# On pointwise decay of waves

W. Schlag  Yale University, Department of Mathematics, 10 Hillhouse Avenue,
New Haven, CT 06511, U.S.A<EMAIL_ADDRESS>
###### Abstract

This note introduces some of the basic mechanisms relating the behavior of the
spectral measure of Schrödinger operators near zero energy to the long-term
decay and dispersion of the associated Schrödinger and wave evolutions. These
principles are illustrated by means of the author’s work on decay of
Schrödinger and wave equations under various types of perturbations including
those of the underlying metric. In particular, we consider local decay of
solutions to the linear Schrödinger and wave equations on curved backgrounds
which exhibit trapping. A particular application are waves on a Schwarzschild
black hole space-time. We elaborate on Price’s law of local decay which
accelerates with the angular momentum, which has recently been settled by
Hintz, also in the much more difficult Kerr black hole setting. While the
author’s work on the same topic was conducted ten years ago, the global
semiclassical representation techniques developed there have recently been
applied by Krieger, Miao, and the author to the nonlinear problem of stability
of blowup solutions to critical wave maps under non-equivariant perturbations.

##  I Introduction

This note mainly serves as an introduction to the techniques used in the
papers DSS1 , DSS2 , which are concerned with the local decay of waves on a
Schwarzschild background. The decay estimates are obtained by separation of
variables and the analysis of the flow for each angular momentum $\ell$ in
DSS1 . By means of a semiclassical WKB analysis in the parameter
$\hbar:=\ell^{-1}$ carried out by means of a global Liouville-Green transform,
as well as semiclassical Mourre theory at energies near the top of the
barrier, DSS2  sums up these fibered estimates over all angular momenta
incurring the loss of finitely many angular derivatives. Note that DSS1 , DSS2
are not entirely self-contained and rely in part on SSS1 , SSS2 , DS , CSST ,
CDST . As shown in these references, the Schrödinger flow can be analyzed
analogously. The original motivation for SSS1 ; SSS2  was to study the long-
term dispersive behavior of solutions to Schrödinger and wave equations on
specific non-compact manifolds exhibiting closed geodesics, such as the
hyperboloid of one sheet. In analogy with the unique periodic geodesic on such
a hyperboloid, which is exponentially unstable, the surface of closed
geodesics around a Schwarzschild black hole is known as photon sphere and
corresponds to the collection of all periodic light rays. The photon sphere is
also unstable.

Recently, in joint work with Krieger and Miao KMS , the semiclassical
techniques leading to a precise representation of the resolvent and the
spectral measure for all energies and all small $\hbar$ developed in CSST ;
CDST , played a crucial role in a nonlinear asymptotic stability question of
blowup solutions to energy critical wave maps into the $2$-sphere. In stark
contrast to the linear case, modes of fixed frequencies interact through the
nonlinearities. Controlling these interaction naturally leads to a
paradifferential calculus involving several simultaneous semi-classical
parameters. The nonlinear work KMS  served as the main motivation for writing
this note, which should not be mistaken for a general review. Numerous
references are missing, which touch in one way or another on the ensuing
discussion. A survey of dispersive decay of Schrödinger, wave, and Klein-
Gordon evolutions involving electric, magnetic, and metric perturbations,
including the semi-classical and gravitational literature, would require many
hundreds of citations. The scope and purpose of this communication is much
more limited. For example, magnetic and time-dependent potentials are not
discussed in detail.

The author’s investigations in this area where largely motivated by Bourgain’s
book Bour  which states at the end of page 27: On the other hand, it would be
most interesting to prove that analogue of (1.99) in low dimensions $d=1,2$.
This is certainly a project of independent importance. Here (1.99) refers to
the pointwise decay of the Schrödinger evolution proved by Journé, Soffer and
Sogge JSS , see the following section.

##  II Lower order perturbations

The free Schrödinger evolution $\psi(t)=e^{-it\Delta}\psi_{0}$ in
${\mathbb{R}}^{d+1}_{t,x}$ satisfies the basic estimates

$\displaystyle\|\psi(t)\|_{H^{s}}$ $\displaystyle=\|\psi_{0}\|_{H^{s}}$ (1)
$\displaystyle\|\psi(t)\|_{\infty}$ $\displaystyle\leq
Ct^{-\frac{d}{2}}\|\psi_{0}\|_{1}$ (2)

as can be seen from the representation

$\displaystyle\psi(t,x)$
$\displaystyle=(2\pi)^{-d}\int_{{\mathbb{R}}^{d}}e^{i(t|\xi|^{2}+x\cdot\xi)}\hat{f}(\xi)\,d\xi$
$\displaystyle=c(d)t^{-\frac{d}{2}}\int_{{\mathbb{R}}^{d}}e^{i\frac{|x-y|^{2}}{4t}}f(y)\,dy$

respectively. For the wave equation $\Box u=\partial_{t}^{2}u-\Delta u=0$ in
$d+1$ dimensions one has constancy of the energy

${\mathcal{E}}(u)=\|\nabla u\|_{2}^{2}+\|\partial_{t}u\|_{2}^{2}$ (3)

as well as the dispersive decay

$\|u(t)\|_{\infty}\lesssim
t^{-\frac{d-1}{2}}(\|u(0)\|_{\dot{B}_{1,1}^{\frac{d+1}{2}}}+\|\partial_{t}u(0)\|_{\dot{B}_{1,1}^{\frac{d-1}{2}}})$
(4)

where $\dot{B}_{1,1}^{\alpha}$ stands for the usual Besov space:
$\|f\|_{\dot{B}^{\alpha}_{1,1}}=\sum_{j\in{\mathbb{Z}}}2^{\alpha
j}\|P_{j}f\|_{1}$ where $P_{j}$ is the Littlewood-Paley projection onto
frequencies of size $2^{j}$. In odd spatial dimensions one can improve the
right-hand side to

$\|u(0)\|_{\dot{W}^{\frac{d+1}{2},1}}+\|\partial_{t}u(0)\|_{\dot{W}^{\frac{d-1}{2},1}}$

where $\dot{W}^{\alpha,p}$ stands for the homogeneous Sobolev spaces. To
obtain (4), one considers a fixed frequency shell $\\{|\xi|\sim 2^{j}\\}$ and
rescales to $j=0$. Then

$e^{it\sqrt{-\Delta}}P_{0}f(x)=\int_{{\mathbb{R}}^{2d}}e^{i((x-y)\cdot\xi+t|\xi|)}\chi(\xi)\,d\xi\,f(y)\,dy$

where $\chi$ is a cut-off function corresponding to $P_{0}$. Passing to polar
coordinates and applying stationary phase to integrals over spheres then
yields the desired $t^{-\frac{d-1}{2}}$ decay.

While (1) and (3) are a result of the time-translation invariance of the
underlying Lagrangians (via Noether’s theorem) and therefore robust under
perturbations that preserve this symmetry, (2) and (4) follow from the form of
the fundamental solutions and are therefore less stable. In fact, much effort
has been devoted to deriving similar dispersive estimates for perturbations of
the free Schrödinger and wave equations in the past thirty years. The starting
point in these investigations was to consider local decay estimates which are
quite different from the global ones as in (2) and (4) (as we shall see
below). Local here refers to the fact that the decay is measured only in
weighted spaces rather than in a uniform sense.

###  II.1 Local decay for $-\Delta+V$

####  II.1.1 The Schrödinger evolution

In JK  Jensen and Kato showed that for $H=-\Delta+V$ in the three-dimensional
case, with real-valued $V$ which is bounded and decays at a sufficient
polynomial rate one has the local decay

$\|\langle
x\rangle^{-\sigma}e^{itH}P_{c}f\|_{L^{2}({\mathbb{R}}^{3})}\lesssim\langle
t\rangle^{-\frac{3}{2}}\|\langle
y\rangle^{\sigma}f\|_{L^{2}({\mathbb{R}}^{3})}$ (5)

for some $\sigma>0$ and with $P_{c}=\chi_{(0,\infty)}(H)$ the projection onto
the continuous spectrum. Moreover, one needs to assume that zero energy is
neither an eigenvalue nor a resonance of $H$ (which is also referred to as
zero energy being regular, the other case being singular).

This latter property refers to the validity of the resolvent estimate

$\displaystyle\sup_{{\,\rm Im\,}z>0}\|\langle
x\rangle^{-\sigma}(-\Delta+V+z)^{-1}\langle x\rangle^{-\sigma}\|_{2\to
2}<\infty$ (6)

with $\sigma>0$ sufficiently large. Alternatively, it is the same as the
nonexistence of $f\not\equiv 0$ with

$\displaystyle Hf=0,\quad
f\in\bigcap_{{\varepsilon}>0}L^{2,-\frac{1}{2}-{\varepsilon}}({\mathbb{R}}^{3})$
(7)

It was already observed by Rauch Rauch  for exponentially decaying potentials,
that a zero energy resonance or eigenvalue, i.e., in the case when (7) admits
a nontrivial solution, destroys the dispersive estimate. More specifically,
one loses one power of $t$ in the decay law in that case.

To see the relevance of zero energy resonances, we expand the resolvent for
$z\to 0$ in ${\,\rm Im\,}z>0$ as follows:

$\displaystyle R(z)$ $\displaystyle:=(-\Delta+V+z)^{-1}$ (8)
$\displaystyle=z^{-1}B_{-1}+z^{-\frac{1}{2}}B_{-\frac{1}{2}}+B_{0}+z^{\frac{1}{2}}B_{\frac{1}{2}}+\rho(z)$

where $B_{-1},\ldots,B_{1}$ are bounded in weighted
$L^{2}({\mathbb{R}}^{3})$-spaces, and with

$\|\langle x\rangle^{-\sigma}\rho(z)f\|_{2}\lesssim|z|\|\langle
x\rangle^{\sigma}f\|_{2}$

for small $z$. Clearly, $B_{-1}$ is the orthogonal projection onto the zero
eigenspace, and zero energy is regular for $H$ iff
$B_{-1}=B_{-\frac{1}{2}}=0$. In general, $B_{-1},B_{-\frac{1}{2}}$ are of
finite rank. As an example, consider the case $V=0$ in three dimensions for
which one has (with $z=-\zeta^{2})$

$(-\Delta-\zeta^{2})^{-1}(x,y)=\frac{e^{i\zeta|x-y|}}{4\pi|x-y|},\quad{\,\rm
Im\,}\zeta>0$

and the Laurent expansion (8) is now obtained by Taylor expanding the
exponential on the right-hand side. It follows that zero energy is neither an
eigenvalue nor a resonance in that case. In contrast, the one-dimensional case
satisfies

$(-\Delta-\zeta^{2})^{-1}(x,y)=\frac{e^{i\zeta|x-y|}}{2i\zeta},\quad{\,\rm
Im\,}\zeta>0$

and zero is a resonance (but not an eigenvalue). We used here that (8) remains
correct in all odd dimensions, whereas in even dimensions a logarithm appears.
Indeed, the free resolvent in $d$-dimensions satisfies

$(-\Delta-\zeta^{2})^{-1}(x,y)=c_{d}\,\zeta^{\frac{d-2}{2}}|x-y|^{-\frac{d-2}{2}}H_{\frac{d-2}{2}}^{+}(\zeta|x-y|)$
(9)

and the Hankel functions of integer order exhibit a logarithmic branch point
at zero.

