A proof of a trace formula by Richard Melrose

The following result was proved by Chazarain [Ch-74] and refined by Duistermaat and Guillemin [DG-75] (see also [CdV-73, CdV-07] and Appendix F for the history of the trace formulae):

This result gives a nicer proof of the main result of my thesis [CdV-73] saying that, in the generic case, the Length spectrum, i.e. the set of lengths of closed geodesics, is a spectral invariant. This formula holds for any elliptic self-adjoint pseudo-differential operator $P$ of degree $1$ by replacing the periodic geodesics by the periodic orbits of the Hamiltonian flow of the principal symbol of $P$ and the Morse index by a Maslov index. Our goal is to prove that the same statement holds for sR Laplacians on a closed contact 3-manifold. What we mean here by ‘Melrose’s trace formula’ is the formulae of theorem 1, interpreted in this contact sR setting. Richard Melrose gave a proof which I found difficult (see Theorem 6.4 in [Me-84]).

For more details on this section, one can look at the paper [CHT-18].

First, we have the following

This result follows from the Riemannian case by passing to the limits (see Section 3 of [Me-86]).

We will also use the following Theorem due to Victor Ivrii, Bernard Lascar and Richard Melrose [Iv-76, La-82, Me-86] and revisited by Cyril Letrouit [Le-21]):

As a preparation, we will prove the local trace formula for the 3D-Heisenberg group $H_{3}$. The Laplacian $\Delta_{3}$ commutes with $Z$. We can hence use a partial Fourier decomposition of $L^{2}(H_{3})$ identifying it with the Hilbert integral

$$L^{2}({\mathbb{R}}^{3})=\int_{\mathbb{R}}^{\oplus}{\cal H}_{\zeta}d\zeta$$

where ${\cal H}_{\zeta}:=\{f|f(x,y,z+a)=e^{ia\zeta}f(x,y,z)\}$ which is identified to $L^{2}({\mathbb{R}}^{2})$ by looking at the value of $f$ at $z=0$. In what follows, we will omit the space ${\cal H}_{0}$ which corresponds to a flat 2D-Euclidian Laplacian. Using this decomposition, the Laplacian rewrites as follows:

$$\Delta_{3}=\sum_{l=0}^{\infty}(2l+1)\int_{\mathbb{R}}^{\oplus}|\zeta|K_{\zeta}^{l}d\zeta$$

where the operator $K_{\zeta}^{l}$ is the projector on the $l-$th Landau level of $\Delta_{\zeta}$, the restriction of $\Delta_{3}$ to ${\cal H}_{\zeta}$, which is a magnetic Schrödinger operator

$$H_{\zeta}:=-\left((\partial_{x}+i\zeta y/2)^{2}+(\partial_{y}-i\zeta x/2)^{2}\right)$$

on ${\mathbb{R}}^{2}$ with magnetic field $\zeta dx\wedge dy$. The Schwartz kernel of $K_{\zeta}^{l}$ satisfies

$$K_{\zeta}^{l}(m,m)=\frac{|\zeta|}{2\pi}$$

(see Appendix B).

Hence the half-wave operator writes

$$e^{it\sqrt{\Delta_{3}}}=\sum_{l=0}^{\infty}\int_{\mathbb{R}}^{\oplus}e^{it\sqrt{(2l+1)|\zeta|}}K_{\zeta}^{l}d\zeta$$

and the local distributional trace is given by

$${\rm Trace}\left(e^{it\sqrt{\Delta_{3}}}f\right)=\frac{1}{2\pi}\left(\int_{{\mathbb{R}}^{3}}f(q)|dq|\right)\sum_{l=0}^{\infty}\int_{\mathbb{R}}e^{it\sqrt{(2l+1)|\zeta|}}|\zeta|d\zeta$$

for $f\in C_{0}^{\infty}(H_{3})$. This trace can be explicitly computed by using the distributional Fourier transform

$$\int_{0}^{\infty}e^{i\tau u}u^{3}du=6(\tau+i0)^{-4}.$$

We get, for $t\neq 0$,

$${\rm Trace}\left(e^{it\sqrt{\Delta_{3}}}f\right)=\frac{6}{\pi t^{4}}\left(\sum_{l=0}^{\infty}\frac{1}{(2l+1)^{2}}\right)\int_{H_{3}}f|dq|.$$

In particular, this trace is smooth outside $t=0$ which is consistent with the fact that there is no periodic geodesic in $H_{3}$.

