In this section, we first revisit the construction of Hadamard projectors for second-order hyperbolic equations acting on sections of Hermitian vector bundles.

Hadamard projectors act on distributional Cauchy data and project on Cauchy data whose solutions have wavefront sets in one of the two energy shells \(\mathcal ^\). If the Hermitian bundle is Hilbertian, i.e., if the fiber scalar product is positive definite, they produce Hadamard states for the associated quantum fields. In general, they produce only Hadamard pseudo-states.

We then consider second-order elliptic equations, typically obtained by Wick rotation of the hyperbolic equations in the time variable, and we construct the associated Calderón projectors. We also study Dirichlet-to-Neumann maps, which will be important in later sections. Finally, we show that Hadamard and Calderón projectors coincide modulo smoothing operators.

5.1 \(\Psi \)DO Calculus on Manifolds of Bounded Geometry

The constructions in this section rely on a global pseudodifferential calculus on a Cauchy surface \(\Sigma \). Namely, we use Shubin’s calculus which we now quickly recall.

Let \((M, \hat)\) be a Riemannian manifold of bounded geometry and \(V\xrightarrow M\) a finite rank complex vector bundle of bounded geometry.

One can then define for \(m\in }}\) the symbol classes \(S^(T^M; L(V))\) of poly-homogeneous symbols, see [52] or [24, Sect. 5]. Using a bounded atlas \(\, \psi _)\}_}}}\) and associated local trivializations of V and partition of unity \(1= \sum _\chi _^\), one can define a quantization map:

$$\begin \,}}: S^(T^M; L(V))\rightarrow L(^\infty _\textrm(M; V)), \end$$

\(\,}}(a)\) being a (classical) pseudodifferential operator of order m. Choosing a different atlas, trivializations or partition of unity produces of course in general a different quantization map \(\,}}'\). However,

$$\begin \,}}(a)- \,}}'(a)\in }}^(M; V), \end$$

where

$$\begin }}^(M; V):=\bigcap _}}}B(H^(M; V), H^(M; V)), \end$$

is an ideal of smoothing operators. Similarly, if \(\Omega \subset M\) is an open set, we set

$$\begin }}^(\Omega ; V):=\bigcap _}}}B(H^(\Omega ; V),H^(\Omega ; V)), \end$$

and if \((M_, \hat_)\) and \(V_\xrightarrow M_\) are of bounded geometry:

$$\begin }}^(M_, M_; V_, V_):=\bigcap _}}}B(H^(M_; V_), H^(M_; V_)). \end$$

One sets then

$$\begin \Psi ^(M; V)= \,}}(S^(T^M; L(V)))+ }}^(M; V), \end$$

and \(\Psi ^(M; V)= \bigcup _}}}\Psi ^(M; V)\).

We refer the reader to [52, App. 1] and [24, Sect. 5] for more details.

5.2 Lorentzian Case

We set \(M= I_\times \Sigma _}\), where \( I\subset }}\) is an interval with \(0\in \mathring\) and \((\Sigma , _)\) a d-dimensional Riemannian manifold of bounded geometry. We set \(\Sigma _= \\times \Sigma \) and identify \(\Sigma _\) with \(\Sigma \). The dual variables to \((t, \textrm)\) are denoted by \((\tau , \textrm)\).

We fix a t-dependent Riemannian metric on \(\Sigma \),

$$\begin :I \ni t\mapsto (t)\in C^_\textrm(I; _^(\Sigma , _)). \end$$

We assume that \((0)= _\) and for ease of notation we often denote \((t)\) by \(_\).

We equip M with the Lorentzian metric

$$\begin :=-dt^+ _d\textrm^. \end$$

5.2.1 Hermitian Bundle

We fix a finite rank complex vector bundle \(V\xrightarrow \Sigma \) of bounded geometry over \((\Sigma , _)\). We still denote by V the vector bundle over M: \(I\times V\xrightarrow M\) which is a vector bundle with the same fibers as V. We have for example

$$\begin C^_\textrm(M; V)\sim C^_\textrm(I; C^_\textrm(\Sigma ; V)). \end$$

We assume that \(V\xrightarrow M\) is equipped with a non-degenerate fiberwise Hermitian structure \((\cdot | \cdot )_\), which is assumed to be independent of t.

We fix a reference fiberwise Hilbertian structure \((\cdot | \cdot )_}\) on the fibers of V, which is also independent of t.

We denote by \((\cdot | \cdot )_}\), \((\cdot | \cdot )_\) the same Hermitian structures acting on the fibers of \(V\xrightarrow \Sigma \). If \(x\in M\) and \(u, v\in V_\), we have

$$\begin (u| v)_= (u| \tau _v)_}, \ \ \tau _\in L(V_), \end$$

and we denote by \(\tau \in C^\infty (M; L(V))\) the corresponding section, which is independent of t.

