Decay of the Local Energy for the Charged Klein–Gordon Equation in the Exterior De Sitter–Reissner–Nordström Spacetime

Abstract

We show decay of the local energy of solutions of the charged Klein–Gordon equation in the exterior De Sitter–Reissner–Nordström spacetime by means of a resonance expansion of the local propagator.

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Notes

  1. 1.

    We will often drop the dependence in s.

  2. 2.

    Note that the norm \(\Vert .\Vert ^2_{_{\dot{{\mathcal {E}}}_\ell }}\) is conserved if \([P_{\ell },sV]=0\); it is the case if \(s=0\).

  3. 3.

    From a geometrical point of view, we are changing the gauge. Namely, \(\frac{Q}{r}\mathrm {d}t\) is replaced by \(\left( \frac{Q}{r}-\frac{Q}{r_+}\right) \mathrm {d}t\) which does not degenerate anymore at \(r=r_+\). To see this, we use the standard Eddington–Finkelstein advanced and retarded coordinates \(u=t-x\), \(v=t+x\) to define the horizons: we have locally near the cosmological horizon \(\mathrm {d}t=\mathrm {d}u+\mathrm {d}x\), \(\mathrm {d}t=\mathrm {d}v-\mathrm {d}x\) and then \(\frac{\mathrm {d}t}{\mathrm {d}r}=\pm F(r)^{-1}\). We eventually use that \(\left( \frac{1}{r}-\frac{1}{r_+}\right) F(r)^{-1}\) remains bounded and does not vanish at \(r=r_+\).

  4. 4.

    Observe that in the example at the beginning of Sect. 3, the kernel R(zxy) is also given in terms of the Jost functions \(e_{\pm }(x,z)=\mathrm {e}^{\pm \mathrm {i}z x}\).

  5. 5.

    We may notice here that the positive mass term \(m^{2}\) allowed us to conclude that \(z=0\) is not a pole. For the wave equation as in [5], we do not have any positivity for \(\ell =0\) and \(z=0\) is shown to be a pole.

  6. 6.

    Lemma 3.1 of course applies if we replace \(k_-\) by \({\tilde{k}}_-\).

  7. 7.

    The factor \(1/\kappa _\pm \) in the second argument of \(f^\pm _\theta \) comes from the fact that \(\kappa _{\pm } x\) corresponds to Zworski’s variable r.

  8. 8.

    See the definition of the operator K at the beginning of the proof of [6, Lem. 6.5].

  9. 9.

    Recall that the set of Fredholm operators in \({\mathcal {L}}({\mathscr {D}},L^2)\) is open for the norm topology.

  10. 10.

    We show it using the sesquilinear form \((\varphi ,\psi )\mapsto \langle \langle x\rangle ^{-\sigma }\langle {\mathcal {A}}\rangle ^{\sigma }\varphi ,\psi \rangle \) first well-defined on \({\mathscr {D}}({\mathcal {A}}^2)\times {\mathscr {D}}({\mathcal {A}}^2)\) because \(\langle x\rangle ^{-2}\in \Psi ^{0,-2}\), and then extended to \(L^2\times L^2\) by maximum principle.

  11. 11.

    We can in fact insert any pseudodifferential operator here provided that hypotheses in [22, §2] are verified.

  12. 12.

    The purpose of Lemma 6.2 is to provide us with integrability in z at the prize of using the weaker spaces \(\dot{{\mathcal {E}}}_{\ell }^{-2}\). The task then consists in showing that all the terms in \(\dot{{\mathcal {E}}}_{\ell }^{-2}\) vanish after deformation of contours and the remaining terms are in \(\dot{{\mathcal {E}}}_{\ell }\).

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Acknowledgements

I am very grateful to the anonymous referee for many suggestions to improve the manuscript, especially concerning the introduction of Sect. 3.

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Correspondence to Nicolas Besset.

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Communicated by Mihalis Dafermos.

Appendices

Appendix A: Analytic Extension of the Coordinate r

In this appendix, we prove Proposition 2.1 which is analogous to [3, Prop IV.2]. Let \(r\in ]r_{-},r_{+}[\). By equation (5), we have

$$\begin{aligned} \exp \left( -\frac{\Lambda }{3A_\pm r_\pm ^2}x\right)&=\prod _{\alpha \in I}\left| \frac{r-r_\alpha }{{\mathfrak {r}}-r_\alpha }\right| ^{\frac{A_\alpha r_\alpha ^2}{A_\pm r_\pm ^2}}. \end{aligned}$$

Call the left-hand side z and the right-hand side \(g_\pm (r)\). Observe that \(g_\pm (r_\pm )=0\). Since \(r\mapsto x(r)\) is increasing and analytic, we can apply the Lagrange’s inversion theorem (see for example [8, §2.2] and references therein) to write

$$\begin{aligned} r&=r_\pm +\sum _{\ell =1}^{+\infty }\frac{z^\ell }{\ell !}\left[ \frac{\mathrm {d}^{\ell -1}}{\mathrm {d}r^{\ell -1}}\left( \frac{r-r_\pm }{g_\pm (r)}\right) ^{\ell }\,\right] _{r=r_\pm }. \end{aligned}$$
(59)

Let us introduce Kronecker’s symbol

$$\begin{aligned} \delta _{\alpha ,\pm }:={\left\{ \begin{array}{ll} 1&{}\quad \text {if }\alpha =\pm \\ 0&{}\quad \text {otherwise} \end{array}\right. } \end{aligned}$$

and the notation

$$\begin{aligned} B_{\pm ,\alpha }:=\frac{A_\alpha r_\alpha ^2}{A_\pm r_\pm ^2}-\delta _{\alpha ,\pm }. \end{aligned}$$

