---
abstract: 'We consider a family of manifolds with a class of degenerating warped product metrics $g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$, with $M$ compact, $\rho$ homogeneous degree one, $a \le -1$ and $b > 0$. We study the Laplace operator acting on $L^{2}$ differential $p$-forms and give sharp accumulation rates for eigenvalues near the bottom of the essential spectrum of the limit manifold with metric $g_{0}$.'
author:
- Jeffrey McGowan
title: Bounds on Accumulation Rates of Eigenvalues on Manifolds with Degenerating Metrics
---

Introduction
============

There are many examples of non-compact manifolds which can be thought of as a ‘limit’ of a sequence of compact manifolds. Particularly nice examples are hyperbolic manifolds in dimensions 2 and 3; the cusp closing theorem of Thurston [@Thurston] then says that every complete, non-compact manifold $M_0$ is the limit of a sequence of hyperbolic manifolds $M_k \to M_0$. Since the Laplacian on $M_0$ has continuous spectrum, one expects the eigenvalues of $M_k$ to accumulate. In dimension 2, Ji, Zworski, and Wolpert ([@Ji; @JiZworski; @Wolpert1; @Wolpert2]) have given bounds for the accumulation rate of eigenvalues near the bottom of the essential spectrum in the hyperbolic case, while in dimension 3 analogous results were obtained by Chavel and Dodziuk ([@ChavelDodziuk]). Dodziuk and McGowan obtained similar results for the Laplacian acting on differential forms ([@DM]).

Colbois and Courtois considered convergence of eigenvalues below the bottom of the essential spectrum in a much more general setting [@CC]. The accumulation rate for eigenvalues of the Laplacian on functions for manifolds $N=\tilde{N}\cup (M^n\times I)$ with ’pseudo-hyperbolic’ metrics on $(M^n\times I)$ was given by Judge [@Judge]. Judge also computes the essential spectrum for a more general class of degenerating metrics, and investigates the convergence of eigenfunctions.

We will consider manifolds $N_{\epsilon}=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with $n=dim(M)$, and a family of metrics $$\label{metric}g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$. Here $\rho = c_1\epsilon +c_2t$, $c_1,c_2>0$, $t\in I=[0,1]$, $a\leq -1$, $b>0$, and $ds_M^2$ is the metric on $M^n$. We identify the boundary of $\tilde{N}$ with $M^{n}\times {1}$. These are the metrics discussed by Melrose in [@Melrose] and considered by Judge in [@Judge]. We consider only non-negative values of $t$ with $t \in [0,1]$, which simplifies the statements of the results, although we must consider manifolds with boundary. The condition $a \leq -1$ means that the limiting manifold $N_{0}$ is complete.

We study the accumulation rate for eigenvalues near the bottom of the essential spectrum of the Laplacian acting on both functions and differential forms. Our main results are

\[th2\] Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho$ as above. Let $$R=\int_0^1\rho(\epsilon,s)^a\,ds$$ be the geodesic distance from the boundary ${0} \times M^{n})$ of $N_{\epsilon}$ to $\tilde{N}$. Let $\Xi_\epsilon(x^2)$ be the number of eigenvalues of the Laplacian acting on coexact $p$-forms (satisfying absolute boundary conditions on the boundary of $N_{\epsilon}$) in $[\sigma,\sigma + x^2)$ where $\sigma$ is the bottom of the essential spectrum for coexact forms of degree $p$ and $0 < p < n$. Then $$\Xi_\epsilon(x^2) =\frac{dxR}{\pi} + O_x(1)$$ where $d$ is the dimension of the space of harmonic forms of degree $p$ on $M$.