Note that \(\tau = \tau ^\). By polar decomposition, after possibly changing \((\cdot | \cdot )_}\), we can assume that

$$\begin \tau ^\tau = \varvec,\hbox \tau \hbox (\cdot | \cdot )_}. \end$$

We assume that the Hermitian structures \((\cdot | \cdot )_\) and \((\cdot | \cdot )_}\), and hence \(\tau \), are of bounded geometry.

If \(a\in L(V_)\) for \(x\in M\), we denote by \(a^\), resp.  \(a^\), the adjoints of a for \((\cdot | \cdot )_}\), resp.  \((\cdot | \cdot )_\). Then,

$$\begin a^= \tau ^_x a^\tau _x \end$$

for some \(\tau _\in L(V_)\).

For \(u, v\in C^\infty _\textrm(M; V)\), we set

$$\begin \begin (u|v)_(\Sigma _)}:=\int _}(u|v)_}|_|^}d\textrm,\\ (u|v)_)}:=\int _}(u|v)_|_|^}d\textrm= (u| \tau v)_(\Sigma _)},\\ (u| v)_(M)}:=\int _(u(t)|v(t))_}|_|^}dtd\textrm,\\ (u| v)_:=\int _(u(t)|v(t))_|_|^}dtd\textrm= (u| \tau v)_(M)}. \end \end$$

If \(\Omega \subset M\) is some open set, we also denote

$$\begin \begin (u| v)_(\Omega )}:=\int _(u(t)|v(t))_}|_|^}dtd\textrm,\\ (u| v)_:=\int _(u(t)|v(t))_|_|^}dtd\textrm= (u| \tau v)_(\Omega )}. \end \end$$

We denote by \(L^(\Sigma ; \tilde)\) the \(L^\) space obtained from the Hilbertian scalar product \((\cdot | \cdot )_(\Sigma )}\).

5.2.2 Adjoints

If \(a\in C^_\textrm(I; \textrm(\Sigma ; V))\), resp.  \(A\in \textrm(M; V)\), we denote by \(a^\) resp.  \(A^\) its formal adjoint for \((\cdot | \cdot )_(\Sigma _)}\) resp. \((\cdot | \cdot )_(M)}\). We set \(\operatorname a= \frac(a+ a^)\).

We denote by \(a^\) resp.  \(A^\) its formal adjoint for \((\cdot | \cdot )_)}\) resp. \((\cdot | \cdot )_\). As above we have:

$$\begin a^= \tau ^ a^\tau , \quad A^= \tau ^A^\tau . \end$$

5.2.3 Hyperbolic Operator

We fix a t-dependent differential operator \(a= a(t, \textrm, D_})\) belonging to \(C^_\textrm(I; \textrm^(\Sigma ; V))\) and denote by \(\sigma _\textrm(a)\in C^\infty (T^\Sigma ; L(V))\) its principal symbol.

We assume the following properties:

$$\begin \begin \mathrm& \ a(t) = a^(t),\ t\in I, \\ \mathrm& \sigma _\textrm(a)(t)(\textrm, \textrm)= \textrm\!\cdot \!_^(\textrm)\textrm\,\varvec_, \ t\in I. \end \end$$

We set

$$\begin D:=\partial _^+ a(t)\hbox ^\infty _\textrm(M; V), \end$$

which is a hyperbolic operator with scalar principal part. Note that \(D= D^\), but of course \(D\ne D^\) in general.

5.2.4 Green’s Formula

We set

$$\begin \varrho u= \begin \\ ^\partial _u(0)}\end, \ \ u\in C^_\textrm(M; V), \end$$

(5.1)

and equip \(^\infty _\textrm(\Sigma ; V\otimes }}^)\) with the Hilbertian scalar product \((\cdot | \cdot )_(\Sigma )\otimes }}^}\) defined by

$$\begin (f|f)_(\Sigma )\otimes }}^}:=(f_| f_)_(\Sigma )}+ (f_| f_)_(\Sigma )}. \end$$

The Green identity in Lemma 2.2 takes the form:

$$\begin (u| Dv)_(\Sigma ))}- (Du| v)_(\Sigma ))}= \pm \textrm^(\varrho u| q \varrho v)_(\Sigma )\otimes }}^}, \end$$

(5.2)

where

$$\begin q= \begin0 & \tau \\ \tau & 0 \end. \end$$

The Hilbertian vector bundle \(V_\) in 2.1.1 equals \(\tilde(\Sigma )\otimes }}^\), since the operator D is of second order.

5.2.5 Symplectic Adjoint

If A is an operator acting on \(^\infty _\textrm(\Sigma ; \tilde\otimes }}^)\), we denote by \(A^\) its adjoint for q, i.e.,

$$\begin A^= q^A^q. \end$$

5.2.6 Square Root of a(t)

We first construct an approximate square root a(t) adapted to our future needs.