To pass to estimates on the evolution one now uses the Laplace transform (as
in the Hille-Yosida theorem) to conclude that

$e^{itH}P_{c}=\frac{1}{2\pi}\int_{p_{0}-i\infty}^{p_{0}+i\infty}e^{tp}R(ip)P_{c}\,dp$
(10)

where $p_{0}>0$ is arbitrary. Assuming for simplicity that $V$ is compactly
supported it follows from the resolvent identity that the Green function
$R(ip)(x,y)$ admits a meromorphic continuation to the left-half plane.

Figure 1: Deforming the contour

One now deforms the contour in (10) as shown in Figure 1. The finitely many
residues $\\{\zeta_{j}\\}$ of the resolvent in the left-half plane (which lie
in ${\mathbb{C}}\setminus(-\infty,0]$) contribute the exponentially decaying
expression

$\sum_{\zeta_{j}}e^{\zeta_{j}t}P_{\phi_{j}}$

where $P_{\phi_{j}}$ is the projection onto the resonant states corresponding
to the complex resonance at $\zeta_{j}$ (the resonant states are commonly
referred to as meta-stable states or quasinormal modes). The more slowly
decaying tail is a result of the branching of the resolvent at $p=0$. More
specifically, it can be read off from (8) via the following standard result
which is known as Watson’s lemma (the notation $\sim$ refers to asymptotic
expansions in the sense of Poincaré).

######  Lemma II.1.

Let $f$ be a complex-valued function of a real variable $x$ such that

  * •

$f$ is continuous on $(0,\infty)$

  * •

$f(x)\sim\sum_{n=0}^{\infty}a_{n}\,x^{\lambda_{n}-1}\qquad\text{\ \ as\ \
}x\to 0+$ with $0<\lambda_{0}<\lambda_{1}<\ldots$.

  * •

$f(x)=O(e^{cx})$ as $x\to\infty$ for some $c>0$

This condition can be removed since Watson’s lemma is really local on some
interval $(0,x_{0})$, but we choose to state it in this global form. Then for
every small $\delta>0$ one has

$\int_{0}^{\infty}e^{-xp}f(x)\,dx\sim\sum_{n=0}^{\infty}\frac{a_{n}}{p^{\lambda_{n}}}\Gamma(\lambda_{n})$

as $|p|\to\infty$ in $|\arg(p)|\leq\frac{\pi}{2}-\delta$.

Therefore, if $B_{-\frac{1}{2}}\neq 0$ in (8), then one obtains
$t^{-\frac{1}{2}}$ local decay, whereas otherwise the rate is
$t^{-\frac{3}{2}}$ which is the same as in (2). Evidently, the global (i.e.,
$L^{\infty}$) decay can never be faster than the local one — whence the need
to exclude zero energy resonance and eigenvalues to preserve (2). We remark
that one can have $B_{-\frac{1}{2}}\neq 0$ even in case the only solutions to
(7) are in $L^{2}$ (in other words, if zero energy is an eigenvalue but not a
resonance). This implies that $t^{-\frac{3}{2}}$ does not result from applying
$P_{c}$ to the evolution even when zero is not a resonance but only an
eigenvalue.

Starting from the spectral representation

$e^{itH}P_{c}=\int_{0}^{\infty}e^{it\lambda}\,E(d\lambda)$ (11)

instead of (10) with the spectral measure

$E(d\lambda)=\frac{1}{2\pi i}[R(\lambda+i0)-R(\lambda-i0)]P_{c}\,d\lambda$

Jensen and Kato derive local decay estimates but under much less severe
restrictions on the decay of $V$ and also on the notion of locality in the
decay estimate. However, it is clear from (11) that the main issue here is
once again the contributions from $\lambda=0$ coming from (8). Indeed, for
energies $\lambda>\lambda_{0}>0$ where $\lambda_{0}>0$ is arbitrary but fixed,
one has the so-called limiting absorption resolvent bounds

$\sup_{\lambda>\lambda_{0}}\big{\|}\langle\cdot\rangle^{-\sigma}\partial_{\lambda}^{k}R(\lambda\pm
i0)\langle\cdot\rangle^{-\sigma}\big{\|}<\infty$

for all $0\leq k\leq k_{0}$ and with $\sigma>0$ depending on $k$ (the value of
$k_{0}$ here depends on the decay of $V$). These bounds allow one to integrate
by parts in (11) in the range $\lambda>\lambda_{0}$ which leads to arbitrary
decay in time.

The most general results on local decay for the Schrödinger evolution were
obtained by Murata Mur . He derived expansions in time for evolutions
$e^{itH}$ in all dimensions and with elliptic $H=-p(D)+V$ where $V$ is a
compact operator in suitable weighted Sobolev spaces. As a general rule, the
coefficients in these expansions corresponding to nongeneric threshold
behavior (i.e., slow decay resulting from threshold eigenvalues or resonances)
are finite rank operators which can be computed in terms of the eigenfunctions
and resonant states. As an example, the one-dimensional free evolution
satisfies

$\displaystyle e^{-it\partial_{x}^{2}}f(x)$
$\displaystyle=ct^{-\frac{1}{2}}\int f(y)\,dy+\rho(t)f(x),$
$\displaystyle\|\langle\cdot\rangle^{-\sigma}\rho(t)f\|_{2}$
$\displaystyle\lesssim t^{-\frac{3}{2}}\|\langle\cdot\rangle^{\sigma}f\|_{2}$

The appearance of the projection $f\mapsto\int f(y)\,dy$ onto the constant
functions is natural in view of the fact that the resonant function of
$-\partial_{x}^{2}$ at zero energy is $f\equiv 1$. This also shows that one
should expect $t^{-\frac{3}{2}}$ local decay for one-dimensional operators
without zero energy resonance (note that, however, the global decay as in (2)
is never faster than $t^{-\frac{1}{2}}$ if $d=1$), at least assuming
sufficient decay of $V$. This is indeed the case, see Mur . In two dimensions,
Murata obtained the faster local $L^{2}({\mathbb{R}}^{2})$ decay
$t^{-1}\log^{-2}t$ for operators without resonance. Erdogan, Green EG2
established the more difficult sharp weighted $L^{1}\to L^{\infty}$ version of
these global bounds in ${\mathbb{R}}^{2}$ assuming that $0$ energy is regular.
These faster local decays (as compared to the global $L^{\infty}$ decay) play
a crucial role in certain applications to nonlinear stability results, see
Buslaev, Perelman BP1 , Krieger, Schlag KS1  for the one-dimensional case, and
Kirr, Zarnescu KZ2  for examples of two-dimensional applications. Loosely
speaking, the point here is that in contrast to the global decay rates these
faster non-resonant local rates are integrable in time which allows one to
close certain bootstrap arguments involving the Duhamel formula.

####  II.1.2 The wave evolution

Similar considerations apply to the wave equation. Indeed, let $\Box u=0$,
with $(u(0),\partial_{t}u(0))=(0,g)$ (initial data $(f,0)$ are then handled by
differentiating in time). Then instead of (10) one has

$u(t)=\frac{\sin(t\sqrt{H})}{\sqrt{H}}P_{c}g=\frac{1}{2\pi
i}\int_{p_{0}-i\infty}^{p_{0}+i\infty}e^{tp}R(p^{2})P_{c}\,gdp$ (12)

where $p_{0}>0$. In contrast to the Schrödinger case the resolvent $R(p^{2})$
in odd dimensions is now analytic around $p=0$ (assuming that there is no zero
energy resonance or eigenvalue) which results in arbitrary local decay of
$u(t)$. More precisely, if $V$ decays exponentially, thus allowing for
analytic continuation of the Green function to the left-half plane, one
obtains exponential decay in time relative to weighted $L^{2}$ in space. This
is of course a consequence of the sharp Huyghens principle in odd dimensions
which states that the fundamental solution of the free wave equation is
localized to a sphere with radius given by the time. We see from this informal
discussion that this principle is robust under perturbations (at least in the
sense that the perturbed wave $u(t)$ will decay very rapidly at distances $\ll
t$ from the origin, which of course is far from being able to describe the
fundamental solution). Note the stark contrast between the strong local decay
of the wave equation as compared to the specific global decay given by (4).

On the other hand, in even dimensions the resolvent will exhibit a $\log p$
singularity, see (9). Due to this branching of the resolvent at $p=0$,
Watson’s lemma implies an explicit power law depending on the dimension
governing the tail of the wave near the origin. This is in agreement with the
fact that there is no sharp Huyghens principle in even dimensions.

To summarize this section, one sees that the local decay for both the
Schrödinger and the wave equation is entirely determined by the singularity
(often but not necessarily by branching) of the resolvent $(-\Delta+V+z)^{-1}$
at $p=0$ where $z=-ip$ in the former case, and $z=p^{2}$ in the latter case,
respectively.

###  II.2 Global decay for $-\Delta+V$

####  II.2.1 The Schrödinger evolution

The first result which proved (2) for $H=-\Delta+V$ in dimensions $d\geq 3$
was obtained by Journé, Soffer, and Sogge JSS . Following unpublished work by
Ginibre, we now give a short proof of a simpler estimate, namely

$\|e^{itH}P_{c}f\|_{L^{\infty}+L^{2}({\mathbb{R}}^{d})}\lesssim\langle
t\rangle^{-\frac{d}{2}}\|f\|_{L^{1}\cap L^{2}({\mathbb{R}}^{d})}$ (13)

assuming that $V$ has sufficient decay and that $H$ has no zero energy
eigenvalue or resonance. The logic here is that the Duhamel formula allows one
to upgrade local decay to global one. More precisely, if

$\|\langle
x\rangle^{-\sigma}e^{itH}P_{c}f\|_{L^{2}({\mathbb{R}}^{d})}\lesssim\langle
t\rangle^{-\frac{d}{2}}\|\langle
y\rangle^{\sigma}f\|_{L^{2}({\mathbb{R}}^{d})}$

and if $V$ decays sufficiently fast, then the same estimate holds without
weights in the sense of (13) (provided $d>2$). More precisely, applying the
Duhamel formula twice yields

$\displaystyle e^{itH}P_{c}$
$\displaystyle=e^{-it\Delta}P_{c}+i\int_{0}^{t}e^{-i(t-s)\Delta}Ve^{isH}P_{c}\,ds$
$\displaystyle=e^{-it\Delta}P_{c}+i\int_{0}^{t}e^{-i(t-s)\Delta}VP_{c}e^{-is\Delta}\,ds$
$\displaystyle\qquad+\int_{0}^{t}\int_{0}^{s}e^{-i(t-s)\Delta}Ve^{i(s-s^{\prime})H}P_{c}Ve^{-is^{\prime}\Delta}\,ds^{\prime}\,ds$