This agrees with the results of Judge [@Judge], Chavel and Dodziuk [@ChavelDodziuk] and Dodziuk and McGowan [@DM] in the special cases they considered.

\[th1\] Suppose $N_\epsilon$ is as in Theorem \[th1\]. Then the essential spectrum of the Laplacian acting on coexact $p$-forms, $0 \leq p \leq n$ on $N_0$ is $$\begin{array}{lcr}
\left[\left(\frac{n-2p}{2}\right)^2c_2^2b^2,\infty\right)&\qquad&a=-1\\\\
\left[0,\infty\right)&\qquad&a<-1
\end{array}$$

Note that this agrees with Judges results ([@Judge]) for functions when $p=0$, and with Mazzeo and Phillips results for the essential spectrum on geometrically finite hyperbolic manifolds ([@MazzeoPhillips], with $c_2=b=1$ and $a=-1$). We have recently learned that these results for the essential spectrum have been obtained independently by Antoci (\[Antoci\]).

This paper is organized as follows. In Section \[geom\] we discuss the geometry of the manifolds under consideration, and rewrite the metric (\[metric\]) in a way which makes the geometry more evident. In Section \[functions\] we illustrate our techniques by computing the essential spectrum and accumulation rates for eigenvalues as $\epsilon \to 0$ in the case of functions ($p=0$). In Section \[upperforms\] we compute the essential spectrum and give lower bounds on the accumulation rate in the $p\neq 0$ case. Finally, in Section \[lowerforms\] we give upper bounds on the accumulation rate for $p \neq 0$, completing the proof of Theorem \[th1\].

We wish to thank Józef Dodziuk for many helpful conversations.

The Geometry {#geom}
============

Metrics of the type (\[metric\]) are discussed by Melrose in [@Melrose]. When $a \leq -1$ such metrics are complete on the limit manifold $N_{0}$. Melrose classifies metrics where $a=-1$, $b=1$ as ’hc’, or hyperbolic cusp metrics, and metrics where $a=-1$, $b=0$ as ’boundary’, or metrics with cylindrical end. Since we will consider metrics where $a \leq -1$, $b > 0$ we rewrite the metric to make the geometry more evident.

Let $\tau$ be the geodesic distance from $t=0$ to $t=1$, in other words the geodesic distance from the a point $(0,p \in M)$ to $\tilde{N}$. Then $$\label{taudef}
\tau = \int_0^t{\rho(\epsilon,s)^a\,ds}= \int_0^t{(c_1\epsilon + c_2s)^a\,ds}$$ and we have two distinct cases, $$\begin{array}{lcc}
\tau =\frac{1}{c_2}\left( \ln\left(\frac{{c_1\epsilon +c_2t}}{c_1\epsilon}\right)\right)&\qquad&a=-1\\
\tau = \frac{1}{c_2}\left(\frac{(c_1\epsilon+c_2t)^{a+1}-(c_1\epsilon)^{a+1}}{c_2(a+1)}\right)&\qquad&a<-1
\end{array}$$

Solving for $t$ and substituting into the metric (\[metric\]) we get $$\label{mymetrics}
\begin{array}{lcc}
ds^2=d\tau^2+(c_1\epsilon)^{2b}e^{2bc_2\tau}ds_M^2&\qquad&a=-1\\
ds^2=d\tau^2+(c_2(a+1)\tau+(c_1\epsilon)^{a+1})^\frac{2b}{a+1}ds_M^2&\qquad&a<-1
\end{array}$$ which is of the form $ds^2=d\tau^2+f_\epsilon(\tau)ds_M^2$ in both cases. As $\epsilon \to 0$, $\tau\to \infty$, and we have a warped product $I \times_{f_\epsilon} M$, with the length of the interval given by $$\label{rcalc}
\tau(1)=\Bigg\{\begin{array}{lr}R = \frac{1}{c_2}\ln\left(\frac{c_1\epsilon+c_2}{c_1\epsilon}\right)&a=-1\\R = \frac{(c_1\epsilon+c_2)^{a+1}-(c_1\epsilon)^{a+1}}{c_2(a+1)}&a<-1\end{array}$$

When $a=-1$, $f_e(\tau)$ gives an essentially hyperbolic metric; one thinks of pinching off a closed geodesic, with $\epsilon$ the length of that geodesic. When $a < -1$, the cross sections $M$ shrink at a slower rate as one recedes from $\tilde{N}$, and the warped product $I \times_{f_\epsilon} M$ is intermediate between a hyperbolic cusp and a flat cylinder.