We refer e.g. to [39, Sect. V.10] for the definition of m-accretive and sectorial operators.

Lemma 5.1

There exist \(r_\in C^_\textrm(I; \Psi ^(\Sigma ; V))\) and \(\epsilon \in C^_\textrm(I; \Psi ^(\Sigma ; V))\) such that:

(1)

\(\epsilon = \epsilon ^, \ \epsilon ^= a+ r_\),

(2)

\(\sigma _\textrm(\epsilon )= (k\!\cdot \!_^(\textrm)k)^}\varvec_\),

(3)

\(\epsilon \) with domain \(H^(\Sigma ; V)\) is m-accretive with \(\operatorname \epsilon \geqslant 1\).

Proof

Note first that a is uniformly elliptic in \(C^_\textrm(I; \Psi ^(\Sigma ; V))\) hence has a parametrix \(a^\in C^_\textrm(I; \Psi ^(\Sigma ; V))\). Therefore, a(t) is closed with domain \(H^(\Sigma ; \tilde)\) and \(\,}}a^(t)= H^(\Sigma ; \tilde)\). We set

$$\begin 2a_\textrm:=\frac(a+ a^)+ \tau ^\frac(a+ a^)\tau . \end$$

It satisfies:

$$\begin \begin i)& a- a_\textrm\in C^_\textrm(I; \textrm^(\Sigma ; V)),\\ ii)& a_\textrm= a_\textrm^= a_\textrm^.\\ \end \end$$

(5.3)

In fact, (5.3) i) follows from (H2), and (5.3) ii) from the fact that \(\tau = \tau ^= \tau ^\).

Clearly \(a_\textrm\) with domain \(H^(\Sigma ; \tilde)\) is self-adjoint for \((\cdot | \cdot )_(\Sigma )}\). We fix \(\chi \in ^\infty _\textrm(}})\) with \(\chi (0)=1\) and set \(\chi _(\lambda )= \chi (R^\lambda )\) for \(R\geqslant 1\). Since \(a_\textrm\) is elliptic, we know that \(\chi _(a_\textrm)\in C^_\textrm(I; \Psi ^(\Sigma ; V))\).

We set now

$$\begin r_= R\chi _(a_\textrm), \end$$

where \(R\gg 1\) will be chosen below. From (5.3), we deduce that

$$\begin r_= r_^= r_^ \end$$

and

$$\begin \operatorname (a+ r_) = a_\textrm+ R\chi _(a_\textrm)+ a_ \end$$

(5.4)

for some \(a_\in C^_\textrm(I; \textrm^(\Sigma ; V))\).

By the self-adjoint functional calculus, we can find \(R\gg 1\) such that:

$$\begin a_\textrm + R\chi (R^a_\textrm)\geqslant \fraca_\textrm+ 1, \end$$

(5.5)

and hence by (5.4)

$$\begin \operatorname (a+ r_)\geqslant 1. \end$$

(5.6)

The same inequality is valid for \((a+ r_)^\). If follows that For \(\operatorname z\geqslant 0 \), \(a+ r_+ z: H^(\Sigma ; \tilde)\rightarrow L^(\Sigma ; \tilde)\) is injective with a dense range. (5.6) also implies that \(\,}}( a+ r_+z)\) is closed, using that a is closed. Therefore, \(a+ r_\) is m-accretive. By [39, Thm. V.3.35], \(a+ r_\) has a unique m-accretive square root:

$$\begin \begin \epsilon&:=\pi ^\int _^ \lambda ^}(a+ r_+ \lambda )^(a+ r_)d\lambda \\&= 2\pi ^\int _^(a+ r_+ s^)^(a+ r_)d\lambda , \end \end$$

(5.7)

where the integrals are strongly convergent on \(\,}}a = H^(\Sigma ; \tilde)\). By [39, Pb. V.3.39], we have

$$\begin \operatorname \epsilon \geqslant 1. \end$$

(5.8)

Arguing as in [24, Subsect. 5.3], using the representation of \(\epsilon \) in the second line of (5.7), we obtain that \(\epsilon \in C^_\textrm(I; \Psi ^(\Sigma ; V))\) with

$$\begin \sigma _\textrm(\epsilon )= (k\!\cdot \!_^(\textrm)k)^}\varvec_. \end$$

The operator \(\epsilon (t)\) with domain \(H^(\Sigma ; \tilde)\) is closed, elliptic, m-accretive and invertible by (5.8), hence \(\epsilon ^\in C^_\textrm(I; \Psi ^(\Sigma ; V))\). From (5.7), we obtain that \(\epsilon = \epsilon ^\). \(\square \)