Applying the local decay for $e^{isH}$ from the previous section (with
$|V|^{\frac{1}{2}}$ acting as weight, say) as well as the bound

$\|e^{-it\Delta}f\|_{L^{2}+L^{\infty}({\mathbb{R}}^{d})}\lesssim\langle
t\rangle^{-\frac{d}{2}}\|f\|_{L^{1}\cap L^{2}({\mathbb{R}}^{d})}$

to this expression yields for $\|f\|_{L^{1}\cap L^{2}({\mathbb{R}}^{d})}=1$

$\displaystyle\|e^{itH}P_{c}f\|_{L^{\infty}+L^{2}({\mathbb{R}}^{d})}\lesssim\langle
t\rangle^{-\frac{d}{2}}+\int_{0}^{t}\langle t-s\rangle^{-\frac{d}{2}}\langle
s\rangle^{-\frac{d}{2}}\,ds$ $\displaystyle+\int_{0}^{t}\int_{0}^{s}\langle
t-s\rangle^{-\frac{d}{2}}\langle s-s^{\prime}\rangle^{-\frac{d}{2}}\langle
s^{\prime}\rangle^{-\frac{d}{2}}\,ds^{\prime}\,ds\lesssim\langle
t\rangle^{-\frac{d}{2}}$

as claimed provided $d\geq 3$. The main gist of JSS  is now to remove the
$L^{2}$-piece from this argument. This is subtle, as the free estimate
involved $(t-s)^{-\frac{d}{2}}$ which is not integrable at $s=t$. To overcome
this difficulty, Journé, Soffer, and Sogge used the bound

$\sup_{1\leq p\leq\infty}\|e^{-it\Delta}Ve^{it\Delta}\|_{p\to
p}\leq\|\hat{V}\|_{1}$

The point here is that the left-hand side for $V=e^{ix\eta}$ is a translation
operator composed with a unimodular factor and therefore $L^{p}$ bounded.

Rodnianski and the author RS  proved that for all $t>0$

$\|e^{itH}f\|_{L^{\infty}({\mathbb{R}}^{3})}\leq
C(V)t^{-\frac{3}{2}}\|f\|_{L^{1}({\mathbb{R}}^{3})}$ (14)

assuming that

$\sup_{x\in{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}\frac{|V(y)|}{|x-y|}\,dy<4\pi$
(15)

as well as that the so-called Rollnick norm of $V$ is less than $4\pi$. The
left-hand side in (15) is commonly referred to as the Kato norm
$\|\cdot\|_{K}$. The Rollnick condition precludes any spectral problems such
as eigenvalues and a zero energy singularity. The approach of RS  to the
pointwise bounds is based on an expansion into an infinite Born series
followed by term-wise estimation of the resulting kernels. The smallness
condition on $V$ guarantees convergence.

Remarkably, Beceanu and Goldberg BecGol  were able to show that the finiteness
of the Kato norm alone suffices. More precisely, they showed that (14) holds
for $e^{itH}P_{c}$ in three dimensions assuming (15) with $4\pi$ replaced by
$\infty$ and that there are no imbedded eigenvalues and resonances in the
continuous spectrum. They accomplished this by means of Beceanu’s Wiener
algebra techniques, see Bec1 . Recall that Wiener’s classical theorem states
that for any $f\in L^{1}({\mathbb{R}})$ the equation
$(\delta_{0}+f)\ast(\delta_{0}+g)=\delta_{0}$ has a (unique) solution with
$g\in L^{1}({\mathbb{R}})$ if and only if $1+\hat{f}\neq 0$ on ${\mathbb{R}}$.
The relevance of this to the decay of solutions to

$(i\partial_{t}-\Delta+V)\psi=F,\quad\psi(0)=\psi_{0}$

can be seen as follows: let $V_{1}V_{2}=V$, $|V_{1}|=|V_{2}|$ and set

$(T_{V_{2},V_{1}}F)(t)=\int_{0}^{t}V_{2}e^{i(t-s)H_{0}}V_{1}F(s)\,ds$

with $H_{0}=-\Delta$. Then, on the one hand, one has

$V_{2}\psi(t)=(\delta_{0}\mathrm{Id}-iT_{V_{2},V_{1}})^{-1}V_{2}\big{(}e^{itH_{0}}\psi_{0}-i\int_{0}^{t}e^{i(t-s)H_{0}}F(s)\,ds\big{)}$

which is to be interpreted in the convolution algebra
$\mathcal{B}(L^{2}({\mathbb{R}}^{3}),\mathcal{M}_{t}L^{2}({\mathbb{R}}^{3}))$
where $\mathcal{M}_{t}$ are the complex measures on the line. On the other
hand, $\widehat{T_{V_{2},V_{1}}}(\lambda)=iV_{2}R_{0}^{-}(\lambda)V_{1}$ with
$R_{0}^{-}(\lambda)=(H_{0}-(\lambda-i0))^{-1}$. Hence, the invertibility of
$\delta_{0}\mathrm{Id}-iT_{V_{2},V_{1}}$ in
$\mathcal{B}(L^{2}({\mathbb{R}}^{3}),\mathcal{M}_{t}L^{2}({\mathbb{R}}^{3}))$
is the same as the pointwise invertibility of the Birman-Schwinger operator
$\mathrm{Id}+V_{2}R_{0}^{-}(\lambda)V_{1}$. This equivalence is delicate, and
requires $V\in L^{\frac{3}{2},1}({\mathbb{R}}^{3})$ the Lorentz space, whence
$V_{1},V_{2}\in L^{3,2}({\mathbb{R}}^{3})$, and also the Keel-Tao Strichartz
endpoint KT . For the abstract Wiener theorem in this context, see (Bec1, ,
Theorem 1.1) and (BecGol, , Theorem 3).

An alternative and very general approach to proving $L^{p}$ bounds on both
wave and Schrödinger evolutions was found by Yajima Y1 , Y2  who proved
$L^{p}$ boundedness of the wave operators, with the limit being taken in the
strong $L^{2}$-sense,

$W=\lim_{t\to\infty}e^{-itH}e^{-it\Delta}$ (16)

for all $1\leq p\leq\infty$ and $d\geq 3$. The fact that these operators exist
and are isometries $L^{2}\to\mathrm{Ran}(P_{c}(H))$ is a classical fact, see
Kato Kato . They intertwine the free evolution with that of $H$ in the sense
that (with $H_{0}=-\Delta$)

$f(H)P_{c}(H)=Wf(H_{0})W^{*}$

for any Borel function $f$ on ${\mathbb{R}}$. In particular,
$e^{itH}P_{c}(H)=We^{itH_{0}}W^{*}$ and (2) therefore implies the bound

$\|e^{itH}P_{c}f\|_{\infty}\leq Ct^{-\frac{d}{2}}\|f\|_{1}$

whenever $W:L^{\infty}\to L^{\infty}$, $W^{*}:L^{1}\to L^{1}$. Yajima obtains
similar results on $W^{k,p}$ assuming more regularity on $V$ (the amount of
regularity depends on $k$). In view of our discussions of the role of zero
energy resonances for local decay, it follows that Yajima’s result Y1  can
only hold under the assumption that zero energy is neither a resonance nor an
eigenvalue. In three dimensions Y1  requires $|V(x)|\lesssim\langle
x\rangle^{-\sigma}$ with $\sigma>5$ and therefore improves on JSS .

Yajima derives his $L^{p}$ bounds by means of a finite Born series expansion
with a remainder term involving the perturbed resolvent. In case of small
potentials, one can sum up the infinite Born expansion leading to more precise
results in terms of conditions on $V$. In view of the preceding discussion of
Wiener theorems as a means of summing divergent series, it is natural to ask
if Yajima’s theorem could be approached by means of a suitable Wiener algebra.
Beceanu and the author BecSch  carried this out and proved that the wave
operators given by (16) in ${\mathbb{R}}^{3}$ are superpositions of
reflections and translations. In fact, assuming that $|V(x)|\lesssim C\langle
x\rangle^{-\frac{5}{2}-\epsilon}$, and that $0$ energy is neither an
eigenvalue nor a resonance, they showed that there exists $g(x,y,\omega)\in
L^{1}_{\omega}\mathcal{M}_{y}L^{\infty}_{x}$ (with $\mathcal{M}_{y}$ being
finite Borel measures in $y$), i.e.,

$\int_{\mathbb{S}^{2}}\|g(x,dy,\omega)\|_{\mathcal{M}_{y}L^{\infty}_{x}}\,d\omega<\infty$

such that for $f\in L^{2}({\mathbb{R}}^{3})$ one has the representation
formula for the wave operator

$(Wf)(x)=f(x)+\int_{\mathbb{S}^{2}}\int_{{\mathbb{R}}^{3}}g(x,dy,\omega)f(S_{\omega}x-y)\,d\omega.$

where $S_{\omega}x=x-2(x\cdot\omega)\omega$ is a reflection. This of course
implies that $W:X\to X$ is bounded for any function space $X$ on
${\mathbb{R}}^{3}$ with a norm which is invariant under translations and
reflections. The proof of this representation formula in BecSch  is not
entirely straightforward. On the one hand, the algebra to which the Wiener
theorem is applied is somewhat delicate and requires casting the finite order
Born series terms in Yajima’s work Y1  (which involve only finitely many
potentials and free resolvents) in some iterative algebraic framework. In
other words, one needs to find the correct algebra $\mathcal{A}$ and
composition law $\circledast$ as well as operator $T$ to write the third Born
term, say, in the form $T\circledast T\circledast T$ in $\mathcal{A}$.
Furthermore, the classical scattering theory based on weighted $L^{2}$ spaces
does not suffice and it is necessary to invoke the the author’s work with
Ionescu IS , which revisits the classical Agmon-Kato-Kuroda theorem in the
context of Fourier restriction and the Stein-Tomas theorem, as well as the
Keel-Tao endpoint KT . This in turn relies on the Carleman theorems and
absence of imbedded eigenvalues obtained in IJ . It is not known whether a
structure theorem holds under a scaling invariant assumption on $V$, see
however BecSch2  for such a result, albeit involving small scaling-invariant
potentials.

In higher dimensions, it turns out that one needs to assume some regularity of
$V$ in order for the expected $L^{1}({\mathbb{R}}^{d})\to
L^{\infty}({\mathbb{R}}^{d})$ bounds to hold. Indeed, Goldberg, Visan GVis
show that the dispersive bound can fail in dimensions $d>3$ for potentials
that belong to the class $C^{\frac{d-3}{2}-}({\mathbb{R}}^{d})$. The logic
here is that the free resolvent takes the form (in odd dimensions)

$(-\Delta-\lambda^{2}+i0)^{-1}(r)=\frac{e^{i\lambda
r}}{r^{d-2}}\sum_{j=0}^{\frac{d-3}{2}}c_{j}\,(\lambda r)^{j}$ (17)

and the highest power $\lambda^{\frac{d-3}{2}}$ here corresponds to a
$\frac{d-3}{2}$ derivative loss on $V$. In the positive direction, Erdoğan and
Green EG  prove the dispersive bound in dimensions $d=5,7$ assuming that $V\in
C^{\frac{d-3}{2}}({\mathbb{R}}^{d})$ (zero energy resonances cannot arise in
dimensions $d\geq 5$).