The same result holds for the trace formula microlocalized near $\Sigma$:

Proof.– We will first prove that

$${\rm Trace}\left(e^{it\sqrt{\Delta_{3}}}\chi\left(\frac{\Delta_{3}}{|Z|^{2}},q\right)\right)~{},$$

with $\chi\in C_{0}^{\infty}({\mathbb{R}}\times H_{3})$, which rewrites as

$$\frac{1}{2\pi}\sum_{l=0}^{\infty}\int_{H_{3}}|dq|\int_{\mathbb{R}}e^{it\sqrt{(2l+1)|\zeta|}}\chi\left(\frac{2l+1}{|\zeta|},q\right)|\zeta|d\zeta~{},$$

is smooth outside $t=0$. Let us define

$$I_{l}(t)=\int_{\mathbb{R}}e^{it\sqrt{(2l+1)|\zeta|}}\chi\left(\frac{2l+1}{|\zeta|},q\right)|\zeta|d\zeta~{}.$$

We split the integral into $\zeta>0$ and $\zeta<0$. Looking at the first integral and using the new variable $s=\sqrt{\zeta/2l+1}$, we get

$$I_{l}(t)=2(2l+1)^{2}\int_{0}^{\infty}e^{it(2l+1)s}\chi\left(\frac{1}{s^{2}},q\right)s^{3}ds~{}$$

Observing that the function $\chi\left(\frac{1}{s^{2}},q\right)s^{3}$ is a smooth classical symbol supported away of $0$, integrations by part give that $I_{l}(t)=O\left(l^{-\infty}\right)$ as well as its derivatives.

Then we introduce

$$\bar{P}:=\frac{1}{2\pi}\int_{0}^{2\pi}e^{it\Omega}Pe^{-it\Omega}dt$$

which is again a pseudo-differential operator. We check that $P-\bar{P}=[Q,\Omega]$ for a pseudo-differential operator Q: if

$$S_{t}:=\int_{0}^{t}e^{is\Omega}(P-\bar{P})e^{-is\Omega}dt,$$

this holds true with

$$Q=\frac{1}{2\pi i}\int_{0}^{2\pi}e^{-it\Omega}S_{t}e^{it\Omega}dt~{}.$$

The corresponding part of the trace vanishes, because $\Omega$ commutes with $\Delta_{3}$. Then the full symbol of $\bar{P}$ can be written as

$$\sum_{j=0}^{\infty}|\zeta|^{-j}p_{j}\left(\frac{I}{\zeta},q\right)$$

with $p_{j}$ compactly supported. This allows to reduce to the first case. $\square$

This section can be skipped. It contains an example with a direct derivation of Melrose’s trace formula.

Let us give a co-compact subgroup $\Gamma$ of $H_{3}$. We will prove Melrose’s trace formula for $M:=\Gamma\backslash H_{3}$ with Laplacian $\Delta_{M}$ which is $\Delta_{3}$, defined in Section 2.2, restricted to $\Gamma-$periodic functions. We fix some time $T>0$ and will look at the trace formula for $|t|\leq T$. We choose $\chi_{D}$ a smoothed fundamental domain, ie $\chi_{D}\in C_{0}^{\infty}(H_{3})$ with $\sum_{\gamma\in\Gamma}\chi_{D}(\gamma q)=1$. We will denote by $\int_{D}\cdots$ the integral $\int_{H_{3}}\chi_{D}\cdots$.