5.2.7 Factorization of D Proposition 5.2

There exists \(b\in C^_\textrm(I; \Psi ^(\Sigma ; V))\) unique modulo \(C^_\textrm(I; \Psi ^(\Sigma ; V))\) such that

$$\begin \begin i)& b= \epsilon + C^_\textrm(I; \Psi ^(\Sigma ; V)),\\ ii)& \textrm\partial _b- b^+ a= r_\in C^_\textrm(I; \Psi ^(\Sigma ; V)),\\ iii)& \begin b+ b^= (2\epsilon )^}(\varvec+ r_)^(2\epsilon )^}, \\ r_\in C^_\textrm(I; \Psi ^(\Sigma ; V)),\ r_= r_^, \ \Vert r_\Vert \leqslant \frac. \end \end \end$$

Proof

We first solve i) and ii). Let \(b_\in C^_\textrm(I, \Psi ^(\Sigma ; V))\). A routine computation shows that

$$\begin \textrm\partial _(\epsilon + b_)- (\epsilon + b_)^+ a= 0 \end$$

iff

$$\begin b_= (2\epsilon )^\textrm\partial _\epsilon + F(b_), \end$$

(5.9)

for

$$\begin F(b_)= (2\epsilon )^(\textrm\partial _b_+ [\epsilon , b_]- b_^). \end$$

By Proposition A.1, we find \(b_\in C^_\textrm(I, \Psi ^(\Sigma ; V))\), unique modulo \(C^_\textrm(I, \Psi ^(\Sigma ; V))\) solving (5.9) modulo \(C^_\textrm(I, \Psi ^(\Sigma ; V))\).

Next we take \(\chi \) as in (5.5) and set

$$\begin b= \epsilon + b_(1- \chi _(a_\textrm)), \end$$

where \(R\gg 1\) will be fixed below. Since \(\chi _(a_\textrm)\in C^_\textrm(I; \Psi ^(\Sigma ; V))\), we see that b satisfies i) and ii).

It remains to check iii). By the same argument as in Lemma 5.1, we can construct the square root \((2\epsilon )^}\), which belongs to \(C^_\textrm(I; \Psi ^}(\Sigma ; V))\) and is invertible with \((2\epsilon )^\star }= (2\epsilon )^}\). We have

$$\begin b+ b^= (2\epsilon )^}(1- s_)(2\epsilon )^}, \end$$

where

$$\begin s_= -(2\epsilon )^}(b_(1-\chi _(a_\textrm))+ (1-\chi _(a_\textrm))b_^)(2\epsilon )^}. \end$$

We see that \(s_= s_^\), \(s_\in C^_\textrm(I; \Psi ^(\Sigma ; V))\) and since \((1-\chi _(a_\textrm))(2\epsilon )^}\) tends to 0 in norm when \(R\rightarrow \infty \) we can fix \(R\gg 1\) such that \(\Vert s_\Vert \leqslant \frac\). We set now

$$\begin 1+r_= (1- s_)^}= \sum _^c_(s_)^, \end$$

where \(c_= \frac(0)}\) with \(f(x)= (1-x)^}\) satisfies \(|c_|\leqslant 2\) for \(n\in }}\). It follows that

$$\begin \Vert r_\Vert \leqslant 2 \sum _^\Vert s_\Vert ^= 2 \Vert s_\Vert (1- \Vert s_\Vert )^\leqslant \frac. \end$$

Moreover,

$$\begin r_\in C^_\textrm(I; \Psi ^(\Sigma ; V)), \quad r_= r_^, \quad (1+ r_)^= (1-s_). \end$$

This proves iii). \(\square \)

We now set

$$\begin b^:=b, \quad b^= - b^, \end$$

(5.10)

and obtain that

$$\begin \begin b^= \pm \epsilon + C^_\textrm(I; \Psi ^(\Sigma ; V)), \\ \textrm\partial _b^- (b^)^+ a= r_^, \end \end$$

for \(r^_= r_\), \(r^_= r_^\). This is equivalent to the two factorizations of D modulo smoothing error terms:

$$\begin (\partial _+ \textrmb^)(\partial _- \textrmb^)= D - r_^. \end$$

(5.11)

5.2.8 Cauchy Evolution

For \(s\in I\), the Cauchy problem

$$\begin Du= 0\hbox M\\ \varrho _u= f\in ^\infty _\textrm(\Sigma ; V\otimes }}^) \end\right. } \end$$

(5.12)

is well-posed, where

$$\begin \varrho _u= \beginu(s) \\ \textrm^\partial _u(s)\end. \end$$

We denote by \(u= U_f\) the unique solution of (5.12), so that \(U_: ^\infty _\textrm(\Sigma ; V\otimes }}^)\rightarrow C^\infty _\textrm(M; V)\). For \(t\in I\), we denote by

$$\begin U(t, s):=\varrho _\circ U_: ^\infty _\textrm(\Sigma ; V\otimes }}^)\rightarrow ^\infty _\textrm(\Sigma ; V\otimes }}^) \end$$

the Cauchy evolution of D. The Green identity (5.2) implies that U(t, s) is pseudo-unitary for \((\cdot |q \cdot )_(\Sigma )\otimes }}^}\):

$$\begin q= U(t,s)^q U(t,s), \hbox U(t,s)^= U(s, t)\ t, s\in I, \end$$

where as before \(A^\) denotes the adjoint of A for \((\cdot | \cdot )_(\Sigma )\otimes }}^}\).