Observe that \(B_{-,-}=B_{+,+}=0\). We then have

$$\begin{aligned} \frac{\mathrm {d}^{\ell -1}}{\mathrm {d}r^{\ell -1}}\left( \frac{r-r_\pm }{g_\pm (r)}\right) ^{\ell }&=\left( \prod _{\alpha \in I{\setminus }\{\pm \}}\left| {\mathfrak {r}}-r_\alpha \right| ^{\ell B_{\pm ,\alpha }}\right) \frac{\mathrm {d}^{\ell -1}}{\mathrm {d}r^{\ell -1}}\left( \prod _{\alpha \in I{\setminus }\{\pm \}}\left| r-r_\alpha \right| ^{-\ell B_{\pm ,\alpha }}\right) . \end{aligned}$$

We now fix \(\pm =+\) (the conclusion will not be changed if we choose \(\pm =-\)). Then

$$\begin{aligned}&\frac{\mathrm {d}^{\ell -1}}{\mathrm {d}r^{\ell -1}}\left( \prod _{\alpha \in I{\setminus }\{+\}}\left( r-r_\alpha \right) ^{-\ell B_{+,\alpha }}\right) \\&\quad =\sum _{0\le k_2\le k_1\le \ell }C_{\ell ,k_1,k_2}\left( \frac{\mathrm {d}^{\ell -k_1}}{\mathrm {d}r^{\ell -k_1}}(r-r_n)^{-\ell B_{+,n}}\right) \nonumber \\&\qquad \times \, \left( \frac{\mathrm {d}^{k_1-k_2}}{\mathrm {d}r^{k_1-k_2}}(r-r_c)^{-\ell B_{+,c}}\right) \left( \frac{\mathrm {d}^{k_2}}{\mathrm {d}r^{k_2}}(r-r_-)^{-\ell B_{+,-}}\right) \end{aligned}$$

where

$$\begin{aligned} C_{\ell ,k_1,k_2}&=\begin{pmatrix} \ell \\ k_1 \end{pmatrix}\begin{pmatrix} k_1\\ k_2 \end{pmatrix}. \end{aligned}$$

Direct computation shows that

$$\begin{aligned} \frac{\mathrm {d}^{p}}{\mathrm {d}r^{p}}(r-r_\alpha )^{-\ell B_{+,\alpha }}&=(-1)^{p}(\ell B_{+,\alpha })(\ell B_{+,\alpha }+1)\ldots (\ell B_{+,\alpha }+p-1)(r-r_\alpha )^{-\ell B_{+,\alpha }-p}. \end{aligned}$$

If we let

$$\begin{aligned} K&:=\prod _{\alpha \in I{\setminus }\{+\}}\left( {\mathfrak {r}}-r_\alpha \right) ^{B_{+,\alpha }},&B_{+}:=\max _{\alpha \in I{\setminus }\{+\}}\{|B_{+,\alpha }|\}, \end{aligned}$$

then it follows that

$$\begin{aligned}&\frac{\mathrm {d}^{\ell -1}}{\mathrm {d}r^{\ell -1}} \left( \frac{r-r_+}{g_+(r)}\right) ^{\ell }\\&\quad =K^{\ell }\sum _{0\le k_2\le k_1\le \ell }C_{\ell ,k_1,k_2}(-1)^{\ell }\nonumber \\&\qquad \times \, (\ell B_{+,n})(\ell B_{+,n}+1)\ldots (\ell B_{+,n}+(\ell -k_1)-1)(r-r_n)^{-\ell B_{+,n}-(\ell -k_1)}\nonumber \\&\qquad \times \, (\ell B_{+,c})(\ell B_{+,c}+1)\ldots (\ell B_{+,c}+(k_1-k_2)-1)(r-r_c)^{-\ell B_{+,c}-(k_1-k_2)}\nonumber \\&\qquad \times \, (\ell B_{+,-})(\ell B_{+,-}+1)\ldots (\ell B_{+,-}+k_2-1)(r-r_\alpha )^{-\ell B_{+,-}-k_2} \end{aligned}$$

and thus

$$\begin{aligned}&\left| \frac{\mathrm {d}^{\ell -1}}{\mathrm {d}r^{\ell -1}}\left( \frac{r-r_+}{g_+(r)}\right) ^{\ell }\right| \nonumber \\&\quad \le K^{\ell }\ell ^{\ell }(B_{+}+1)^{\ell }\left( \prod _{\alpha \in I{\setminus }\{+\}}(r_+-r_\alpha )^{-B_{+,\alpha }}\right) ^{\ell }\nonumber \\&\qquad \times \, \sum _{0\le k_2\le k_1\le \ell }C_{\ell ,k_1,k_2}(r_+-r_n)^{-(\ell -k_1)}(r_+-r_c)^{-(k_1-k_2)}(r_+-r_-)^{-k_2}\nonumber \\&\quad =K^{\ell }\ell ^{\ell }(B_{+}+1)^{\ell }\left( \prod _{\alpha \in I{\setminus }\{+\}}(r_+-r_\alpha )^{-B_{+,\alpha }}\right) ^{\ell }\left( \sum _{\alpha \in I{\setminus }\{+\}}(r_+-r_\alpha )^{-1}\right) ^{\ell }\\&\quad =\left( K(B_{+}+1)\prod _{\alpha \in I{\setminus }\{+\}}(r_+-r_\alpha )^{-B_{+,\alpha }}\sum _{\alpha \in I{\setminus }\{+\}}(r_+-r_\alpha )^{-1}\right) ^{\ell }\ell ^{\ell }\\&\quad =:{\tilde{K}}^{\ell }\ell ^{\ell }. \end{aligned}$$