The case of low dimensions $d=1$ and $d=2$ always requires a separate analysis
since the free resolvent in those cases exhibits a zero energy singularity
(more precisely, there is a zero energy resonance given by the constant state
$f=1$). We refer the reader to GS1 , Weder Weder1 , d’Ancona, Fanelli DaFa1
for the one-dimensional case, and Sch1  for dispersive estimates for the two-
dimensional case provided zero energy is regular. Erdogan and Green ErdGreen
carried out a more complete analysis of the dispersive decay in
${\mathbb{R}}^{2}$ allowing for $s$ and $p$-wave resonances at zero energy.
This classification refers to nonzero solutions $\psi$ of $H\psi=0$ which (i)
are asymptotic to a nonzero constant at spatial $\infty$ for $s$-waves (ii)
are in $L^{q}({\mathbb{R}}^{2})$ for all $q>2$ for $p$-waves. They showed that
the $s$-wave resonance, which arises in the $V=0$ case, leads to the same
$t^{-1}$ decay as in the free evolution, whereas the $p$-wave destroys this
rate of decay. With Goldberg these authors also obtained such a classification
in ${\mathbb{R}}^{4}$. Finally, more recently Erdoğan, Green, and Toprak have
applied spectral methods to analyze the delicate dispersive decay of the Dirac
operator, see EGT .

####  II.2.2 The wave equation

Starting with Beals and Strauss BS , Bea  many authors have considered the
problem of proving the dispersive estimate (4) for equations $(\Box+V)u=0$,
$(u,\partial_{t}u)(0)=(f,g)$ (it will suffice to set $f=0$). In BS  and Bea
the potential is assumed to be either nonnegative or small (which excludes any
spectral problems), as well as rapidly decaying and smooth. The result is of
the form (4) but with slightly more derivatives on the data. Georgiev,
Visciglia GV  assume that $0\leq V\leq\langle x\rangle^{-2-\epsilon}$ in three
dimensions and obtain (4) for energies away from zero as well as Strichartz
estimates for all energies. Cuccagna Cuc1  proves Strichartz estimates in
three dimensions assuming that $|\partial^{\alpha}V(x)|\lesssim\langle
x\rangle^{-3-\epsilon}$ for $|\alpha|\leq 2$ and that zero energy is regular.
D’Ancona and Pierfelice DAnPier  prove global dispersive (4) for $d=3$
assuming that $\|\min(V,0)\|_{K}<2\pi$ but with $\dot{B}^{1}_{1,1}$ on the
right-hand side. Pierfelice Pie  obtains the same result under the smallness
assumption (15) (the arguments in KS3  yield the same but with $\|\nabla
g\|_{1}$ instead of the Besov norm). D’Ancona, Fanelli DaFa2  consider the
wave and Dirac equations in three dimensions

$\displaystyle u_{tt}-(\nabla+iA)^{2}u+Vu$ $\displaystyle=0$ $\displaystyle
iU_{t}-{\mathcal{D}}U+MU$ $\displaystyle=0$

respectively. Assuming smallness of $A,V,M$ but allowing nearly scaling-
invariant singularities of these functions both at zero and infinity (which
are $|x|^{-1}$, $|x|^{-2}$, and $|x|^{-\frac{1}{2}}$, respectively) the
$t^{-1}$ global decay is obtained but for data in weighted Soboloev and Besov
spaces. By the aforementioned results of Yajima et al. on the
$W^{k,p}$-boundedness of the wave operators one can obtain $L^{p}$ decay
estimates for the wave equation from the free estimates (4). Note that the
Besov spaces are then defined relative to $H$ rather than the free Laplacian,
but it is often possible to pass between the two. For a more recent reference
on the integrated decay of waves, which also allows for magnetic
perturbations, see d’Ancona’s work DAn .

####  II.2.3 The case of singular zero energy

Certain stability problems in physics lead to linear operators with a zero
energy eigenvalue or resonance. Examples are the energy critical wave equation
$\Box u-u^{5}=0$ in ${\mathbb{R}}^{1+3}$ which admits the stationary solutions
$W_{\lambda}(x):=\lambda(1+\lambda^{2}|x|^{2}/3)^{-\frac{1}{2}}$ for
$\lambda>0$. Linearizing around $W_{\lambda}$ leads to
$H=-\Delta-5W_{\lambda}^{4}$ which has $\partial_{\lambda}W_{\lambda}$ as a
resonant mode of zero energy. Another example is the critical Yang-Mills
problem in dimensions $4+1$. It is therefore necessary to obtain dispersive
bounds in this context as well. Note that the local decay of Section II.1
easily allows for this as the asymptotic expansions in time (as derived in Mur
, JK  for example) isolate the contributions of the threshold singularities
and identifies them as being of finite rank. In case of $L^{1}\to L^{\infty}$
this required some additional work, see ES1 , ES2 , and Y4  for the case of
the Schrödinger evolution. Yajima Y4  obtains explicit expressions for the
term $Bt^{-\frac{1}{2}}$ which needs to be subtracted to obtain the
$t^{-\frac{3}{2}}$ decay of the bulk (explicit here means that $B$ can be
computed from the zero energy and resonance states). The wave equation in
three dimensions is analyzed in KS3 . We recall the main linear result from
the latter reference.

######  Proposition II.2.

Assume that $V$ is a real-valued potential such that $|V(x)|\lesssim\langle
x\rangle^{-\kappa}$ where $\kappa>3$ is fixed but arbitrary. If zero energy is
regular for $H$, then

$\Big{\|}\frac{\sin(t\sqrt{H})}{\sqrt{H}}P_{c}f\Big{\|}_{\infty}\lesssim
t^{-1}\|f\|_{W^{1,1}({\mathbb{R}}^{3})}$

for all $t>0$. Now assume that zero is a resonance but not an eigenvalue of
$H=-\Delta+V$. Let $\psi$ be the unique resonance function normalized so that
$\int V\psi(x)\,dx=1$. Then there exists a constant $c_{0}\neq 0$ such that

$\Big{\|}\frac{\sin(t\sqrt{H})}{\sqrt{H}}P_{c}f-c_{0}(\psi\otimes\psi)f\Big{\|}_{\infty}\lesssim
t^{-1}\|f\|_{W^{1,1}({\mathbb{R}}^{3})}$ (18)

for all $t>0$.

Several results exist on the boundedness of the wave operators on $L^{p}$ in
case zero energy is singular. However, they are limited to a smaller range of
$p$ (in $d=3$ one needs $\frac{3}{2}<p<3$), and are less useful for nonlinear
applications, at least in three dimensions. On the other hand, in
${\mathbb{R}}^{2}$, Erdogan, Goldberg and Green EGG  showed that the wave
operators remain bounded in the full range $1<p<\infty$ if $0$ energy exhibits
only an $s$-wave resonance or only a zero energy eigenvalue.

For the Klein-Gordon equation on the line with a non-generic decaying
potential (i.e., the associated Schrödinger operator exhibits a zero energy
resonance), an analogue of Proposition II.2 was obtained in LLSS , albeit for
local decay. This is part of a larger body of work aiming at understanding
kink stability.

##  III Metric perturbations

If one replaces $-\Delta$ by the elliptic operator
$H:=-\sum_{j,k=1}^{d}\partial_{j}(a_{jk}(x)\partial_{k})$ then one encounters
a new obstruction to proving decay estimates in addition to the zero energy
resonance or eigenvalue of Section II: the phenomenon of trapping, which is a
large energy problem. Trapping refers to the possibility that the classical
Hamiltonian

$h(x,\xi):=\frac{1}{2}\sum_{j,k=1}^{d}a_{jk}(x)\xi^{k}\xi^{j}$

exhibits closed trajectories. More precisely, assuming symmetry
$a_{jk}=a_{kj}$ one has the Hamiltonian equations

$\dot{x}:=\sum_{j=1}^{d}a_{j\cdot}(x)\xi^{j},\qquad\dot{\xi}=\frac{1}{2}\sum_{j,k=1}^{d}\nabla_{x}a_{jk}(x)\xi^{k}\xi^{j}$

which might exhibit time-periodic trajectories. To understand the crucial
effect of the existence of closed geodesics, we consider the method of proving
decay estimates using energy estimates:

$\frac{d}{dt}\langle u,A(t)u\rangle=\langle u,i[H,A(t)]u\rangle+\langle
u,\frac{\partial A(t)}{\partial t}u\rangle$

where $u=u(x,t)$ is the solution of the Schrödinger equation, with Hamiltonian
$H$. A similar identity can be applied for the wave equation, see BSof . Next,
suppose the expectation of $A(t)$ is bounded from above, uniformly in $t$, by
$\|u\|_{2}$ and moreover, that the commutator is positive, in the sense that

$i[H,A(t)]+\frac{\partial A(t)}{\partial t}\geq\theta B^{\star}B$

for some $\theta>0$ and some operator $B$. Upon integration over time, we
obtain an integrated decay estimate for $B$:

$\int_{0}^{\infty}\|Bu\|^{2}\,dt\leq c\|u(0)\|_{2}^{2}$

The operator family $A(t)$ is variably called a multiplier, or a propagation
observable, or an escape function, or conjugate operator.

To illustrate this further, let $h(x,\xi)$ be a classical Hamiltonian on
${\mathbb{R}}^{2d}$. If $(x(t),\xi(t))$ is an orbit under the Hamiltonian flow
of $h$, then

$\frac{d}{dt}a(x(t),\xi(t))=\\{h,a\\}(x(t),\xi(t))$

where the right-hand side is the Poisson bracket. For the Euclidean case,
i.e., $h(x,\xi)=\frac{1}{2}\xi^{2}+V(x)$ one can take
$a(x,\xi)=x\cdot\xi=\\{h,\frac{1}{2}|x|^{2}\\}$ which gives
$\\{h,a\\}=2h-2V-x\cdot\nabla V$. Now suppose that $-2V(x)-x\cdot\nabla
V(x)\geq 0$ for $|x|\geq R>0$, say. Since $h$ is conserved, we conclude that a
trajectory with $h=\alpha>0$ which remains in $|x|\geq R$ satisfies

$\frac{d^{2}}{dt^{2}}\frac{1}{2}|x(t)|^{2}\geq 2\alpha$

and therefore $|x(t)|$ grows linearly in $t$. This indicates that
$(x(t),\dot{x}(t))$ undergoes scattering like a free particle. Under a short-
range condition on $V(x)$, i.e., $|V(x)|\leq C\langle x\rangle^{-1-\epsilon}$
this is indeed the case; i.e., all trajectories which are not trapped are
asymptotically free. See the book by Derezinski and Gerard DerGer  for a
systematic development of these techniques in both classical and quantum
mechanics.