5.2.9 Factorization of the Cauchy Evolution

For a solution \(u\in C^\infty _\textrm(M; V)\) of \(Du=0\), we set \(\psi (t)= \beginu(t) \\ \textrm^\partial _u(t)\end\) so that

$$\begin \partial _\psi (t)= \textrmA(t)\psi (t), \quad A(t)= \begin0 & \varvec \\ a(t) & 0 \end, \end$$

and \(\psi (t)= U(t,s)\psi (s)\). Next, we define S(t) by

$$\begin S^(t)\psi (t)= \begin(\partial _-\textrmb^(t)) u(t) \\ (\partial _-\textrmb^(t))u(t)\end, \end$$

which yields

$$\begin S= \textrm^\begin\varvec & -\varvec \\ b^ & -b^ \end(b^- b^)^, \quad S^= \textrm\begin- b^ & \varvec \\ -b^ & \varvec \end. \end$$

From (5.11), we obtain that

$$\begin \partial _S^(t)\psi (t)= \textrmB(t) S^(t)\psi (t), \end$$

for

$$\begin B= \begin- b^ & 0 \\ 0 & - b^ \end+B_, \end$$

where

$$\begin B_= \beginr_^ & - r_^ \\ r_^ & -r_^ \end(b^- b^)^\in C^_\textrm(I; \Psi ^(\Sigma ; V\otimes }}^)). \end$$

We find

$$\begin S^q S= \tau (b^- b^)^\begin\varvec & 0 \\ 0 & -\varvec \end. \end$$

Let \(c= (2\epsilon )^}(1+ r_)\), where \(r_\) is as in Proposition 5.2iii). Then \(c^(b^- b^)^c=1\), hence \(c^\tau (b^-b^)^c= \tau \). Setting

$$\begin T:=S\circ \beginc & 0 \\ 0 & c \end, \end$$

we have

$$\begin T^q T= \begin\tau & 0 \\ 0 & -\tau \end. \end$$

(5.13)

Moreover, we have

$$\begin \partial _T^(t)\psi (t)= \textrmC(t)T^(t)\psi (t), \end$$

for

$$\begin C= \begin\epsilon ^ & 0 \\ 0 & \epsilon ^ \end+ C_(t) \end$$

where

$$\begin \epsilon ^= \pm c^b^c+ \textrmc^\partial _c= \pm \epsilon + C^_\textrm(I; \Psi ^(\Sigma ; V)), \end$$

and

$$\begin C_= c^B_c\in C^_\textrm(I; \Psi ^(\Sigma ; V\otimes }}^)). \end$$

In the next proposition, we use the notation recalled in 1.5.8. The hypotheses of Kato’s theorem are easy to check using \(\Psi \)DO calculus.

Proposition 5.3

For all \(t,s\in I\), we have

$$\begin \begin U(t,s)&= T(t)\textrm(\textrm\int _^C(\sigma )d\sigma )T^(s)\\&= T(t)\begin\textrm(\textrm\int _^\epsilon ^(\sigma )d\sigma ) & 0 \\ 0 & \textrm(\textrm\int _^\epsilon ^(\sigma )d\sigma ) \endT^(s)+ R_(t,s). \end \end$$

5.2.10 Hadamard Projectors

We set \(\pi ^= \begin\varvec & 0 \\ 0 & 0 \end\), \(\pi ^= \begin0 & 0 \\ 0 & \varvec \end\), and

$$\begin c^:=T(0)\pi ^T^(0)= \begin\mp (b^- b^)^b^ & \pm (b^- b^)^ \\ \mp b^(b^- b^)^b^ & \pm b^(b^- b^)^ \end(0).\qquad \end$$

(5.14)

We recall that \(A^\) is the symplectic adjoint of A, see 5.2.5.

Proposition 5.4

The operators \(c^\pm \) defined in (5.14) satisfy:

(1)

\(c^+ c^= \varvec\),

(2)

\((c^)^= c^\),

(3)

\(\textrm(U(\cdot , 0)c^)'\subset (\mathcal ^\cup \mathcal )\times T^\Sigma \) for \(\mathcal = \\subset T^M.\)

Proof

(1) is straightforward; (2) follows from (5.13). We set \(Q_= (\partial _- \textrm\epsilon ^(t, \textrm, \partial _}))\), considered as an operator acting on \(M\times \Sigma \) on the first group of variables and let \(A(t, \textrm, \textrm')\) the distributional kernel of \(U(\cdot , 0)c^\).