We start with

$$e_{M}(t,q,q^{\prime})=\sum_{\gamma\in\Gamma}e_{3}(t,q,\gamma q^{\prime})$$

where $e_{3}$ is the half-wave kernel in $H_{3}$. Because of the finite speed of propagation and the fact that $\Gamma$ is discrete, we have only to consider a finite sum for the trace:

$${\rm Trace}\left(e^{it\sqrt{\Delta_{M}}}\right)=\int_{D}e_{3}(t,q,q)|dq|+\sum_{\gamma\in\Gamma\setminus{\rm Id},~{}\min_{q}d(q,\gamma q)\leq T}\int_{D}e_{3}(t,q,\gamma q)|dq|$$

The first term is smooth outside $t=0$ while the second one is given by the CDG trace formula.

With more details: let $c>0$ be small enough so that the cone $C_{c}=\{g^{\star}<c\zeta^{2}\}$ has the property $(\cal P)$ that there is no $\gamma\in\Gamma\setminus{\rm Id}$ with a geodesic from $q$ to $\gamma q$ of length smaller than $T$ starting with Cauchy data in this cone. This is possible thanks to Lemma 2.1. We can hence split the integrals $I_{\gamma}(t)=\int_{D}e_{3}(t,q,\gamma q)|dq|$ into two pieces

$$I_{\gamma}(t)={\rm Trace}\left(\left(e^{it\sqrt{\Delta_{3}}}\right)\tau_{\gamma}\chi_{D}P\right)+{\rm Trace}\left(\left(e^{it\sqrt{\Delta_{3}}}\right)\tau_{\gamma}\chi_{D}({\rm Id}-P)\right)$$

with $\tau_{\gamma}(f)=f\circ\gamma^{-1}$ and $P=\psi(\Delta_{3}/|Z|^{2})$ where $\psi$ belongs to $C_{o}^{\infty}({\mathbb{R}})$, is equal to $1$ near $0$ and is supported in $]-{c},{c}[$. The first term is smooth by Theorem 3.2 and property $(\cal P)$. The second term corresponds to the elliptic region and hence we use the parametrix for the wave equation given by “FIO”s as given in the CDG trace formula. We get then that the singularities of the wave trace locate on the length spectrum.

Note that the ”heat trace” can be computed from the explicit expression of the spectrum. This is worked out in Appendix E.

In what follows, $M$ is a closed 3D sR manifold of contact type equipped with a smooth volume. We denote by $\Delta$ the associated Laplacian. The proof of the Melrose formula will be done by using a normal form allowing a reduction to the case of Heisenberg.

Let us fix $T>0$ and try to prove Melrose’s trace formula for times $t\in J:=[-T,T]$. Let us fix, for each $\sigma\in\Sigma$, a conical neighbourhood   $U_{\sigma}$ of $\sigma$ as in section 6.2. We then take $W_{\sigma}\Subset V_{\sigma}\Subset U_{\sigma}$ so that, for any $z\in V_{\sigma}$ and any $t\in J$, $G_{t}(z)\in U_{\sigma}$ where $G_{t}$ is the geodesic flow. This is clearly possible using the classical normal form and the Lemma 2.1. We then take a finite cover of $\Sigma$ by open cones $W_{\alpha}:=W_{\sigma_{\alpha}}$ and a finite pseudo-differential partition of unity $(\chi_{0},\chi_{\alpha}(\alpha\in B))$ so that $WF^{\prime}(\chi_{0})\cap\Sigma=\emptyset$, $\chi_{\alpha}={\rm Id}$ in $W_{\alpha}$ and $WF^{\prime}(\chi_{\alpha})\subset V_{\alpha}$. We have then to compute the traces of $\left(\cos t\sqrt{\Delta}\right)\chi_{0}$ and $\left(\cos t\sqrt{\Delta}\right)\chi_{\alpha}$. We will prefer to use the wave equation now because the operator $\sqrt{\Delta}$ is not a pseudo-differential operator! We know from Theorem 3.2 that, for $t\in J$, $WF^{\prime}\left(\cos(t\sqrt{\Delta})\chi_{\alpha}\right)$ is a subset of

$$\{(z,z,t,0)|z\in V_{\alpha}\}\cup\{(z,G_{\pm t}(z),t,\tau=\pm g^{\star}(z))|z\in V_{\alpha}\}.$$

