed the core of the second author’s Master’s project at Texas A&M University. DB acknowledges partial support from NSF grants DMS-1500646 and DMS-1654056. The authors also thank David Borthwick, Tanya Christiansen, Colin Guillarmou, and Jeremy Marzuola for helpful conversations.