Proposition 5.3 follows that \(Q_A\in C^\infty (M\times \Sigma , L(V\otimes }}^, V))\). If \(Q_\) were classical \(\Psi \)DOs on \(M\times \Sigma \), this would imply that \(\textrm(U(\cdot , 0)c^)'\subset \mathcal ^\times T^\Sigma \) by elliptic regularity. We reduce ourselves to this situation by an argument from [12, Lem. 6.5.5], see for example [24, Prop. 6.8] for details. \(\square \)

We call the maps \(c^\) Hadamard projectors.

Remark 5.5

From (5.14), we obtain immediately that \(c^\) are projections indeed. The terminology is justified by the fact that Proposition 5.4 implies that \(\lambda ^= \pm q\circ c^\) are a pair of Cauchy surface Hadamard pseudo-covariances for D. Note, however, that the positivity condition \(\lambda ^\geqslant 0\) for \((\cdot | \cdot )_(\Sigma )\otimes }}^}\) is in general not satisfied. Moreover, different choices of b in Proposition 5.2 lead to different projections \(c^\), differing by a term in \(\Psi ^(\Sigma ; V\otimes }}^)\).

5.3 Euclidean Case

We now consider a Euclidean analogue of the setting considered so far. We set \(\tilde= I_\times \Sigma _}\), where \(I\subset }}\) is an interval with \(0\in \mathring\) and \(\Sigma \) a d-dimensional manifold. As before, we identify \(\\times \Sigma \) with \(\Sigma \). We fix an s-dependent sesquilinear form:

$$\begin \tilde}: I\ni s\mapsto C^_\textrm(I; L(T\Sigma , T\Sigma ^)), \end$$

such that \(\tilde}(0)\) is a Riemannian metric on \(\Sigma \), i.e., \(\tilde}(0)= \tilde}(0)^\), \(\tilde}(0)>0\). For ease of notation, \(\tilde}(s)\) is often denoted by \(\tilde}_\).

We assume that \(\tilde}\) is uniformly coercive, i.e., there exists \(C>0\) such that:

$$\begin \begin C^\tilde}(0)\leqslant \operatorname \tilde}(s)\leqslant C \tilde}(0) \\ \big |\! \operatorname \tilde}(s)\big |\leqslant C \operatorname \tilde}(s), \ \ s\in I. \end \end$$

5.3.1 Hilbertian Bundle

We equip \(\tilde\) with the Hilbertian bundle \(\tilde\) as in 5.2.1. For \(u,v\in ^\infty _\textrm(\Sigma ; \tilde)\) resp. \(^\infty _\textrm(}}; \tilde)\), we set

$$\begin \begin (u| v)_(\Sigma )}:=\int _(u| v)_}|\tilde}_|^}d\textrm, \\ (u| v)_(}})}:=\int _}}}(u| v)_}|\tilde}_|^}dtd\textrm. \end \end$$

5.3.2 Adjoints

As in Sect. 5.2.2 if \(\tilde(s)\in C^_\textrm(I; \textrm(\Sigma ; \tilde)\), resp. \(\tilde\in \textrm(}}; \tilde)\) we denote by \(\tilde^(s)\) resp. \(\tilde^\) its formal adjoint for \((\cdot | \cdot )_(\Sigma )}\) resp. \((\cdot | \cdot )_(}})}\).

5.3.3 Elliptic Operator

We fix an s-dependent differential operator \(\tilde(s)= \tilde(s, \textrm, D_})\) belonging to \(C^_\textrm(I; \textrm^(\Sigma ; \tilde))\) and denote by \(\sigma _\textrm(\tilde)(s)\) its principal symbol. We assume the following property:

$$\begin \begin \mathrm1)} \quad \sigma _\textrm(\tilde)(s)(\textrm, \textrm)= \textrm\!\cdot \!\tilde}_^(\textrm)\textrm\varvec_}. \end \end$$

We set

$$\begin }:=-\partial _^+ \tilde(s)\hbox ^\infty _\textrm(}}; \tilde), \end$$

which is an elliptic differential operator.

5.3.4 Factorization of \(}\)

As in Lemma 5.1, we see that \(\tilde(s)\) is closed with domain \(H^(\Sigma ; \tilde)\) and \(\,}}\tilde^(s)= H^(\Sigma ; \tilde)\).