Positive commutator methods are also used to prove refined average decay
estimates which hold on subsets of the phase space. Such estimates for the
wave and Schrödinger equation were first derived by Morawetz, using the radial
derivative operator and the generator of the conformal group as multipliers.
These multipliers also work if repulsive interactions are added. However,
modifications are needed if trapped geodesics are present, and usually only
lead to weaker estimates. A major step in this direction is the use of a sharp
localization of the energy, due to Mourre Mourre . The energy estimate can be
obtained by taking the derivative with respect to time of the expectation
value of some operator, also called propagation observable as in Sigal, Soffer
SSof1 . The remarkable paper by Hunziker, Sigal, and Soffer HSS  presents a
time-dependent approach to Mourre theory based on the commutator expansion
lemma of Sigal and Soffer. The latter refers to expressing $[f(A),B]$ through
a series of Taylor type involving higher-order commutators between $A$ and
$B$.

A parallel development to this approach was based on $\Psi$DO methods. In this
approach one constructs a function on the phase space which has positive
Poisson bracket with the principal symbol of the Hamiltonian. Then, one uses
the quantized symbol of this function as a propagation observable, and by
means of $\Psi$DO theory, and in particular, Garding’s inequality, passes to
the desired smoothing (or limiting absorption) bound. Some of the earliest
implementations of this approach are Colin , Doi  and since then a vast
literature has developed in this direction.

The importance of a nontrapping condition is readily understood: it allows for
the construction of monotonic propagation observables, globally in the phase
space. In the presence of closed trajectories this is not possible. However,
when the trajectories are closed but (strongly) unstable, there is now
substantial evidence that the decay estimates continue to hold in some sense.

On the level of the resolvents, one considers $(H-z)^{-1}$ with $H$ a variable
coefficient operator as above, with $a_{ij}$ a short-range perturbation of a
constant elliptic symbol. Furthermore, assume that all classical Hamiltonian
orbits of large initial velocity are not trapped. Then the limiting absorption
principle

$\sup_{{\,\rm Im\,}z>0,{\,\rm Re\,}z\geq
N}\|\langle\cdot\rangle^{-\sigma}(H-z)^{-1}\chi_{I}(H)\langle\cdot\rangle^{-\sigma}\|_{2\to
2}<\infty$ (19)

holds with $N>0$ and $\sigma>0$ sufficiently large, see Murata Mur1 . In fact,
the nontrapping condition is necessary, see Theorem 2 in loc. cit., and one
also obtains (19) for derivatives in $z$ of the resolvent. The latter property
then clearly implies local decay on the time-evolution restricted to high
energies.

In fact, while Doi Doi , Murata Mur  show that smoothing estimates and the
usual decay estimates do not hold in the presence of trapping, Ikawa Ik  shows
that one still obtains local decay estimates for the Laplacian dynamics on
${\mathbb{R}}^{n}$ with several convex obstacles removed. In the meantime, the
microlocal analysis on manifolds with unstable closed geodesics, of the
resolvent of the Laplacian on the one hand, and the Schrödinger evolution on
the other hand, has grown into a vast area in and of itself which is
intimately connected to the semiclassical analysis of scattering resonances.
See for example the recent research monograph Bony  on Resonances for
homoclinic trapped sets, or Dyatlov’s introduction to the fractal uncertainty
principle Dyat .

In general relativity, unstable closed geodesics arise naturally in the study
of the linear wave evolution on the background of both Schwarzschild and Kerr
black holes. A substantial amount of work has accumulated around this topic,
see for example the early works by Blue and Soffer, see BSof  as well the very
recent study of Price’s law by Hintz Hintz . The latter paper was preceded by
the work of Tataru Tat_GR , as well as the results by Donninger, Soffer and
the author DSS2  on the spatially local, but temporally global, decay of
linear waves on Schwarzschild. Metcalfe, Tataru, and Tohaneanu MTT12
subsequently established Price’s law on nonstationary spacetimes with
sufficient decay in a suitable sense. Very recently, Angelopoulos, Aretakis,
Gajic AAG  presented a “physical space" approach to Price’s law on Kerr space-
times in contrast to Hintz’s microlocal technique. We now set out to describe
the author’s results in more detail.

###  III.1 Asymptotically conical surfaces of revolution

A model case for the Schwarzschild manifold, Soffer, Staubach and the author
SSS1 ; SSS2  studied wave evolutions on surfaces of revolution with conic
ends. Let $\Omega\subset{\mathbb{R}}^{N}$ be an embedded compact
$d$-dimensional Riemannian manifold with metric $ds_{\Omega}^{2}$ and define
the $(d+1)$-dimensional manifold

$\displaystyle\mathcal{M}$
$\displaystyle:=\\{(x,r(x)\omega)\>|\>x\in{\mathbb{R}},\;\omega\in\Omega\\},$
$\displaystyle ds^{2}$
$\displaystyle=r^{2}(x)ds_{\Omega}^{2}+(1+r^{\prime}(x)^{2})dx^{2}$

where $r\in C^{\infty}({\mathbb{R}})$ and $\inf_{x\in{\mathbb{R}}}r(x)>0$. We
say that there is a conical end at the right (or left) if

$r(x)=|x|\,(1+h(x)),\quad h^{(k)}(x)=O(x^{-2-k})\quad\forall\;k\geq 0$ (20)

as $x\to\infty\;(x\to-\infty)$.

Of course one can consider cones with arbitrary opening angles but this adds
nothing of substance. Examples of such manifolds are given by surfaces of
revolution with $\Omega=S^{1}$ such as the one-sheeted hyperboloid which
satisfies $r(x)=\sqrt{1+x^{2}}$. They have the property that the entire
Hamiltonian flow on $\mathcal{M}$ is trapped on the set
$(x_{0},r(x_{0})\Omega)$ when $r^{\prime}(x_{0})=0$. From now we will only
consider $S^{1}$ as cross-section $\Omega$ for the sake of simplicity. The
only difference from the general case is that instead of $\big{\\{}e^{\pm
i\ell\theta},\ell^{2}\big{\\}}_{\ell=0}^{\infty}$ one has a complete system
$\\{Y_{n},\mu_{n}\\}_{n=0}^{\infty}$ of $L^{2}$-normalized eigenfunctions and
eigenvalues, respectively, of $\Delta_{\Omega}$. In other words,
$-\Delta_{\Omega}Y_{n}=\mu_{n}^{2}Y_{n}$ where
$0=\mu_{0}^{2}<\mu_{1}^{2}\leq\mu_{2}^{2}\leq\ldots$.

Note that we do not specify the local geometry of $\mathcal{M}$, but only the
asymptotic one at the ends. This allows for very different behaviors of the
geodesics. For the case of the one-sheeted hyperboloid, for example, the
geodesic flow around the unique periodic geodesic is hyperbolic in the sense
of dynamical systems, whereas if we place a section of $S^{2}$ in the middle
of $\mathcal{M}$ then we encounter a set of positive measure in the cotangent
bundle leading to stable periodic geodesics. These two scenarios are depicted
in Figure 2. It is natural to ask to what extent this local geometry affects
the dispersion of the flow. The following result summarizes what is proved in
DSS1 , DSS2  for the case of $\Omega=S^{1}$ (see those references for general
compact $\Omega$).

######  Theorem III.1.

Let $\mathcal{M}$ be a surface which is asymptotically conical at both ends as
defined above. For each $\ell\geq 0$ and all $0\leq\sigma\leq\sqrt{2}\ell$,
there exist constants $C(\ell,\mathcal{M},\sigma)$ and
$C_{1}(\ell,\mathcal{M},\sigma)$ such that for all $t>0$

$\displaystyle\|w_{\sigma}\,e^{it\Delta_{\mathcal{M}}}\,f\|_{L^{\infty}(\mathcal{M})}$
$\displaystyle\leq\frac{C(\ell,\mathcal{M},\sigma)}{t^{1+\sigma}}\Big{\|}\frac{f}{w_{\sigma}}\Big{\|}_{L^{1}(\mathcal{M})}$
(21)
$\displaystyle\|w_{\sigma}\,e^{it\sqrt{-\Delta_{\mathcal{M}}}}\,f\|_{L^{\infty}(\mathcal{M})}$
$\displaystyle\leq\frac{C_{1}(\ell,\mathcal{M},\sigma)}{t^{\frac{1}{2}+\sigma}}\Big{(}\Big{\|}\frac{\partial_{x}f}{w_{\sigma}}\Big{\|}_{L^{1}(\mathcal{M})}$
$\displaystyle\qquad+\Big{\|}\frac{f}{w_{\sigma}}\Big{\|}_{L^{1}(\mathcal{M})}\Big{)}$
(22)

provided $f=f(x,\theta)=e^{i\ell\theta}\tilde{f}(x)$ where $\tilde{f}$ does
not depend on $\theta$. Here $w_{\sigma}(x):=\langle x\rangle^{-\sigma}$ are
weights on $\mathcal{M}$.

Figure 2: Unstable versus stable geodesic flow

In (22) one can obtain somewhat finer results by distinguishing between
$\cos(t\sqrt{-\Delta_{\mathcal{M}}})$ and
$\frac{\sin(\sqrt{-\Delta_{\mathcal{M}}})}{\sqrt{-\Delta_{\mathcal{M}}}}$, see
DSS1  for statements of that kind. Needless to say, $\sigma=0$ is the analog
of the usual dispersive decay estimate for the Schrödinger and wave evolutions
on ${\mathbb{R}}^{2}$. We remark that as in the case of the plane
${\mathbb{R}}^{2}$, the free Laplacian $\Delta_{\mathcal{M}}$ exhibits a zero
energy resonance which is, however, only visible at $\ell=0$ (this case is
treated separately in SSS1 , whereas SSS2  studies $\ell>0$).

Clearly, the local decay given by $\sigma>0$ has no analog in the Euclidean
setting and it also has no meaning for $\ell=0$. The restriction
$\sqrt{2}{\ell}$ is optimal in Theorem III.1, at least for the Schrödinger
equation, and no faster decay can be obtained than the one stated in (21). The
$\sqrt{2}$-factor comes from the opening angle of $\frac{\pi}{4}$ and changing
that angle leads to different constants, namely $\frac{1}{\cos(\theta)}$ where
$\theta$ is the opening angle of the asymptotic cone.

A heuristic explanation for the existence of this accelerated local decay is
given by the geodesic flow combined with the natural dispersion present in
these equations. Indeed, the former will push any nontrapped geodesics into
the ends (with $\ell$ playing the role of the velocity of the geodesics),
whereas the latter will spread any data which is initially highly localized
around a periodic geodesic away from it thus making it susceptible to the
mechanism we just described.

What is not clear from this heuristic is whether or not the localized decay
law should depend on the local geometry (by which we mean the geometry which
is not described by the asymptotic cones). Theorem III.1 shows that this is
not so, since the local decay is fixed and given by a specific power.
Therefore, one sees that the local geometry manifests itself exclusively
through the constants $C(\ell,\mathcal{M},\sigma)$. This is natural, as one
would expect a much longer waiting time before the large $t$ behavior of the
theorem sets in if $\mathcal{M}$ exhibits stable geodesics. In fact, the
constant $C(\ell)$ grows exponentially in that case as can be seen by
solutions which are highly localized (microlocally) around a periodic
geodesic, see Plamen  and Sbie .