Clearly, for any fixed $\epsilon$ and $\tau$, the cross section $\{\tau\}\times f_\epsilon(\tau)M$ has injectivity radius bounded below by some constant. Moreover, $f_\epsilon(\tau)$ is an increasing function of $\tau$ for all $a \le -1$. Since $M$ is compact,a scaling argument shows that the first non-zero eigenvalue of the Laplacian acting on coexact forms of degree $p$ on $M_{\epsilon,\tau}$, say $\nu_{p,\epsilon}(\tau)$, is a decreasing function of $\tau$. Hence, as the geodesic distance of a given cross section from $\tilde{N}$ increases, $\nu_{p,\epsilon}(\tau)$ increases. This allows us, for technical reasons, to restructure our decomposition of $N$ as follows;$$N=\tilde{N}'\cup\left(M\times [0,R-r_0+1]\right),$$ where $\tilde{N}' = \tilde{N} \cup M \times [R-r_0,R]$. $r_0$ will be chosen so that $\nu_{p,\epsilon}(r_0)$ is relatively large.

Functions
=========

We follow essentially the argument in [@DM]. First, we choose a function $f$ whose restriction to $\tilde{N}'$ is orthogonal to a basis of eigenfunctions with eigenvalues less than or equal to $\sigma + x^2$. This will only change the counting function $N_\epsilon(x^2)$ by a bounded amount which can be absorbed into the $O_x(1)$ term ([@ChavelDodziuk Lemma 3.6]). Next, we decompose $f$ on $M\times [0,R-r_0]$ as $f=\bar{f}+\bar{\bar{f}}$, where $\bar{f}$ depends only on $\tau$ and $\bar{\bar{f}}$ is orthogonal to constants on $M$. $\bar{f}$ is computed by averaging over each cross section.

Now, if we choose $r_0$ so that $\nu_{p,\epsilon}(r_0)>\sigma + x^2$, then $\bar{\bar{f}}$ does not contribute to the counting function $N_\epsilon(x^2)$. Concentrating on $\bar{f}$, a straightforward calculation shows that $$\label{funSL}
\Delta\bar{f}=*d*d\bar{f}=\frac{1}{f_\epsilon^{\frac{n}{2}(\tau)}}\frac{d}{d\tau}\left(\frac{d\bar{f}}{d\tau}f_\epsilon^\frac{n}{2}(\tau)\right)$$ is a classical Sturm-Liouville problem, and we can convert to the form (see [@CourantHilbert1]) $$u''-ru=\lambda u$$ with $$\begin{aligned}
u& = & f_\epsilon^\frac{n}{4}(\tau)\bar{f} \\
r & = &\frac{( f_\epsilon^\frac{n}{4}(\tau))''}{f_\epsilon^\frac{n}{4}(\tau)}\end{aligned}$$ When $a=-1$, $$r=\left(\frac{nbc_2}{2}\right)^2,$$ and (\[funSL\]) becomes $$u''=\left(\lambda + \left(\frac{nbc_2}{2}\right)^2\right)u.$$ We get

\[fun1\] Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{-2}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho = c_1\epsilon + c_2 t$. Then the essential spectrum of the Laplacian acting on functions on $N_0$ is $$\left[\left(\frac{nc_2b}{2}\right)^2=\sigma,\infty\right).$$ Let $R$ be as in (\[rcalc\]) and let $N_\epsilon(x^2)$ be the number of eigenvalues of the Laplacian acting on function in $[\sigma,\sigma + x^2)$. Then $$N_\epsilon(x^2) = \frac{xR}{\pi}+O_x(1).$$

This is as in [@Judge], with slightly different notation.