We add to \(\tilde(s)\) a self-adjoint term \(\tilde_\in C^_\textrm(I; \Psi ^(\Sigma ; \tilde))\) such that

$$\begin \operatorname \tilde(s)+ r_(s)\geqslant \delta \varvec, \ \ \delta >0, \end$$

(5.15)

and \(\tilde(s)+ r_(s)\) is m-accretive. We denote by

$$\begin \tilde:=(\tilde+ r_)^}\in C^_\textrm(I; \Psi ^(\Sigma ; \tilde)) \end$$

its unique m-accretive square root given by (5.7), which satisfies:

$$\begin \operatorname \tilde(s)\geqslant \delta ^}\varvec. \end$$

(5.16)

As in Sect. 5.2.6 we have:

$$\begin \sigma _\textrm(\tilde(s))= (\sigma _\textrm(\tilde(s)))^}. \end$$

The operator \(\tilde(s)\) with domain \(H^(\Sigma ; \tilde)\) is closed, elliptic and invertible by (5.16), hence \(\tilde^(s)\in C^_\textrm(I; \Psi ^(\Sigma ; \tilde))\).

Proposition 5.6

There exists \(\tilde^(s)\in C^_\textrm(I; \Psi ^(\Sigma ; \tilde))\), unique modulo a term in \(C^_\textrm(I; \Psi ^(\Sigma ; \tilde))\), such that:

$$\begin \begin i)& \tilde^(s)= \pm \tilde(s)+ C^_\textrm(I; \Psi ^(\Sigma ; \tilde)),\\ ii)& \partial _\tilde^(s)- (\tilde^)^(s)+ \tilde(s)= \tilde^_(s)\in C^_\textrm(I; \Psi ^(\Sigma ; \tilde)),\\ iii)& \pm \operatorname \tilde^\geqslant \varvec,\\ iv)& \tilde^\!\!-\! \tilde^\!:\! H^(\Sigma ; \tilde)\!\rightarrow \! L^(\Sigma ; \tilde),\,(\tilde^\!\!-\! \tilde^)^\!\!\in \! C^_\textrm(I; \Psi ^(\Sigma ; \tilde)). \end \end$$

Proof

We look for \(\tilde^\) under the form \(\tilde^= \pm \tilde+ \tilde_\), \(\tilde_\in C^_\textrm(I; \Psi ^(\Sigma ; \tilde))\) and obtain the equation

$$\begin \tilde_= (2 \tilde)^\partial _\tilde+ \tilde^(\tilde_), \end$$

for

$$\begin \tilde^(\tilde_)= (2\tilde)^(\pm \partial _\tilde_+ [\tilde, \tilde_]\mp \tilde_^). \end$$

We use the same fixed point argument as in Proposition 5.2 and obtain \(\tilde^\) satisfying i) and ii). To obtain iii) we use that \(\tilde^= \pm \tilde+ C^_\textrm(I; \Psi ^(\Sigma ; \tilde))\) and add to \(\tilde^\) elements \(r_^\in C^_\textrm(I, \Psi ^(\Sigma ; \tilde))\) so that \(\pm \operatorname \tilde^\geqslant 1\). Then \(\tilde^- \tilde^\) is m-accretive with \(0\not \in \sigma (\tilde^- \tilde^)\), which implies iv). \(\square \)

We obtain the following factorization of \(}\), analogous to (5.11):

$$\begin (-\partial _+ \tilde^)(\partial _+ \tilde^)= }- \tilde^_. \end$$

(5.17)

Remark 5.7

Suppose that the interval I is symmetric with respect to 0 and that

$$\begin \tilde^(s)= \tau \tilde(-s)\tau ^, \ \ s\in I. \end$$

Then, from (5.7) we have \(\tilde^(s)= \tau \tilde(-s)\tau ^\). We set

$$\begin \tilde(\tilde_)= (2\tilde)^(\partial _\tilde_+ [\tilde, \tilde_]- \tilde_^), \end$$

we solve the fixed point equation

$$\begin \tilde_= (2\tilde)^\partial _\tilde+ \tilde(\tilde_), \end$$

and construct \(\tilde= \tilde+ \tilde_\) such that

$$\begin \begin \partial _\tilde- \tilde^+ \tilde= \tilde_\in C^_\textrm(I; }}^(\Sigma ; \tilde)),\\ \operatorname \tilde\geqslant \varvec. \end \end$$

Then, we can take:

$$\begin \begin \tilde^(s)= \tilde(s), \quad \tilde^_(s)= \tilde_(s),\\ \tilde^(s)= - \tau ^\tilde^(-s)\tau , \quad \tilde_^(s)= \tau ^\tilde_^(-s)\tau . \end \end$$

5.3.5 Parametrix for \(}\)

By Proposition 5.6, we know that \(\pm \tilde^(s)\) is m-accretive with \(0\in \,}}(\tilde^(s))\) (the resolvent set) and \(\,}}\tilde^(s)= H^(\Sigma ; \tilde)\) for all \(s\in I\). Since \(\tilde^\in C^_\textrm(I; \Psi ^(\Sigma ; \tilde))\), we can check the hypotheses of [38] (in the version presented in [51]) and conclude that

$$\begin V^(s, s'):=\textrm(\int _^\tilde^(\sigma )d\sigma )\hbox \mp (s-s')\geqslant 0, \ s,s'\in I. \end$$

Lemma 5.8 (1)

\(V^(s, s'): H^(\Sigma ; \tilde)\rightarrow H^(\Sigma ; \tilde)\) is uniformly bounded for \(s, s'\in I\) and \(\pm (s-s')\geqslant 0\);

(2)

\(V^(s, s')\in }}^(\Sigma ; \tilde)\) for \(\mp (s-s')\geqslant \delta >0\).