In contrast, the methods of DSS2  show that this constant grows like
$\ell^{C}$ if the manifold $\mathcal{M}$ has a unique periodic geodesic and is
uniformly convex near it. This then allows one to sum up the estimates for
each angular momentum as described by the following theorem.

######  Theorem III.2.

Let $\mathcal{M}$ be asymptotically conical at both ends as above and suppose
that $\mathcal{M}$ has a unique periodic geodesic and is uniformly convex near
it. Then for all $t>0$, and any ${\varepsilon}>0$, and with
$\mathcal{D}:=1-\partial_{\theta}^{2}$,

$\displaystyle\|w_{1+{\varepsilon}}e^{it\Delta_{\mathcal{M}}}\,w_{1+{\varepsilon}}f\|_{L^{2}(\mathcal{M})}$
$\displaystyle\leq\frac{C(\mathcal{M},{\varepsilon})}{\langle
t\rangle}\big{\|}{\mathcal{D}\,f}\big{\|}_{L^{2}(\mathcal{M})}$ (23)
$\displaystyle\|w_{1}e^{it\Delta_{\mathcal{M}}}\,w_{1}f\|_{L^{\infty}(\mathcal{M})}$
$\displaystyle\leq\frac{C(\mathcal{M},{\varepsilon})}{t}\big{\|}{\mathcal{D}^{2+{\varepsilon}}\,f}\big{\|}_{L^{1}(\mathcal{M})}$
(24)

provided $f=f(x,\theta)$ is Schwartz on $\mathcal{M}$, say. For the wave
equation one has

$\displaystyle\|w_{\frac{1}{2}+{\varepsilon}}e^{\pm
it\sqrt{-\Delta_{\mathcal{M}}}}\,w_{\frac{1}{2}+{\varepsilon}}f\|_{L^{2}(\mathcal{M})}$
$\displaystyle\leq\frac{C_{1}(\mathcal{M},{\varepsilon})}{\langle
t\rangle^{\frac{1}{2}}}\Big{(}\big{\|}\mathcal{D}^{\frac{5}{4}}\,f^{\prime}\big{\|}_{L^{2}(\mathcal{M})}+\big{\|}\mathcal{D}^{\frac{5}{4}}\,f\big{\|}_{L^{2}(\mathcal{M})}\Big{)}$
(25) $\displaystyle\|w_{\frac{1}{2}+{\varepsilon}}e^{\pm
it\sqrt{-\Delta_{\mathcal{M}}}}\,w_{\frac{1}{2}+{\varepsilon}}f\|_{L^{\infty}(\mathcal{M})}$
$\displaystyle\leq\frac{C_{1}(\mathcal{M},{\varepsilon})}{t^{\frac{1}{2}}}\Big{(}\big{\|}\mathcal{D}^{\frac{9}{4}+{\varepsilon}}\,\partial_{x}f\big{\|}_{L^{1}(\mathcal{M})}+\big{\|}\mathcal{D}^{\frac{9}{4}+{\varepsilon}}\,f\big{\|}_{L^{1}(\mathcal{M})}\Big{)}$
(26)

The weights $w_{1}$ and $w_{\frac{1}{2}+\epsilon}$ appearing in (24) and (26),
respectively, are a by-product of our proof and can most likely be removed.
The origin of the weights in our method will be explained in Section III.2
below.

Figure 3: Two planes joined by a neck

One also obtains the accelerated decay rates which are better by $t^{-\sigma}$
as in Theorem III.1 provided one puts in the weights as before, makes the
number of derivatives required on the right-hand side depend on $\sigma$, and
provided the data are perpendicular to $e^{i\ell\theta}$ for
$\sigma>\sqrt{2}\ell\geq 0$. We remark that one can think of the surfaces in
Theorem III.1 as two planes joined by a neck, see Figure 3. On the other hand,
the methods which are currently used to prove Theorems III.1 and III.2 do not
extend to the case of more necks, as then there is no clear way of separating
variables.

There is no reason to expect that the number of derivatives required on the
data in Theorem III.2 is optimal, in fact it most certainly is not.
Heuristically speaking, these derivatives measure the spreading or non-
concentration of solutions near hyperbolic orbits in dependence on the angular
momentum, which is a quantum effect. See for example Christianson’s work Chr
on this topic.

Doi Doi  proved that the presence of trapping destroys the so-called local
smoothing estimate for the Schrödinger evolution. More precisely, he showed
that one loses (even locally in time) the $\frac{1}{2}$-derivative gain
present in $e^{it\Delta}$. Note that this does not constitute a contradiction
to Theorem III.2 as the latter does not claim any gain of regularity (on the
contrary, we lose angular derivatives). In a similar vein, Burq, Guillarmou,
Hassell BGH  proved that Strichartz estimates may remain valid on metrics with
trapping.

We now describe the method of proof leading to Theorem III.1. Later we will
discuss how to obtain Theorem III.2, which requires considerably more work. We
will then also describe the result DSS2  for linear waves on Schwarzschild,
which is very close to Theorem III.2.

To begin with, let $\xi$ be arclength along a generator of $\mathcal{M}$. Then
the Laplacian takes the form

$\Delta_{\mathcal{M}}=\frac{1}{r(\xi)}\partial_{\xi}(r(\xi)\partial_{\xi})+\frac{1}{r^{2}(\xi)}\Delta_{S^{2}}$

Now

$e^{-i\ell\theta}r^{\frac{1}{2}}(\xi)\Delta_{\mathcal{M}}(r^{-\frac{1}{2}}(\xi)e^{i\ell\theta}f(\xi))=\mathcal{H}_{\ell}f$

with

$\mathcal{H}_{\ell}=-\partial_{\xi}^{2}+V_{\ell},\qquad
V_{\ell}(\xi)=\frac{2\ell^{2}-\frac{1}{4}}{\langle\xi\rangle^{2}}+O(\langle\xi\rangle^{-3})$
(27)

where each $\xi$-derivative of the $O(\cdot)$-term gives one extra power of
$\xi$ as decay.

Figure 4: Potentials corresponding to the surfaces in Figure 2

We remark that the leading $\langle\xi\rangle^{-2}$ decay is critical for
several reasons. For us most relevant is the behavior of the Jost solutions as
the energy $\lambda^{2}$ tends to zero; in fact these Jost solutions are
continuous in $\lambda$ around $\lambda=0$ provided the decay of the potential
is at least $\langle\xi\rangle^{-2-\epsilon}$ for some $\epsilon>0$. At
$\epsilon=0$ this property is lost – which is precisely what allows for the
accelerated decay of Theorem III.1. To be more specific, one first reduces
Theorem III.1 (at least the Schrödinger bound (21), the wave equation being
similar) via the spectral theorem to the point-wise bound

$\displaystyle\sup_{\infty>\xi\geq\xi^{\prime}>-\infty}$
$\displaystyle(\langle\xi\rangle\langle\xi^{\prime}\rangle)^{-\frac{1}{2}}\Big{|}\int_{0}^{\infty}e^{it\lambda^{2}}{\,\rm
Im\,}\Big{[}\frac{f_{+,\ell}(\xi,\lambda)f_{-,\ell}(\xi^{\prime},\lambda)}{W_{\ell}(\lambda)}\Big{]}$
$\displaystyle\qquad\,\lambda\,d\lambda\Big{|}\leq C_{\ell}\,t^{-1-\sigma}$
(28)

where $C_{\ell}$ is a uniform constant and $\sigma=\sqrt{2}\ell$. Here
$f_{\pm,\ell}$ are the (outgoing) Jost solutions, which satisfy
$\mathcal{H}_{\ell}f_{\pm,\ell}=\lambda^{2}f_{\pm,\ell}$ and $f_{\pm,\ell}\sim
e^{\pm i\lambda\xi}$ as $\xi\to\pm\infty$. Moreover, $W_{\ell}(\lambda)$ is
the Wronskian of $f_{+},f_{-}$. We remark that the quantity inside the
absolute values in (28) is exactly

$\int_{0}^{\infty}e^{it\lambda^{2}}E(d\lambda^{2})(\xi,\xi^{\prime})$

where $E(d\lambda^{2})(\cdot,\cdot)$ is the kernel of the spectral resolution
of $\mathcal{H}_{\ell}$. As usual,

$f_{+}(\xi,\lambda)=e^{i\xi\lambda}+\int_{\xi}^{\infty}\frac{\sin(\lambda(\xi^{\prime}-\xi))}{\lambda}V(\xi^{\prime})f_{+}(\xi^{\prime},\lambda)\,d\xi^{\prime}$

From this formula, one immediately sees the aforementioned discontinuity at
$\lambda=0$ since $\xi V(\xi)\not\in L^{1}(0,\infty)$. Setting
$\xi=\xi^{\prime}=0$, (21) of Theorem III.1 reduces to the standard stationary
phase type bound (with $\nu:=\sqrt{2}\,\ell$)

$\Big{|}\int_{0}^{\infty}e^{it\lambda^{2}}\lambda^{1+2\nu}\chi(\lambda)\,d\lambda\Big{|}\leq
Ct^{-1-\nu}$

where $\chi$ is a smooth cut-off function to the interval $[0,1]$, say. To see
why the spectral measure should be as flat as $\lambda^{1+2\nu}\,d\lambda$,
let us first give an informal proof of the fact that

$W_{\ell}(\lambda)=c\lambda^{1-2\nu}(1+o(1))\qquad\lambda\to 0$ (29)

where $c\neq 0$. Since this Wronskian appears in the denominator of the
resolvent, it at least serves as an indication that the spectral measure might
be this small for small $\lambda$ (one has to be very careful here, since the
numerator is of the same size – however, the imaginary part of the resolvent
has the desired size $O(\lambda^{2\nu})$). To begin with, recall from basic
scattering theory that the Wronskian is given by

$W(\lambda)=\frac{-2i\lambda}{T(\lambda)}$ (30)

where $T(\lambda)$ is the transmission coefficient, see Figure 5 (in that
figure the dashed line is supposed to indicate an energy level $k^{2}$, and
the turning points are defined as the projections of the intersection of the
graph with that line). By the so-called WKB approximation, one has to leading
order that $T(\lambda)=e^{-S(\lambda)}$ with the action $S$ given by

$\displaystyle S(\lambda)$
$\displaystyle=\int_{x_{0}}^{x_{1}}\sqrt{\nu^{2}\langle
y\rangle^{-2}-\lambda^{2}}\,dy$
$\displaystyle=\hbar^{-1}\int_{x_{0}}^{x_{1}}\sqrt{2\langle
y\rangle^{-2}-\hbar^{2}\lambda^{2}}\,dy$

with $x_{0}<0<x_{1}$ being the turning points which are defined as
$V(x_{0})=V(x_{1})=\lambda^{2}$. Note that we modified the potential by
removing the cubic corrections as well as the
$-\frac{1}{4}\langle\xi\rangle^{-2}$ part of the potential (the latter
obviously requiring some justification). Furthermore, we used that
$\nu=\sqrt{2}\ell$ and assumed $\ell>0$. As a result,