Therefore, the convergence of the original series is absolute for \(z\in {\mathbb {C}}\) if

$$\begin{aligned} \frac{(|z|\ell {\tilde{K}})^{\ell }}{\ell !}&<\ell ^{-(1+\varepsilon )} \end{aligned}$$

for any \(\varepsilon >0\). Using Stirling approximation \(\ell !\sim \sqrt{2\pi }\ell ^{\ell +1/2}\) for large values of \(\ell \), we see that it is sufficient to have

$$\begin{aligned} {\tilde{K}}|z|&<\frac{\mathrm {e}^{-(1/2+\varepsilon )\ln \ell /\ell }}{\sqrt{2\pi }^{\ell }}<1. \end{aligned}$$

This condition is fulfilled if

$$\begin{aligned} \mathfrak {R}x&>\frac{3A_+ r_+^2}{\Lambda }\ln {\tilde{K}}. \end{aligned}$$

Appendix B: Localization of High-Frequency Resonances

We provide in this section an asymptotic approximation of resonances near the maximal energy \(W_0(0)=\max _{x\in {\mathbb {R}}}\{W_0(x)\}\) as \(h\rightarrow 0\). This a generalization of the main Theorem in [28] to the case \(Q\ne 0\). More precisely, we show that the resonances associated with the meromorphic extension of \(p(z,s)^{-1}\) are close to the ones associated with the extension of \((P-z^2)^{-1}\), provided that Q is sufficiently small. This is a direct consequence of the fact that the extra term hsV in the semiclassical quadratic pencil is \({\mathcal {O}}(hs)\).

As in the paragraph 5.2, we set \(h:=(\ell (\ell +1))^{-1/2}\) with \(\ell >0\) and consider \(z\in \left[ \ell /R,R\ell \right] +\mathrm {i}\left[ -C_{0},C_{0}\right] \). We then define the spectral parameter \(\lambda :=h^2z^2\) and also \({\tilde{P}}_h\) the semiclassical operator associated with \(P_\ell \). Recall also that \({\mathfrak {r}}=\frac{3M}{2}\left( 1+\sqrt{1-\frac{8Q^2}{9M^2}}\right) \) is the radius of the photon sphere and \(W_0(0)=F({\mathfrak {r}})/{\mathfrak {r}}^2\) with our definition of the Regge–Wheeler coordinate x (see (5)).

Theorem B.1

Let

$$\begin{aligned} \Gamma _0(h)&:=\left\{ W_0(0)+h\left( 2\sqrt{W_0(0)}sV(0)+\mathrm {i}^{-1}\sqrt{W_0''(0)/2}\left( k+\frac{1}{2}\right) \right) \mid k\in {\mathbb {N}}\right\} . \end{aligned}$$

For all \(C_0>0\) such that \(\partial D(W_0(0),C_0h)\cap \Gamma _0(h)=\emptyset \), there is a bijection \(b\equiv b(h)\) from \(\Gamma _0(h)\) onto the set of resonances of \({\tilde{P}}_h\) in \(D(W_0(0),C_0h)\) (counted with their natural multiplicity) such that

$$\begin{aligned} b(h)(\mu )-\mu&=o_{h\rightarrow 0}(h)&\mathrm {uniformly\ for\ }\mu \in \Gamma _0(h). \end{aligned}$$

Proof

This is a direct application of the results of Sá Barreto–Zworski [28] which are based on the work of Sjöstrand [30] (see Theorem 0.1 therein), the latter dealing with resonances generated by non-degenerate critical points when the trapping set is reduced to a single point (the difference for us is \(W_0(0)\ne 0\)).

We recall that in the zone III the symbol of the semiclassical quadratic pencil is the function \((x,\xi )\!\mapsto \!\xi ^2\!+\!W_0(x)\!+\!h^2W_1(x)\!-\!(\sqrt{\lambda }-hsV(x))^2\!=:\!p(x,\xi )\!-\!\lambda \). We also recall the hypothesis in [30] for the case of a Schrödinger operator of the form (0.1) in the reference:

  • The trapping set is reduced to the point \(\{(0,0)\}\) ((0.3) in [30]),

  • 0 is a non-degenerate critical point ((0.4) in [30], which implies in the Schrödinger case the more general assumptions (0.7) and (0.9) in the reference).

Although the symbol p depends on \(\lambda \), its principal part \(p_0\) and subprincipal part \(p_{-1}\) do not: indeed, for \(\lambda \in D\left( W_0(0),C_0 h\right) \) with \(C_0>0\), we can write when \(h\ll 1\)

$$\begin{aligned} p(x,\xi )&=\underbrace{\xi ^2+W_0(x)}_{=p_0(x,\xi )}+h\underbrace{2\sqrt{W_0(0)}sV(x)}_{=p_{-1}(x,\xi )} +\text { lower order terms in } h. \end{aligned}$$

This is enough to apply [30, Thm. 0.1]: using formula (0.14) in the reference, we get the result for the set

$$\begin{aligned} \left\{ p_0(0,0)+h\left( p_{-1}(0,0)+\mathrm {i}^{-1}\sqrt{W_0''(0)/2}\left( k+\frac{1}{2}\right) \right) \mid k\in {\mathbb {N}}\right\} \end{aligned}$$

which is \(\Gamma _0(h)\). \(\square \)

Approximation of high-frequency resonances \(\Gamma _0(h)\ni z^2=\lambda /h^2\) is obtained as in [28], by taking the square root of any element of \(\Gamma _0(h)\) and using Taylor expansion for \(0<h\ll 1\) (corresponding to \(\ell \gg 0\)) as well as symmetry with respect to the imaginary axis (for the choice of the sign of the square root). In our setting, we obtain the set \(\Gamma \) of Theorem 4.1.