When $a<-1$, $$r=\frac{(a+1)bn(bn-2a-2)c_{2}^{2}}{2(c_{2}(a_+1)\tau+(c_{1}\epsilon)^{a+1})^{2}}.\label{integrable}$$ The potential (\[integrable\]) is integrable, and [@ChavelDodziuk Theorem 4.1] tells us the counting function for the corresponding Sturm-Liouville problem has the same asymptotics as if the potential were identically 0. Hence,

Suppose $N$ is as in Proposition \[fun1\] with metric $$g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, where $a < -1$ and $\rho$ as above. Then the essential spectrum of the Laplacian acting on functions on $N_0$ is $
\left[0,\infty\right).$ Let $R$ be as in (\[rcalc\]) and let $N_\epsilon(x^2)$ be the number of eigenvalues of the Laplacian acting on function in $[0,x^2)$. Then $$\label{Nfun}N_\epsilon(x^2) = \frac{xR}{\pi}+O_x(1).$$

The essential spectrum in this case was given by Judge ([@Judge]). The accumulation estimate (\[Nfun\]) can also be obtained using Judge’s techniques ([@Judge2]).

Upper eigenvalue bounds for forms {#upperforms}
=================================

We consider the sequence of eigenvalues of the Laplacian acting on coexact forms of degree $p$, $0 < \nu_{1} \le \nu_{2}\le \cdots \to \infty$. If we can give an upper bound $y\ge \nu_{j}$ for some $j$, we will obtain a lower bound for the counting function $\Xi_{\epsilon}(y)  \le j$. We work in the space $\mathcal{E}$ of $C^{\infty}$ coexact forms of degree $p$ on $N_{\epsilon}$ with support contained in $\{x|1 \le d(x,\tilde{N})\le R\}$, with coefficients which depend only on $\tau$. Any form $\omega \in \mathcal{E}$ is zero on $\tilde{N}$, and we choose forms $$\label{omega}\omega=\sum_{i=1}^{d}b_{i}d\tau\wedge H_{i}$$ where $d$ is the dimension of the space of harmonic $p$ forms on the cross section $M$ and $H_{i},i=1,2,\ldots,d$ is a basis of harmonic $p$ forms on $M$.

Using Courant’s min-max principle the eigenvalues $\nu_{j}$ are no greater than the critical values of the Rayleigh-Ritz quotient $(\Delta \omega,\omega)/(\omega,\omega))$ with $\omega \in \mathcal{E}$. Since $\omega$ is coexact, we have $$\frac{(\Delta \omega,\omega)}{(\omega,\omega))}=\frac{(d\omega,d\omega)}{(\omega,\omega)}$$ Since the $b_{i}$ depend only on $\tau$, and an application of $d$ to the sum in (\[omega\]) involves only derivatives with respect to other basis elements, we compute $$d{\omega} = \sum_{i=1}^{d}b_{i}'d\tau\wedge H_{i}$$ where the prime indicates differentiation with respect to $\tau$. Computing the respective $L^{2}$ norms we have $$\begin{aligned}
(\omega,\omega)&=&C_M\sum_{i=1}^{d}\int_{1}^{R}b_{i}^{2}f_{\epsilon}^{\frac{n-2p}{2}}(\tau)\,d\tau\label{bottom}\\
(d\omega,d\omega)&=&C_M\sum_{i=1}^{d}\int_{1}^{R}(b_{i}')^{2}f_{\epsilon}^{\frac{n-2p}{2}}(\tau)\,d\tau
\label{top}\end{aligned}$$ $C_{M}$ can be computed by integrating the basis elements of the cross section $M$.

Using integration by parts in the numerator of the Rayleigh-Ritz quotient, we get $d$ copies of a Sturm-Liuoville problem very similar to the one in section \[functions\],$$-\frac{1}{f_{\epsilon}^{\frac{n-2p}{2}}}\left(b_{i}'f_{\epsilon}^{\frac{n-2p}{2}}\right)'=\lambda b_{i}.$$ As in Section \[functions\], we reduce to the form $$u''-ru=\lambda u.$$ In the pseudo-hyperbolic case, when $a=-1$, we have $$r=\left(\frac{n-2p}{2}\right)^{2}c_{2}^{2}b^{2}$$ and we get