If $u(t)=\cos(t\sqrt{\Delta})\chi_{\alpha}u_{0}$, we have $u_{tt}+\Delta u=0,u(0)=\chi_{\alpha}u_{0},u_{t}(0)=0.$ We can hence use the normal form and denote by $\equiv$ the equality “modulo smooth functions of $t\in J$” to get

$$Z_{\alpha}(t):={~\rm Trace}(\cos(t\sqrt{\Delta})\chi_{\alpha})\equiv{~\rm Trace}(\cos(t\sqrt{\Delta_{3}+R_{0}})\widetilde{\chi_{\alpha}}),$$

where $\widetilde{\chi_{\alpha}}$ is the PDO obtained by Egorov theorem when we take the normal form. Then, because $R_{0}$ commutes with $\Omega$,

$$Z_{\alpha}(t)\equiv\sum_{l=0}^{\infty}{~\rm Trace}\left(\cos\left(t\sqrt{(2l+1)\left(|Z|+\frac{1}{2l+1}R_{0}\right)}\right)\Pi_{l}\widetilde{\chi_{\alpha}}\right)$$

and

$$Z_{\alpha}(t)\equiv\sum_{l=0}^{\infty}{~\rm Trace}\left(A_{l}^{-1}\cos\left(t\sqrt{(2l+1)|Z|}\right)A_{l}\Pi_{l}\widetilde{\chi_{\alpha}}\right)$$

$$Z_{\alpha}(t)\equiv\sum_{l=0}^{\infty}{~\rm Trace}\left(\cos\left(t\sqrt{(2l+1)|Z|}\right)A_{l}\Pi_{l}\widetilde{\chi_{\alpha}}A_{l}^{-1}\right).$$

We can assume that $A_{l}$ is invertible on $WF^{\prime}(\chi_{\alpha})$ and put $\widetilde{\widetilde{\chi_{\alpha}^{l}}}=A_{l}{\widetilde{\chi_{\alpha}}}A_{l}^{-1}$. We get

$$Z_{\alpha}(t)\equiv\sum_{l=0}^{\infty}{~\rm Trace}\left(\cos\left(t\sqrt{(2l+1)|Z|}\right)A_{l}\Pi_{l}A_{l}^{-1}\widetilde{\widetilde{\chi_{\alpha}^{l}}}\right)$$

Using the fact $A_{l}$ commutes with $\Omega$ and hence with $\Pi_{l}$, we get finally

$$Z_{\alpha}(t)\equiv\sum_{l=0}^{\infty}{~\rm Trace}\left(\cos\left(t\sqrt{(2l+1)|Z|}\right)\Pi_{l}\widetilde{\widetilde{\chi_{\alpha}^{l}}}\Pi_{l}\right)$$

We can then apply a variant of the Proposition 4.1, more precisely of its proof, where $P$ is replaced by $\oplus_{l=0}^{\infty}\Pi_{l}\widetilde{\widetilde{\chi_{\alpha}^{l}}}\Pi_{l}$ and using the fact that the $A_{l}$’s and hence the $\widetilde{\widetilde{\chi_{\alpha}^{l}}}$ too are uniformly bounded pseudo-differential operators.

It remains to study the part $Z_{0}(t)={\rm Trace}\left(\cos(t\sqrt{\Delta})\chi_{0}\right)$ which involves the elliptic part of the dynamics for which we can use the FIO parametrix as in [DG-75]. More precisely, for $t\in J$, the geodesic flow maps the microsupport of $\chi_{0}$ away of $\Sigma$, therefore there exists a parametrix for $U(t)\chi_{0}$ given by Fourier integral operators as in the paper [DG-75] and the calculation of the trace therefore follows the same path.

Appendices