Remark B.2

  1. 1.

    Let \(\Gamma _{\mathrm {DSS}}\) be the set of pseudo-poles in the De Sitter–Schwarzschild case (see the Theorem at the end of [28]). Then \(\Gamma _{\mathrm {DSS}}\) is the limit of \(\Gamma \) as \(Q\rightarrow 0\) in the sense of the sets, i.e. for all \(z\in \Gamma \), there exists \(z_0\in \Gamma _{\mathrm {DSS}}\) such that \(z\rightarrow z_0\) as \(Q\rightarrow 0\).

  2. 2.

    The pseudo-poles in the charged case are shifted with respect to the uncharged case. If the charges of the Klein–Gordon field and the black hole have the same sign (that is if \(qQ>0\)), then all the pseudo-poles go to infinity with a real part which never vanishes. However, if the charges have opposite sign (\(qQ<0\)), then all the pseudo-poles real part cancels precisely when \(qQ=-(k+1/2)\sqrt{F({\mathfrak {r}})}\), \(k\in {\mathbb {N}}{\setminus }\{0\}\), before going to infinity. Notice that no pseudo-pole goes to \({\mathbb {C}}^+\) as \(|s|\rightarrow +\infty \).

  3. 3.

    We can provide a physical interpretation of the set of pseudo-poles. First observe that \(\sqrt{F({\mathfrak {r}})}/{\mathfrak {r}}\) is nothing but the inverse of the impact parameter \(b=|E/L|\) of trapped null geodesics with energy E and angular momentum L. Theorem 4.1 shows that resonances near the real line in the zone III are qQ-dependent multiples of this quantity: they thus correspond to impact parameters of trapped photons with high energy and angular momentum.

  4. 4.

    Observe that in Newtonian mechanics, the electromagnetism and gravitation do not interact with chargeless and massless photons. As a consequence, photons are not deviated and only ones with impact parameter \(|b|\le r_-\) can “fall“ in the black hole. Hence, high-frequency resonances in zone III are expected to be multiple of \(r_-^{-1}\). As \(r_-\rightarrow 0\), all resonances go to infinity: the trajectory are now classical straight lines as there is no obstacle anymore see Figs. 5.

Fig. 5
figure5

On the left: a relativistic trapped null geodesic. On the right: classical null geodesic trajectories

Appendix C: Abstract Semiclassical Limiting Absorption Principle for a Class of Generalized Resolvents

We show in this section an abstract semiclassical limiting absorption principle for perturbed resolvents.

Abstract setting Let \(\big ({\mathcal {H}},\langle \cdot ,\cdot \rangle \big )\) be a Hilbert space, \(J:=[a,b]\subset {\mathbb {R}}\), \(J_\mu ^+:=\{\omega \in {\mathbb {C}}^+\mid \mathfrak {R}\omega \in J,\,\mathfrak {I}\omega <\mu \}\) for some \(\mu >0\) fixed and \(h_0>0\). The norm associated with \(\langle \cdot ,\cdot \rangle \) will be denoted by \(\Vert \cdot \Vert \). We consider families of self-adjoint operators \(P\equiv P(h)\) and \({\mathcal {A}}\equiv {\mathcal {A}}(h)\) acting on \({\mathcal {H}}\) for \(0<h<h_0\). We set

$$\begin{aligned} L^{\infty }_{\ell \mathrm {oc}}(P):=\big \{A:{\mathcal {H}}\rightarrow {\mathcal {H}} \text { linear}\mid \forall \chi \in {\mathcal {C}}^{\infty }_{\mathrm {c}}({\mathbb {R}},{\mathbb {R}}),\,\forall u\in {\mathscr {D}}(P),\,\Vert \chi (P)Au\Vert <+\infty \big \} \end{aligned}$$

and \(\Vert .\Vert _{_{P}}\) will be the operator norm on \({\mathcal {B}}({\mathscr {D}}(P),{\mathcal {H}})\). We also define the local version of the operator P:

$$\begin{aligned} P_{\tau }&:=\tau (P)P\qquad \qquad \forall \tau \in {\mathcal {C}}^\infty _{\mathrm {c}}({\mathbb {R}},{\mathbb {R}}). \end{aligned}$$

Let then \(f:{\mathbb {C}}\times L^{\infty }_{\ell \mathrm {oc}}(P)\rightarrow L^{\infty }_{\ell \mathrm {oc}}(P)\) satisfying the following continuity-type relation near \(0_{_{L^{\infty }_{\ell \mathrm {oc}}(P)}}\): there exist \(\delta _{_{J,\mu }}:{\mathbb {R}}_+\rightarrow {\mathbb {R}}\) satisfying \(\delta _{_{J,\mu }}(r)\rightarrow 0\) as \(r\rightarrow 0\) and \(\varepsilon _{_{J,\mu }}:L^{\infty }_{\ell \mathrm {oc}}(P)\rightarrow L^{\infty }_{\ell \mathrm {oc}}(P)\) such that, for all \((z,A)\in J^+_\mu \times L^{\infty }_{\ell \mathrm {oc}}(P)\) with \(\Vert A\Vert _{_{P}}\) small,

figureb

We make the following assumptions:

figurec

Recall that \(P\in {\mathcal {C}}^{2}\left( {\mathcal {A}}\right) \) means for all \(z\in {\mathbb {C}}{\setminus }\sigma (P)\) that the map

$$\begin{aligned} {\mathbb {R}}\ni t\mapsto \mathrm {e}^{\mathrm {i}t{\mathcal {A}}}(P-z)^{-1}\mathrm {e}^{-\mathrm {i}t{\mathcal {A}}} \end{aligned}$$

is \({\mathcal {C}}^2\) for the strong topology of \(L^2\). Recall also that for all linear operators \(L_1,L_2\) acting on \({\mathcal {H}}\), \(\mathrm {ad}^{0}_{L_1}(L_2):=L_2\) and \(\mathrm {ad}^{k+1}_{L_1}(L_2):=[L_1,\mathrm {ad}^k_{L_1}(L_2)]\). Our goal is to show the following result:

Theorem C.1

Assume hypotheses (C), (I), (P), (M) and (A). Then for all \(\sigma >1/2\),

$$\begin{aligned} \sup _{z\in J^+_\mu }\Vert \langle {\mathcal {A}}\rangle ^{-\sigma }(P-f(z,hB))^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert&\lesssim h^{-1}. \end{aligned}$$
(60)

In the sequel, we will write \(R(z,hB):=(P-f(z,hB))^{-1}\) and call it the generalized resolvent (of P). Also, since J and \(\mu \) are now fixed, we will simply write \(J,\,\delta \) and \(\varepsilon \) instead of \(J_{\mu },\,\delta _{_{J,\mu }}\) and \(\varepsilon _{_{J,\mu }}\).

Preliminary results The purpose of this paragraph is to show preliminary results used to prove Theorem C.1. We first prove an adapted version of [13, Lem. 2.1] to our situation.

Lemma C.2

Let \(0\le \sigma \le 1\), \(z\in J^+\) and let \(\chi \in {\mathcal {C}}^{\infty }_{\mathrm {c}}\left( {\mathbb {R}},{\mathbb {R}}\right) \). If h is small enough, then R(zhB) and \(\chi (P)\) are bounded on \({\mathscr {D}}(\langle {\mathcal {A}}\rangle ^{\sigma })\).

Proof

The result is true for \((P-z)^{-1}\) and \(\chi (P)\) by [13, Lem. 2.1]. Let us show that \(R(z,hB){\mathscr {D}}(\langle {\mathcal {A}}\rangle ^\sigma )\subset {\mathscr {D}}(\langle {\mathcal {A}}\rangle ^\sigma )\):

$$\begin{aligned}&\Vert \langle {\mathcal {A}}\rangle ^{\sigma }R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \le \Vert \langle {\mathcal {A}}\rangle ^{\sigma }(P-z)^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \\&\qquad +\Vert \langle {\mathcal {A}}\rangle ^{\sigma }(R(z,hB)-(P-z)^{-1})\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \\&\quad \lesssim 1+\Vert \langle {\mathcal {A}}\rangle ^{\sigma }R(z,hB)(z-f(z,hB))(P-z)^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \end{aligned}$$

and (using that \(\varepsilon (hB)\in {\mathcal {B}}({\mathscr {D}}({\mathcal {A}}))\) by Assumption (A) for \(k=0\))

$$\begin{aligned}&\Vert \langle {\mathcal {A}}\rangle ^{\sigma }R(z,hB)(z-f(z,hB))(P-z)^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \\&~\le \Vert \langle {\mathcal {A}}\rangle ^{\sigma }R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \Vert \langle {\mathcal {A}}\rangle ^{\sigma }(z-f(z,hB))\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \Vert \langle {\mathcal {A}}\rangle ^{\sigma }(P-z)^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \\&~\lesssim \delta (h\Vert B\Vert _{_{P}})\Vert \langle {\mathcal {A}}\rangle ^{\sigma }\varepsilon (hB)\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \Vert \langle {\mathcal {A}}\rangle ^{\sigma }R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert . \end{aligned}$$

We then use the uniformity in assumption (A) for \(k=0\) to write for h very small

$$\begin{aligned} \delta (h\Vert B\Vert _{_{P}})\Vert \langle {\mathcal {A}}\rangle ^{\sigma }\varepsilon (hB)\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \Vert \langle {\mathcal {A}}\rangle ^{\sigma }R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert&<\frac{1}{2}\Vert \langle {\mathcal {A}}\rangle ^{\sigma }R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert . \end{aligned}$$

The proof is complete. \(\square \)

Corollary C.3

Let \(0\le \sigma \le 1\), \(z\in J^+\) and \(\tau ,\chi \in {\mathcal {C}}^{\infty }_{\mathrm {c}}\left( {\mathbb {R}},[0,1]\right) \) such that \(\chi \equiv 1\) on I and \(\tau \chi =\chi \). If h is small enough, then \((P_\tau -f(z,hB))\chi (P)\), \((P_\tau -f(z,hB))\chi (P)(P+\mathrm {i})^{-1}\) and \((P-f(z,hB))(P+\mathrm {i})^{-1}\) preserve \({\mathscr {D}}(\langle {\mathcal {A}}\rangle ^{\sigma })\).

Proof

We have

$$\begin{aligned} \langle {\mathcal {A}}\rangle ^{\sigma }(P_\tau -f(z,hB))\chi (P)\langle {\mathcal {A}}\rangle ^{-\sigma }&=\langle {\mathcal {A}}\rangle ^{\sigma }(P_\tau -z)\chi (P)\langle {\mathcal {A}}\rangle ^{-\sigma }\\&\quad +\, \langle {\mathcal {A}}\rangle ^{\sigma }(z-f(z,hB))\langle {\mathcal {A}}\rangle ^{-\sigma }\langle {\mathcal {A}}\rangle ^{\sigma }\chi (P)\langle {\mathcal {A}}\rangle ^{-\sigma } \end{aligned}$$