\[lowerform\] Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{-2}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho=c_{1}\epsilon+c_{2}t$. If $N_\epsilon(x^2)$ is the number of eigenvalues of the Laplacian acting on coexact $p$-forms in $[\left(\frac{n-2p}{2}\right)^{2}c_{2}^{2}b^{2},\left(\frac{n-2p}{2}\right)^{2}c_{2}^{2}b^{2}+ x^2)$ , then $$N_\epsilon(x^2) \ge \frac{dxR}{\pi} + O_x(1)$$ where $d$ is the dimension of the space of harmonic forms of degree $p$ on $M$. In this case, $R = \frac{1}{c_2}\ln\left(\frac{c_1\epsilon+c_2}{c_1\epsilon}\right)$, and letting $\epsilon \to 0$, we see that the essential spectrum of the Laplacian acting on coexact $p$-forms, $0 \leq p \leq n$ on $N_0$ is $\left[\left(\frac{n-2p}{2}\right)^2c_2^2b^2,\infty\right)$ if $d \ne 0$.

When $a < -1$ a messy but straightforward calculation give $$r=\frac{c_{2}^{2}b(n-2p)(\frac{b(n-2p)}{2}-1}{2(c_{2}(a+1)\tau + (c_{1}\epsilon)^{a+1})^{2}}.$$ This is an integrable potential, and we get

\[lowerform2\] Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho=c_{1}\epsilon+c_{2}t$. If $N_\epsilon(x^2)$ is the number of eigenvalues of the Laplacian acting on coexact $p$-forms in $[0, x^2)$ , then $$N_\epsilon(x^2) \ge \frac{dxR}{\pi} + O_x(1)$$ where $d$ is the dimension of the space of harmonic forms of degree $p$ on $M$. In this case, $R = \frac{(c_1\epsilon+c_2)^{a+1}-(c_1\epsilon)^{a+1}}{c_2(a+1)}$, and letting $\epsilon \to 0$, we see that the essential spectrum of the Laplacian acting on coexact $p$-forms, $0 \leq p \leq n$ on $N_0$ is $\left[0,\infty\right)$ if $d \ne 0$.

Lower eigenvalue bounds for forms {#lowerforms}
=================================

We will use the method of [@McGowan Lemma ?] to get global lower eigenvalue bounds for forms on $N_{\epsilon}$ based on lower eigenvalue bounds on local eigenvalue bounds on (overlapping) pieces of $N_{\epsilon}$. In particular, we use the idea of constructing a globally defined form while keeping control of the Rayleigh-Ritz quotient as in [@DM]. For details on the underlying Čech-de Rham formalism see [@BottTu Chapter 2]. The pieces we will consider might have mildly singular boundaries, but all the familiar results of Hodge theory hold ([@Cheeger1; @Cheeger2], see also [@McGowan Section ?] and [@DM Section 4]). We omit many details here, but refer the reader especially to [@DM] if they wish to fill in the blanks.

First, we pick a simple open cover of $N_{\epsilon}$ consisting of two pieces; $U_{1}=\tilde{N}'\setminus\partial\tilde{N}'$, and $U_{2}=M\times [0,R-r_{0}+1]$. Recall that $\tilde{N}' = \tilde{N} \cup M \times [R-r_0,R]$, so $U_{1}$ and $U_{2}$ overlap, with $U_{1}\cap U_{2}=M\times [R-r_{0},R-r_{0}+1]$. Next, we choose a coexact $p$ form $\phi$ so that the restriction $\phi|_{U_{1}}=\phi_{1}$ is orthogonal to the finite dimensional space of exact eigenforms (on $U_{1}$) of degree $p+1$ with eigenvalue less than or equal to $y^{2}$. This is possible using [@DM Proposition 5.1]; the proof must be modified somewhat to account for the more general setting here, but the modifications are simple if messy. We will specify values for $r_{0}$ and $y^{2}$ later.