$S(\lambda)=2\nu|\log\lambda|(1+o(1))\qquad\lambda\to 0$

which then gives (29) to leading order. To justify the removal of the
$\frac{1}{4}\langle\xi\rangle^{-2}$-part of the potential $V_{\ell}$, we
simply note that the usual WKB ansatz for the zero energy solutions of
$\mathcal{H}_{\ell}$, viz. $\mathcal{H}_{\ell}f=0$ is the approximate equality

$f(\xi)\simeq
V_{\ell}^{-\frac{1}{4}}(\xi)e^{\pm\int_{1}^{\xi}\sqrt{V_{\ell}(\eta)}\,d\eta}$

In view of (27) one obtains the asymptotic behavior
$\xi^{\frac{1}{2}\pm\sqrt{\nu^{2}-\frac{1}{4}}}$ as $\xi\to\infty$. On the
other hand, the exact solutions of

$-f^{\prime\prime}(\xi)+\frac{\nu^{2}-\frac{1}{4}}{\xi^{2}}f(\xi)=0$

are of the form $\xi^{\frac{1}{2}\pm\nu}$. The WKB approximation can therefore
only be correct provided the $-\frac{1}{4}\xi^{-2}$ term is removed from the
potential $V_{\ell}$ (for a precise rendition – with control of error terms –
of this heuristics discussion see Section 2 of SSS2 ). Another important
comment concerning $V_{\ell}$ is that (30), while true to leading order semi-
classically as $\hbar=\ell^{-1}\to 0$, provided the energy
$\lambda>\lambda_{0}>0$ (where the latter is fixed), does not necessarily hold
as $\lambda\to 0$. The key property here is that $\mathcal{H}_{\ell}$ does not
have a zero energy resonance which means that there is no globally subordinate
(or recessive) solution. This refers to solutions of the slowest allowed
growth at both ends. For example, consider the operator
$H=-\frac{d^{2}}{dx^{2}}+V$ where $V=\bar{V}$ satisfies $(1+|x|)V(x)\in
L^{1}({\mathbb{R}})$. Then by the usual Jost/Volterra perturbation analysis,
there is a fundamental system of solutions to $Hf=0$ consisting of
$f_{1}(x)\sim x$, respectively $f_{2}(x)\sim 1$ as $x\to\infty$. Thus $f_{2}$
is the unique (up to nonzero factors) subordinate solution at $+\infty$. A
resonance at $0$ energy therefore occurs if $Hf=0$ admits a solution $f\neq 0$
which is asymptotic to a constant for both $x\to\pm\infty$. Since the only
other option would be some linear growth at either end, this is equivalent to
$f\in L^{\infty}({\mathbb{R}})$. This is a universal characterization of $0$
energy resonances through solutions of $Hf=0$ even if $V$ violates
$(1+|x|)V(x)\in L^{1}({\mathbb{R}})$ as in Bessel-type potentials arising in
most problems discussed in this note, or for that matter, for $V$ which are
strongly singular. The latter means that $V$ is locally bounded (for
simplicity) but $\int_{-\infty}^{\infty}|V(x)|\,dx=\infty$, and we assume that
$H$ is limit point at both ends. Depending on the specific choice of $V$, one
needs to find a fundamental system of $Hf=0$ at both $\pm\infty$ and then
select the subordinate solution. A resonance is characterized by a solution
which is globally subordinate.

Figure 5: Reflection and transmission coefficients

While our discussion has been largely heuristic, we emphasize that (29) is
proved in SSS2  by means of an asymptotic description of the Jost solutions as
$\lambda\to 0$. Moreover, it is shown there that the constant $c$ in (29)
vanishes in case of a zero energy resonance which indicates that the WKB
approximation fails in that case as $\lambda\to 0$. Finally, we emphasize that
the only natural small parameter in SSS2  for fixed $\ell\geq 1$ is the energy
$\lambda$. This is in contrast to the summation problem in $\ell$ where
$\hbar:=\ell^{-1}$ represents another (and most important) small parameter. In
fact, for large $\ell$ the errors in the WKB approximations are controlled in
terms of this small parameter rather than in terms of the small energy (we
will return to this matter below). In order to be able to distinguish the two
potentials in Fig. 4, or manifolds with distinct local geometries in Fig. 2 we
therefore need to obtain precise asymptotics for the Jost solutions and the
spectral measure for both small energies $0<\lambda<\lambda_{0}$ and all large
$|\ell|$, simultaneously. This sets these problems apart from most of the
semi-classical literature in several ways: (i) it is not enough to compute the
limit $\hbar\to 0$. In fact, we need a precise asymptotic representation of
the Jost solutions uniformly in small $\hbar$ and all energies. This will be
explained in more detail below in the summation section. (ii) The need for
uniform control for all small energies is also in stark contrast to the
literature which typically restricts any semi-classical analysis to positive
or large energies.

The rigorous proof of (29) proceeds by means of a classical matching method.
To be more specific, consider the Schrödinger operator on the line (for
notational convenience we write $x$ instead of $\xi$)

$\displaystyle\mathcal{H}_{\nu}$
$\displaystyle=-\partial_{x}^{2}+(\nu^{2}-\frac{1}{4})\langle
x\rangle^{-2}-U_{\nu}(x),$ $\displaystyle\frac{d^{k}U_{\nu}(x)}{dx^{k}}$
$\displaystyle=O(x^{-3-k})$

for all $k\geq 0$ as $x\to\pm\infty$ and with $\nu>0$ fixed. To decribe the
Jost solution $f_{+,\nu}(x)$ on the interval $x\geq 1$ we start from the zero
energy solutions

$\displaystyle u_{0,\nu}^{+}(x)$
$\displaystyle=x^{\frac{1}{2}+\nu}(1+O(x^{-\alpha})),$ $\displaystyle
u_{1,\nu}^{+}(x)$ $\displaystyle=x^{\frac{1}{2}-\nu}(1+O(x^{-1}))\text{\ \ as\
}x\to\infty$

which form a fundamental system of $\mathcal{H}_{\nu}f=0$ (and with
$\alpha:=\min(1,2\nu)$). Next, one perturbs these solutions with respect to
the energy $\lambda$. More specifically, one shows via Volterra iteration that
there is a basis $\\{u_{0,\nu}^{+}(x,\lambda),u_{1,\nu}^{+}(x,\lambda)\\}$ of
solutions to the equation $\mathcal{H}_{\nu}f=\lambda^{2}f$ which satisfy (at
least for $\nu>1$)

$u_{j,\nu}^{+}(x,\lambda)=u_{j,\nu}^{+}(x)(1+O(\lambda^{2}x^{2}))$ (31)

on the interval $1\leq x\ll\lambda^{-1}$ (we are only considering small
$\lambda$ for now). Clearly, one has

$f_{+,\nu}(x,\lambda)=a_{+,\nu}(\lambda)u_{0,\nu}^{+}(x,\lambda)+b_{+,\nu}(\lambda)u_{1,\nu}^{+}(x,\lambda)$

where the coefficients are given by

$\displaystyle a_{\pm,\nu}(\lambda)$
$\displaystyle=-W(f_{\pm,\nu}(\cdot,\lambda),u_{1,\nu}^{\pm}(\cdot,\lambda)),$
(32) $\displaystyle b_{\pm,\nu}(\lambda)$
$\displaystyle=W(f_{\pm,\nu}(\cdot,\lambda),u_{0,\nu}^{\pm}(\cdot,\lambda))$

The aforementioned matching means nothing else than computing these
Wronskians. The point where they are computed is chosen to be
$\lambda^{-1+\epsilon}$ with $\epsilon>0$ small and fixed. On the one hand,
this choice guarantees that the errors in (31) are $O(\lambda^{2\epsilon})$,
which is admissible. On the other hand, it requires that we obtain a
sufficiently accurate description of the Jost solutions on
$[\lambda^{-1+\epsilon},\infty)$. The latter is accomplished by comparing the
outgoing Jost solution of the operator $\mathcal{H}_{\nu}$ to that of
$\mathcal{H}_{0,\nu}$ given by

$\mathcal{H}_{0,\nu}:=-\partial_{x}^{2}+(\nu^{2}-\frac{1}{4})x^{-2}$

The outgoing Jost solution of this operator on $\xi\geq 1$ equals

$\sqrt{\frac{\pi}{2}}\,e^{i(2\nu+1)\pi/4}\sqrt{\xi\lambda}\,H^{(+)}_{\nu}(\xi\lambda)$

which is asymptotic to $e^{i\xi\lambda}$ as $\xi\to\infty$. Here
$H^{(+)}_{\nu}(z)=J_{\nu}(z)+iY_{\nu}(z)$ is the usual Hankel function.
Carrying out the perturbative analysis with $\mathcal{H}_{0,\nu}$ as giving
the leading order allows one to approximate $f_{+}(\xi,\lambda)$ with small
errors on the interval $(\lambda^{-1+\epsilon},\infty)$. With this asymptotic
representation in hand, one now has the following result, see Proposition 3.12
in SSS2 .

######  Proposition III.3.

Let $\beta_{\nu}:=\sqrt{\frac{\pi}{2}}\,e^{i(2\nu+1)\pi/4}$. With nonzero real
constants $\alpha_{0,\nu}^{+},\beta_{0,\nu}^{+}$, and some sufficiently small
${\varepsilon}>0$,

$\displaystyle a_{+,\nu}(\lambda)$
$\displaystyle=\lambda^{\frac{1}{2}+\nu}\beta_{\nu}(\alpha_{0,\nu}^{+}+O(\lambda^{\varepsilon})+iO(\lambda^{(1-2\nu){\varepsilon}}))$
(33) $\displaystyle b_{+,\nu}(\lambda)$
$\displaystyle=i\lambda^{\frac{1}{2}-\nu}\beta_{\nu}(\beta_{0,\nu}^{+}+O(\lambda^{\varepsilon})+iO(\lambda^{(1+2\nu){\varepsilon}}))$

as $\lambda\to 0+$ with real-valued $O(\cdot)$ which behave like symbols under
differentiation in $\lambda$. The asymptotics as $\lambda\to 0-$ follows from
that as $\lambda\to 0+$ via the relations
$a_{+,\nu}(-\lambda)=\overline{a_{+,\nu}(\lambda)}$,
$b_{+,\nu}(-\lambda)=\overline{b_{+,\nu}(\lambda)}$.