which is bounded by assumption (A) for \(k=0\), Lemma C.2 and the fact that \(P_\tau \chi (P)=\varphi (P)\) with \(\varphi \in {\mathcal {C}}^{\infty }_{\mathrm {c}}({\mathbb {R}},{\mathbb {R}})\) by functional calculus. Next, [13, Lem. 2.1] implies that \((P+\mathrm {i})^{-1}\) preserves \({\mathscr {D}}({\mathcal {A}})\), so we can write

$$\begin{aligned}&\langle {\mathcal {A}}\rangle ^{\sigma }(P_\tau -f(z,hB))\chi (P)(P+\mathrm {i})^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma }\\&\quad =\langle {\mathcal {A}}\rangle ^{\sigma }(P_\tau -f(z,hB))\chi (P)\langle {\mathcal {A}}\rangle ^{-\sigma }\langle {\mathcal {A}}\rangle ^{\sigma }(P+\mathrm {i})^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma } \end{aligned}$$

which is clearly bounded thanks to the above computation. Finally,

$$\begin{aligned}&\langle {\mathcal {A}}\rangle ^{\sigma }(P-f(z,hB))(P+\mathrm {i})^{-1} \langle {\mathcal {A}}\rangle ^{-\sigma }\\&\quad =\langle {\mathcal {A}}\rangle ^{\sigma }(P+\mathrm {i}-\mathrm {i}-z+z-f(z,hB))(P+\mathrm {i})^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma }\\&\quad =\mathrm {Id}-(\mathrm {i}+z)\langle {\mathcal {A}}\rangle ^{\sigma }(P+\mathrm {i})^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma }\\&\qquad +\, \langle {\mathcal {A}}\rangle ^{\sigma }(z-f(z,hB))\langle {\mathcal {A}}\rangle ^{-\sigma }\langle {\mathcal {A}}\rangle ^{\sigma }(P+\mathrm {i})^{-1}\langle {\mathcal {A}}\rangle ^{-\sigma } \end{aligned}$$

and we again use [13, Lem. 2.1] and assumption (A) for \(k=0\). \(\square \)

The next result is an adaptation of [13, Lem. 3.1] to our setting.

Lemma C.4

Let \(0<\sigma \le 1\) and let \(\tau ,\chi \in {\mathcal {C}}^{\infty }_{\mathrm {c}}\left( {\mathbb {R}},[0,1]\right) \) such that \(\chi \equiv 1\) on I and \(\tau \chi =\chi \). Consider the following three statements:

  1. (i)

    \(\displaystyle \sup _{z\in J^{+}}\Vert \langle {\mathcal {A}}\rangle ^{-\sigma }R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \lesssim h^{-1};\)

  2. (ii)

    For all \(z\in J^{+}\) and all \(u\in (P+\mathrm {i})^{-1}{\mathscr {D}}(\langle {\mathcal {A}}\rangle ^{\sigma })\),

    $$\begin{aligned} \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert&\lesssim h^{-1}\Vert (P-f(z,hB))u\Vert +h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{\sigma }(P-f(z,hB))\chi (P)u\Vert ; \end{aligned}$$
  3. (iii)

    For all \(z\in J^{+}\) and all \(u\in {\mathscr {D}}(\langle {\mathcal {A}}\rangle ^{\sigma })\),

    $$\begin{aligned} \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }\chi (P)u\Vert&\lesssim h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{\sigma }(P_\tau -f(z,hB))\chi (P)u\Vert . \end{aligned}$$

    If h is sufficiently small, then (iii) implies (ii) and (ii) implies (i).

Proof

First of all, observe that (i) makes sense by Lemma C.2, and (ii), (iii) make sense by Corollary C.3 and because \(P\chi (P)=P_{\tau }\chi (P)\).

  • We show that (ii) implies (i). Let \(u\in {\mathcal {H}}\) and let \(v:=R(z,hB)\langle A\rangle ^{-\sigma }u\). Then

    $$\begin{aligned} w&:=u-\langle {\mathcal {A}}\rangle ^{\sigma }(f(z,hB)-\mathrm {i})R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }u\in {\mathcal {H}}. \end{aligned}$$

    This makes sense if h is small enough because R(zhB) preserves \({\mathscr {D}}(\langle {\mathcal {A}}\rangle ^\sigma )\) by Lemma C.2 and because

    $$\begin{aligned} \langle {\mathcal {A}}\rangle ^{\sigma }(f(z,hB)-\mathrm {i})\langle {\mathcal {A}}\rangle ^{-\sigma }&=\langle {\mathcal {A}}\rangle ^{\sigma }(f(z,hB)-z)\langle {\mathcal {A}}\rangle ^{-\sigma }+(z-\mathrm {i}) \end{aligned}$$

    is bounded by assumption (A) for \(k=0\). Next, using the resolvent identity \((P+\mathrm {i})^{-1}-R(z,hB)=(P+\mathrm {i})^{-1}(f(z,hB)-\mathrm {i})R(z,hB)\), we see that

    $$\begin{aligned} (P+\mathrm {i})^{-1}\langle A\rangle ^{-\sigma }w&=\big ((P+\mathrm {i})^{-1}-(P+\mathrm {i})^{-1}(f(z,hB)-\mathrm {i})\,R(z,hB)\big )\langle {\mathcal {A}}\rangle ^{-\sigma }u\\&=R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }u\\&=v \end{aligned}$$

    so that \(v\in (P+\mathrm {i})^{-1}{\mathscr {D}}(\langle {\mathcal {A}}\rangle ^{\sigma })\). Hence, applying (ii) to v yields

    $$\begin{aligned}&\Vert \langle {\mathcal {A}}\rangle ^{-\sigma }R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert =\Vert \langle {\mathcal {A}}\rangle ^{-\sigma }v\Vert \nonumber \\&\quad \lesssim h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert +h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{\sigma }(P-f(z,hB))\chi (P)R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert \nonumber \\&\quad \lesssim h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert +h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{\sigma }[P-f(z,hB),\chi (P)]R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert \nonumber \\&\qquad +h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{\sigma }\chi (P)\langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert . \end{aligned}$$