Now, since $\phi_{1}$ is assumed to be exact with eigenvalue greater than or equal to $y^{2}$, there exists a unique coexact form $\psi_{1}$ of degree $p$ on $U_{1}$ with $d\psi_{1}=\phi_{1}$ and $$\frac{(\phi_{1},\phi_{1})}{(\psi_{1},\psi_{1})}=\frac{(d\psi_{1},d\psi_{1})}{(\psi_{1},\psi_{1})}\ge y^{2}.$$ Likewise, by exactness, there exists a unique coexact form $\psi_{2}$ on $U_{2}$ with $d\psi_{2}=\phi_{2}$, but we do not yet have any bounds on the Rayleigh-Ritz quotient (and hence on eigenvalues) $$\label{RR}\frac{(\phi_{2},\phi_{2})}{(\psi_{2},\psi_{2})}=\frac{(d\psi_{2},d\psi_{2})}{(\psi_{2},\psi_{2})}.$$

Next, we wish to decompose $\phi_{i}$, $i=1,2$ on $M\times [0,R]$ in a similar fashion to our decomposition for functions at the beginning of Section \[functions\]. We will model ourselves on the argument in [@DM], but since the cross section ${\tau} \times M^{n}$ is arbitrary here, we cannot just average coefficients. Rather, we use a harmonic projection. First, decompose $\phi_{i} = \alpha \wedge d\tau + \beta$, where $\beta$ does not contain $d\tau$. Next, use harmonic projection on $\alpha$ to get, $$\alpha = \sum_{i=1}^{d}a_{i}d\tau\wedge H_{i}  + \gamma$$ where $d$ is the dimension of the space of harmonic $p$ forms on $M$, $H_{i}$ is a basis of harmonic $p$ forms on $M$, and the $a_{i}$ depend only on $\tau$.

Now, we can write $\phi_{i}=\bar{\phi_{i}}+\bar{\bar{\phi_{i}}}$, with $$\begin{aligned}
\label{decomp}
\bar{\phi_{i}} &=&  \sum_{i=1}^{d}a_{i}d\tau\wedge H_{i}\\
\bar{\bar{\phi_{i}}} &=& \phi_{i}-\bar{\phi_{i}}  = \beta+\gamma\end{aligned}$$ We do the same for $\psi_{i}$. By construction, the coefficients of $\bar{\phi_{i}}$ and $\bar{\psi_{i}}$ depend only on $\tau$. A straightforward calculation (see, for example, [@Dodziuk]), shows that as the metric on the cross sections scales by a factor $f_{\epsilon}(\tau)$, the Rayleigh-Ritz quotient scales as $$\frac{(d\bar{\bar{\phi_{i}}},d\bar{\bar{\phi_{i}}})|_{g_{_{\epsilon}}}} {(\bar{\bar{\phi_{i}}},\bar{\bar{\phi_{i}}})|_{g_{_{\epsilon}}}} = \frac{1}{f_{\epsilon}(\tau)}\left(\frac{(d\bar{\bar{\phi_{i}}},d\bar{\bar{\phi_{i}}})|_{g_{_{1}}}}{(\bar{\bar{\phi_{i}}},\bar{\bar{\phi_{i}}})|_{g_{_{1}}}}\right).$$ For small $\tau$, $f_{\epsilon}(\tau)$ is small, and thus if $r_{0}$ is chosen appropriately, $\bar{\bar{\phi_{i}}}$ will not contribute to any accumulation of eigenvalues.

So far, we have put only a finite number of conditions, depending only on $x$, on our original selection of $\phi$. These conditions guarantee that $\phi_{1}$ is orthogonal to the finite dimensional space of exact eigenforms (on $U_{1}$) of degree $p+1$ with eigenvalue less than or equal to $y^{2}$. We still need to determine how many additional choices we must make to gain control of the Rayleigh-Ritz quotient (\[RR\]). By construction, we can write $$\bar{\phi_{2}}=\sum_{i=1}^{d}a_{i}d\tau\wedge H_{i}$$ where $d$ is the dimension of the space of harmonic $p$ forms on $M$, $H_{i}$ is a basis of harmonic $p$ forms on $M$, and the $a_{i}$ depend only on $\tau$. Consequently, we can write $$\bar{\psi_{2}}=\sum_{i=1}^{\zeta}f_{i}\,d\tau\wedge\alpha_{i,p-1}+\sum_{i=1}^{d}b_{i}H_{i}$$ with $d\bar{\psi_{2}}=\bar{\phi_{2}}$, $a_{i}=b_{i}'$, and the prime denoting differentiation with respect to $\tau$.