Proof

To prove (1), it suffices to apply Kato’s theorem in [38] to the Hilbert space \(H^(\Sigma ; \tilde)\). The hypotheses follow from \(\Psi \)DO calculus. If we denote by \(V^_(s, s')\) the resulting semi-group on \(H^(\Sigma ; \tilde)\), then \(V_^\) is an extension resp. restriction of \(V^\) if \(m<0\) resp. \(m>0\). Let us prove (2) in the \(+\) case. We fix \(s'\in I\), \(\chi \in ^\infty _\textrm(I)\) with \(\,}}\chi \subset \\), \(\chi =1\) in \(\\). Then \((\partial _- \tilde^(s))\chi (s)V^(s, s')u= \chi '(s)V^(s, s')u\). The operator \(\partial _- \tilde^(s)\) has principal symbol \((\textrm\sigma - \sigma _\textrm(\tilde(s))(\textrm)^})\varvec_}\) hence is elliptic in \(\Psi ^(I\times \Sigma ; \tilde)\). If \(u\in H^}(\Sigma ; \tilde)\), then \(V^(\cdot , s')u\in H^(I\times \Sigma ; \tilde)\) for some \(n_\) and by elliptic regularity \(\chi (\cdot )V^(\cdot , s')u\in H^+1}(I\times \Sigma ; \tilde)\). By iterating this argument we obtain that \(\chi (\cdot )V^(\cdot , s')u\in H^(I\times \Sigma ; \tilde)\) for any n so \(V^(s, s')u\in H^(\Sigma ; \tilde)\) for any \(m\in }}\). \(\square \)

For \(v\in C^_\textrm(I; ^\infty _\textrm(\Sigma ; \tilde))\), we set

$$\begin T^v(s):=\pm \int _}}}H(\mp (s-s'))V^(s, s')v(s')ds', \end$$

where \(H(t)= \varvec_}}^}(t)\) is the Heaviside function, so that

$$\begin (-\partial _+ \tilde^)\circ T^=T^\circ (-\partial _+ \tilde^)= \varvec. \end$$

(5.18)

Proposition 5.9

Let

$$\begin }^= \left( (\tilde^- \tilde^)^(T^- T^)\right) . \end$$

Then

$$\begin }\circ }^= \varvec+ R_, \end$$

for some \(R_\in }}^(}}; \tilde)\).

Proof

We obtain using (5.18) and Proposition 5.6ii):

$$\begin \begin&\partial _\left( (\tilde^- \tilde^)^(T^- T^)\right) \\&=(\tilde^- \tilde^)^\left( \tilde^T^- \tilde^T^- (\tilde^- \tilde^)(\tilde^- \tilde^)^(T^- T^)\right) + r_, \end \end$$

for

$$\begin r_= - (\tilde^- \tilde^)^(\tilde^_- \tilde^_)(\tilde^- \tilde^)^(T^- T^). \end$$

Next,

$$\begin \begin \tilde^- (\tilde^- \tilde^)(\tilde^-\tilde^)^= -(\tilde^- \tilde^)\tilde^(\tilde^- \tilde^)^\\ -\tilde^+ (\tilde^- \tilde^)(\tilde^- \tilde^)^= (\tilde^- \tilde^\tilde^)(\tilde^- \tilde^)^, \end \end$$

hence

$$\begin \begin&\partial _\left( (\tilde^- \tilde^)^(T^- T^)\right) \\&\quad = - \tilde^(\tilde^- \tilde^)^T^+ \tilde^(\tilde^- \tilde^)^T^+ r_\\&\quad =T^- \tilde^\left( (\tilde^- \tilde^)^(T^- T^)\right) + r_. \end \end$$

(5.19)

Using again (5.18), we obtain

$$\begin (- \partial _+ \tilde^)(\partial _+ \tilde^)\left( (\tilde^- \tilde^)^(T^- T^)\right) = \varvec- (\partial _- \tilde^)r_. \end$$

Hence, setting

$$\begin }^v(s)= (\tilde^(s)- \tilde^(s))^(T^- T^)v(s), \end$$

(5.20)

by (5.17) we get

$$\begin }}^= \varvec+ R_, \ \ R_\in C^_\textrm(I; \P