    By assumption (A) for \(k=1\) and Lemma C.2, we have

    $$\begin{aligned}&\Vert \langle {\mathcal {A}}\rangle ^{\sigma }[P-f(z,hB),\chi (P)]R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert \\&\quad =\Vert \langle {\mathcal {A}}\rangle ^{\sigma }[z-f(z,hB),\chi (P)]R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert \\&\quad \le \delta (h\Vert B\Vert _{_{P}})\Vert \langle {\mathcal {A}}\rangle ^{\sigma }[\varepsilon (hB),\chi (P)]\langle {\mathcal {A}}\rangle ^{-\sigma }\Vert \Vert \langle {\mathcal {A}}\rangle ^{\sigma }R(z,hB)\langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert \\&\quad \lesssim h\delta (h\Vert B\Vert _{_{P}}). \end{aligned}$$

    Therefore, (i) follows from (ii) if h is small enough.

  • We show that (iii) implies (ii). Let \({\tilde{\chi }}:=1-\chi \) and let \(u\in (P+\mathrm {i})^{-1}{\mathscr {D}}\left( \langle {\mathcal {A}}\rangle ^{\sigma }\right) \). We write

    $$\begin{aligned} \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }u\Vert&\le \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }\chi (P)u\Vert +\Vert \langle {\mathcal {A}}\rangle ^{-\sigma }{\tilde{\chi }}(P)u\Vert \end{aligned}$$
    (61)

    and (iii) implies that

    $$\begin{aligned} \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }\chi (P)u\Vert&\lesssim h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{\sigma }(P-f(z,hB))\chi (P)u\Vert \end{aligned}$$

    because \(\tau \equiv 1\) on \(\mathrm {Supp\,}\chi \). In order to control the term involving \({\tilde{\chi }}(P)\) in (61), we write \({\tilde{\chi }}=\psi _{-}+\psi _{+}\) with \(\psi _{\pm }\in {\mathcal {C}}^{\infty }\left( {\mathbb {R}},[0,1]\right) \) such that \(\mathrm {Supp}\ \psi _{-}\subset ]-\infty ,\alpha ]\) and \(\mathrm {Supp}\ \psi _{+}\subset [\beta ,+\infty [\). We also pick \(\rho \in {\mathcal {C}}^\infty _{\mathrm {c}}({\mathbb {R}},{\mathbb {R}})\) such that \(\rho \psi _-=\psi _-\). Since \(B\in L^{\infty }_{\ell \mathrm {oc}}(P)\), we have for any \(v\in {\mathscr {D}}(P)\)

    $$\begin{aligned}&\mathfrak {R}\big \langle \psi _{-}(P)^{2}(f(z,hB)-P)v,v\big \rangle \nonumber \\&\quad =\mathfrak {R}\big \langle \psi _{-}(P)^{2}zv,v\big \rangle +\mathfrak {R}\big \langle \psi _{-}(P)^{2}\delta (h\Vert B\Vert _{_{P}})\varepsilon (hB)v,v\big \rangle -\mathfrak {R}\big \langle \psi _{-}(P)^{2}Pv,v\big \rangle \nonumber \\&\quad \ge a\Vert \psi _{-}(P)v\Vert ^2-\delta (h\Vert B\Vert _{_{P}})\Vert \rho (P)\varepsilon (hB)\Vert _{_{P}}\Vert \psi _{-}(P)v\Vert ^2-\alpha \Vert \psi _{-}(P)^2v\Vert ^2\nonumber \\&\quad \ge c_-\Vert \psi _{-}(P)v\Vert ^{2} \end{aligned}$$
    (62)

    where \(c_->0\) if h is sufficiently small. Using Cauchy-Schwarz inequality, we get \(\Vert \psi _{-}(P)(P-f(z,hB))v\Vert \ge c_-\Vert \psi _{-}(P)v\Vert \) and thus \(\Vert \psi _{-}(P)R(z,hB)v\Vert \lesssim \Vert \psi _{-}(P)v\Vert \). Similarly, one can show \(\Vert \psi _{+}(P)R(z,hB)v\Vert \lesssim \Vert \psi _{+}(P)v\Vert \). These inequalities and \({\tilde{\chi }}^{2}=(\psi _{-}+\psi _{+})^{2}=\psi _{-}^{2}+\psi _{+}^{2}\) then imply

    $$\begin{aligned} \Vert {\tilde{\chi }}(P)R(z,hB)v\Vert \lesssim \Vert {\tilde{\chi }}(P)v\Vert \end{aligned}$$

    which in turn implies for \(u\in {\mathscr {D}}(P)\)

    $$\begin{aligned} \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }{\tilde{\chi }}(P)u\Vert&\lesssim \Vert {\tilde{\chi }}(P)u\Vert \\&=\Vert {\tilde{\chi }}(P)R(z,hB)(P-f(z,hB))u\Vert \\&\lesssim \Vert {\tilde{\chi }}(P)(P-f(z,hB))u\Vert \\&\lesssim \Vert (P-f(z,hB))u\Vert . \end{aligned}$$

\(\square \)

Proof of Theorem C.1.