To evaluate the Rayleigh-Ritz quotient on $U_{2}$, we use $$\begin{aligned}
(\bar{\psi_{2}},\bar{\psi_{2}})&=&C_M\sum_{i=1}^{d}\int_{0}^{R-r_{0}}b_{i}^{2}f_{\epsilon}^{\frac{n-2p}{2}}(\tau)\,d\tau\\
(\bar{\phi_{2}},\bar{\phi_{2}})&=&C_M\sum_{i=1}^{d}\int_{0}^{R-r_{0}}(b_{i}')^{2}f_{\epsilon}^{\frac{n-2p}{2}}(\tau)\,d\tau\end{aligned}$$ and we again have $d$ copies of $$-\frac{1}{f_{\epsilon}^{\frac{n-2p}{2}}}\left(b_{i}'f_{\epsilon}^{\frac{n-2p}{2}}\right)'=\lambda b_{i}.$$ Letting $\sigma$ be the bottom of the essential spectrum for $N_{0}$ we see that the number of eigenvalues in the interval $[\sigma,\sigma+x^{2})$ for the equation $\Delta_{p}\psi_{2}= \nu \psi_{2}$ is given by $\frac{dxR}{\pi}+O_{x}(1)$. Thus, we can choose $\phi$ in such a way that $\psi_{2}$ is orthogonal in $L^{2}$ to the basis of eigenforms with eigenvalues less than $\sigma + x^{2}$ on $U_{2}$ by imposing $\frac{dxR}{\pi}+O_{x}(1)$ conditions. The number of conditions imposed on the choice of $\phi$ depends only on $x$.

We have chosen $\phi$ in such a way that we have control over the relevant Rayleigh-Ritz quotients on both $U_{1}$ and $U_{2}$, but it is not the case that $\psi_{1}$ and $\psi_{2}$ must match on $U_{1}\cap U_{2}$. Since $d\psi_{1}=\phi|_{U_{1}}$ and $d\psi_{2}=\phi|_{U_{2}}$ it is clear that the difference $\psi_{2}-\psi_{1}$ must be exact, and we use the method of [@McGowan Section ?] to build a globally defined form $\psi$ with $d\psi=\phi$ and with control over the Rayleigh-Ritz quotient. Since our open cover has only two pieces, this produces no difficulties. If we choose $y^{2}$ above in such a way that summing the relevant Rayleigh-Ritz inequalities gives the correct lower eigenvalue bound, we have

Suppose $N_\epsilon=\tilde{N}\cup (M^n\times I)$, $\tilde{N}$ and $M^n$ compact, with metric $$g_\epsilon=\rho(\epsilon,t)^{2a}dt^2 +\rho(\epsilon,t)^{2b}ds_M^2$$ on $M^n \times I$, with $\rho$ as above, and $d$ the dimension of the space of harmonic $p$-forms on $M^{n}$. Let $$R=\int_0^1\rho(\epsilon,s)^a\,ds.$$ Let $N_\epsilon(x^2)$ be the number of eigenvalues of the Laplacian acting on coexact $p$-forms in $[\sigma,\sigma + x^2)$ where $\sigma$ is the bottom of the essential spectrum for coexact forms of degree $p$ and $0 < p < n$. Then $$N_\epsilon(x^2) =\frac{dxR}{\pi} + O_x(1)$$ where $d$ is as in theorem \[th1\].

In the special case when $d=0$, i.e. when there are no harmonic forms on the cross section, we have the following corollary,

Suppose $N$ is as above, with $d=0$ for some $p$. Then the essential spectrum of the Laplacian acting on exact forms of degree $p$ is empty.

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