We show that the regularity (P) and the Mourre estimate (M) are enough to establish (60). As pointed out at the beginning of [13], the key point is the following energy estimate: for any self-adjoint operators H acting on \({\mathcal {H}}\), \(u\in {\mathscr {D}}\left( H\right) \), \(\tau \in {\mathcal {C}}^{\infty }_{\mathrm {c}}\left( {\mathbb {R}},[0,1]\right) \) and \(P_{\tau }:=\tau (P)P\), we have

$$\begin{aligned} 2\mathfrak {I}\big \langle Hu,(P_{\tau }-f(z,hB))u\big \rangle&=\big \langle u,[P_{\tau },\mathrm {i}H]u\big \rangle -2\mathfrak {I}\big \langle u,f(z,hB)Hu\big \rangle \end{aligned}$$
(63)

where the commutator must be understood as a quadratic form on \({\mathscr {D}}(H)\).

We follow the proof of [13, Thm. 1]. Let \(\tau ,\chi \in {\mathcal {C}}^{\infty }_{\mathrm {c}}\left( {\mathbb {R}},[0,1]\right) \) such that \(\chi \equiv 1\) on I and \(\tau \chi =\chi \) and let

$$\begin{aligned} F\left( \xi \right)&:=-\int _{\xi }^{+\infty }g(\zeta )^{2}\mathrm {d}\zeta \end{aligned}$$

with \(g\in {\mathcal {C}}^{\infty }\left( {\mathbb {R}},[0,1]\right) \) satisfying \(g\left( \xi \right) =0\) for \(\xi \ge 2\) and \(g\left( \xi \right) =1\) for \(\xi \le 1\). By Lemma C.4, it is sufficient to prove the following estimate: for any \(z\in J^+\) and \(u\in {\mathscr {D}}(\langle {\mathcal {A}}\rangle ^{\sigma })\),

$$\begin{aligned} \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }\chi (P)u\Vert&\lesssim h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{\sigma }(P_\tau -f(z,hB))\chi (P)u\Vert . \end{aligned}$$

As \(P\in {\mathcal {C}}^{2}\left( {\mathcal {A}}\right) \), P and \({\mathcal {A}}\) are self-adjoint and satisfy the Mourre estimate (M) on I, we can apply the estimate (3.30) in the proof of [13, Thm. 1]:

$$\begin{aligned} \chi (P)[P_\tau ,\mathrm {i}F({\mathcal {A}})]\chi (P)\gtrsim h\chi (P)\langle {\mathcal {A}}\rangle ^{-2\sigma }\chi (P). \end{aligned}$$
(64)

Now we apply the identity (63) with \(H=F({\mathcal {A}})\): for all \(u\in {\mathscr {D}}({\mathcal {A}})\),

$$\begin{aligned} 2\mathfrak {I}\big \langle F({\mathcal {A}})u,(P_{\tau }-f(z,hB))u\big \rangle&=\big \langle u,[P_{\tau },\mathrm {i}F({\mathcal {A}})]u\big \rangle +2\mathfrak {I}\big \langle f(z,hB)u,F({\mathcal {A}})u\big \rangle . \end{aligned}$$

Since \(F<0\) is bounded and \(\mathfrak {I}z>0\), we can write for all h sufficiently small

$$\begin{aligned}&2\mathfrak {I}\big \langle F({\mathcal {A}})u,(P_\tau -f(z,hB))u\big \rangle \\&\quad =\big \langle u,[P_{\tau },\mathrm {i}F({\mathcal {A}})]u\big \rangle -2(\mathfrak {I}z)\big \langle u,F({\mathcal {A}})u\big \rangle -2\delta (h\Vert B\Vert _{_{P}})\mathfrak {I}\big \langle u,\varepsilon (hB)F({\mathcal {A}})u\big \rangle \\&\quad >\big \langle u,[P_\tau ,\mathrm {i}F({\mathcal {A}})]u\big \rangle -2\delta (h\Vert B\Vert _{_{P}})\Vert \varepsilon (hB)u\Vert \Vert F({\mathcal {A}})u\Vert \end{aligned}$$

where we used that that \(\varepsilon (hB)\in {\mathcal {B}}({\mathscr {D}}({\mathcal {A}}))\) by Assumption (A). It thus follows

$$\begin{aligned} 2\mathfrak {I}\big \langle F({\mathcal {A}})u,(P_\tau -f(z,hB))u\big \rangle&\ge \big \langle u,[P_\tau ,\mathrm {i}F({\mathcal {A}})]u\big \rangle . \end{aligned}$$
(65)

Plugging the estimate (64) into inequality (65) and putting \(\chi (P)u\) instead of u yield

$$\begin{aligned} \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }\chi (P)u\Vert ^{2}&=\langle u,\chi (P)\langle {\mathcal {A}}\rangle ^{-2\sigma }\chi (P)u\rangle \nonumber \\&\lesssim h^{-1}\big \langle u,\chi (P)[P_\tau ,\mathrm {i}F({\mathcal {A}})]\chi (P)u\big \rangle \nonumber \\&\le h^{-1}\big |\big \langle F({\mathcal {A}})\chi (P)u,(P_\tau -f(z,hB))\chi (P)u\big \rangle \big |. \end{aligned}$$

Using again the boundedness of F, we get

$$\begin{aligned} \Vert \langle {\mathcal {A}}\rangle ^{-\sigma }\chi (P)u\Vert ^{2}&\lesssim h^{-1}\Vert \langle {\mathcal {A}}\rangle ^{-\sigma }\chi (P)u\Vert \Vert \langle {\mathcal {A}}\rangle ^{\sigma }(P_\tau -f(z,hB))\chi (P)u\Vert \end{aligned}$$

which establishes the point (iii) and thus the point (i) in Lemma C.4. \(\square \)

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Besset, N. Decay of the Local Energy for the Charged Klein–Gordon Equation in the Exterior De Sitter–Reissner–Nordström Spacetime. Ann. Henri Poincaré (2020). https://doi.org/10.1007/s00023-020-00919-z

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