Analogous expressions hold for $a_{-,\nu}$ and $b_{-,\nu}$ which of course
refers to the solutions on $x\leq-1$. From these expansions, one then
concludes the following statement for the Wronskian between
$f_{+}(\cdot,\lambda)$ and $f_{-}(\cdot,\lambda)$:

$W_{\nu}(\lambda)=ie^{i\nu\pi}\,\lambda^{1-2\nu}(W_{0,\nu}+O_{{\mathbb{C}}}(\lambda^{\varepsilon}))\text{\
\ as\ \ }\lambda\to 0+$

Here $W_{0,\nu}$ is a real constant and
$O_{\mathbb{C}}(\lambda^{\varepsilon})$ is complex valued and of symbol type
(meaning that each derivative loses one power). Most importantly,
$W_{0,\nu}=0$ if and only if zero is a resonance of $\mathcal{H}_{\nu}$. For
the case of surfaces of revolutions, it is easy to exclude zero energy
resonances of the associated Schrödinger operator, at least for $\ell\geq 1$.
In fact, with $\mathcal{H}_{\ell}$ denoting the operator obtained for fixed
angular momentum $\ell\geq 1$,

$\mathcal{H}_{\ell}(r^{\frac{1}{2}}e^{\pm\ell y})=0,\quad
y(\xi)=\int_{0}^{\xi}\frac{d\eta}{r(\eta)}$

Because $y$ is odd, the smaller branch at $\xi=\infty$ has to be the larger
one at $\xi=-\infty$ which places us in the nonresonant case. It is perhaps
worth mentioning that the potentials arising from surfaces of revolution do
not need to be nonnegative (for positive potentials it is evident that zero is
not a resonance). In fact, if $\mathcal{M}$ has very large curvature then the
potential can be negative. We remark that for $\ell=0$ it is proved in SSS1
that

$W_{0}(\lambda)=2\lambda\left(1+ic_{3}+i\frac{2}{\pi}\log\lambda\right)+O(\lambda^{\frac{3}{2}-{\varepsilon}})\text{\
\ as\ \ }\lambda\to 0+$

On a technical level, the logarithmic term in $\lambda$ makes the $\ell=0$
case somewhat harder to analyze than the cases $\ell\geq 1$. Not surprisingly,
in proving dispersive estimates for $-\Delta_{{\mathbb{R}}^{2}}+V$ one
encounters similar logarithmic issues, see Sch1 .

In conclusion, we would like to stress that the estimates in SSS2  produce
constants that grow very rapidly in $\ell$, somewhat faster than
$e^{\ell^{2}}$, to be precise. This is due to a number of sources. First, for
the small energy analysis we just described to work, one needs to chose the
energy cut-off $\lambda_{0}=\lambda_{0}(\ell)$ to depend on $\ell$ which
already introduces large constants into the proof. Second, for energies
$\lambda>\lambda_{0}(\ell)>0$ one uses a very crude method, namely termwise
estimation of a Born series which cannot distinguish the sign of the
potential. Even replacing the crude Born series by something more elaborate
would not make much of a difference. Indeed, by the preceding discussion the
two manifolds in Figure 2 behave very differently as far as the dependence of
the constant on $\ell$ is concerned.

Since the small energy matching method outlined above cannot easily
distinguish between these manifolds, we shall now discuss an approach that is
capable of differentiating between them, albeit only for large $\ell$. For
this reason, the finite $\ell$ analysis of SSS1  and SSS2  is needed in the
proof of Theorem III.2.

###  III.2 Summation over all angular momenta

We shall now prove Theorem III.2. We will follow DSS2  and sketch how to
obtain (23) and (24), with the case of the wave equation being similar. With
$V_{\ell}$ as in (27), we claim the following bound:

$\int_{-\infty}^{\infty}\langle\xi\rangle^{-2}\big{|}e^{it\mathcal{H}_{\ell}}u(\xi)\big{|}^{2}\,d\xi\lesssim\langle
t\rangle^{-2}\ell^{4}\int_{-\infty}^{\infty}\langle\xi\rangle^{2}|u(\xi)|^{2}\,d\xi$
(34)

The proof of (34) will be discussed below. Taking it for granted, suppose that
$f$ is a Schwartz function on $\mathcal{M}$ and write

$f(\xi,\theta)=\sum_{\ell=-\infty}^{\infty}e^{i\ell\theta}f_{\ell}(\xi)=\sum_{\ell=-\infty}^{\infty}e^{i\ell\theta}r^{-\frac{1}{2}}(\xi)u_{\ell}(\xi)$

Then

$\displaystyle e^{it\Delta_{\mathcal{M}}}f$
$\displaystyle=\sum_{\ell=-\infty}^{\infty}e^{it\Delta_{\mathcal{M}}}\big{[}e^{i\ell\theta}r^{-\frac{1}{2}}(\xi)u_{\ell}(\xi)\big{]}$
$\displaystyle=\sum_{\ell=-\infty}^{\infty}e^{i\ell\theta}r^{-\frac{1}{2}}(\xi)\big{[}e^{it\mathcal{H}_{\ell}}u_{\ell}\big{]}(\xi)$

whence

$\displaystyle\big{\|}w_{1}e^{it\Delta_{\mathcal{M}}}f\big{\|}^{2}_{L^{2}(\mathcal{M})}$
$\displaystyle=\frac{1}{2\pi}\\!\int_{-\infty}^{\infty}\\!\int_{0}^{2\pi}\\!\\!w_{1}^{2}(\xi)\Big{|}\\!\sum_{\ell=-\infty}^{\infty}\\!\\!\\!e^{i\ell\theta}r^{-\frac{1}{2}}(\xi)\\!\big{[}\\!e^{it\mathcal{H}_{\ell}}u_{\ell}\big{]}(\xi)\Big{|}^{2}\\!\\!r(\xi)\,d\xi
d\theta$
$\displaystyle\lesssim\sum_{\ell=-\infty}^{\infty}\int_{-\infty}^{\infty}\langle\xi\rangle^{-2}\Big{|}e^{it\mathcal{H}_{\ell}}u_{\ell}(\xi)\Big{|}^{2}\,d\xi$
$\displaystyle\lesssim\sum_{\ell=-\infty}^{\infty}\langle
t\rangle^{-2}\langle\ell\rangle^{4}\int_{-\infty}^{\infty}\langle\xi\rangle^{2}|f_{\ell}(\xi)|^{2}r(\xi)\,d\xi$
$\displaystyle\lesssim\langle
t\rangle^{-2}\\!\\!\sum_{\ell=-\infty}^{\infty}\\!\int_{-\infty}^{\infty}\\!\langle\ell\rangle^{4}w_{-1}(\xi)^{2}\Big{|}\\!\int_{0}^{2\pi}\\!f(\xi,\theta)e^{-i\ell\theta}\,d\theta\Big{|}^{2}r(\xi)\,d\xi$
$\displaystyle\lesssim\langle
t\rangle^{-2}\|w_{-1}(1-\partial_{\theta}^{2})f\|_{L^{2}(\mathcal{M})}^{2}$

which is (23). To prove (34), it is clear from Theorem III.1 that it suffices
to consider $\ell$ large, say $|\ell|\geq\ell_{0}\gg 1$. Fixing such an
$\ell$, one switches to a semi-classical representation, via the identity

$e^{it\mathcal{H}_{\ell}}=e^{i\frac{t}{\hbar^{2}}\mathcal{H}(\hbar)},\qquad\mathcal{H}(\hbar):=-\hbar^{2}\partial_{\xi}^{2}+\hbar^{2}V_{\ell}$

where $V_{\ell}$ is as in (27) and with $\hbar:=\ell^{-1}$. By construction,
$V(\xi,\hbar):=\hbar^{2}V_{\ell}(\xi)$ has the property that its maximal
height is now essentially fixed at $V_{\rm max}(\hbar)=V_{\rm
max}(0)+O(\hbar^{2})$ with $V_{\rm max}(0)\simeq 1$. The essential property of
the potential is that it has a unique nondegenerate maximum, i.e., it looks
like the one on top in Figure 4.

For the remainder of this section, $\hbar$ will be small. From the spectral
representation one has

$e^{i\frac{t}{\hbar^{2}}\mathcal{H}(\hbar)}=\frac{2}{\pi}\hbar^{-2}\int_{0}^{\infty}e^{i\frac{t}{\hbar^{2}}E^{2}}{\,\rm
Im\,}\Big{[}\frac{f_{+}(x,E;\hbar)f_{-}(x^{\prime},E;\hbar)}{W(f_{+}(\cdot,E;\hbar),f_{-}(\cdot,E;\hbar))}\Big{]}\,E\,dE$
(35)

with $f_{\pm}$ being the outgoing Jost solutions for the semi-classical
operator $\mathcal{H}(\hbar)$ which means that

$\displaystyle(-\hbar^{2}\partial_{x}^{2}+V(x;\hbar))f_{\pm}(x,E;\hbar)$
$\displaystyle=E^{2}f_{\pm}(x,E;\hbar)$ $\displaystyle f_{\pm}(x,E;\hbar)$
$\displaystyle\sim e^{\pm\frac{i}{\hbar}Ex}\qquad x\to\pm\infty$

With $\epsilon>0$ fixed and small (independently of $\hbar$), one now
considers energies $0<E<\epsilon$ (low), $\epsilon<E<100$ (intermediate), and
$E>100$ (large) separately. The middle interval is further split into energies
$\epsilon<E<V_{\rm max}(0)-\epsilon$, $V_{\rm max}(0)-\epsilon<E<100$,
respectively. The latter interval is to some extent the most important of all
as it contains the nondegenerate maximum of the potential $V(\hbar)$. We shall
see that it is precisely this maximum which determines the number of
derivatives lost in the process of summing over $\ell$.

The easiest region is $E>100$. Indeed, for these energies the potential is
essentially negligible and a classical WKB approximation reduces matters to
the free case. This means (again heuristically) that (23) is a consequence of
the $L^{1}\to L^{\infty}({\mathbb{R}}^{2})$ bound on
$e^{it\Delta_{{\mathbb{R}}^{2}}}$ which explains the weights $w_{1+\epsilon}$.

####  III.2.1 WKB in the doubly asymptotic limit $\hbar\to 0$ and $E\to 0$

The low-lying energies $0<E<\epsilon$ are also treated by means of WKB, but
there one faces the difficulty that the WKB approximation of the generalized
eigenfunctions needs to be accurate in the entire range $0<E<\epsilon$ and
$0<\hbar<\hbar_{0}$. There exists an extensive literature on the validity of
the WKB approximation provided the energy stays away from zero, i.e.,
$E>E_{0}>0$ uniformly in $\hbar$, see for example Olver  or Ramond Ram .
However, the issue of controlling all errors in the WKB method uniformly in
small $\hbar$ and small $E$ does not seem to have been considered before. For
the problem of sending $E\to 0$ it is of course most relevant that the
potential has the (critical) inverse square decay, as was already apparent in
the discussion of the matching method in the previous section.

This lead Costin, Schlag, Staubach, and Tanveer CSST  to carry out a
systematic analysis of this two-parameter WKB problem for inverse square
potentials. More specifically, they considered the scattering matrix

$\Sigma(E;\hbar)=\left[\begin{matrix}t(E;\hbar)&r_{-}(E;\hbar)\\\
r_{+}(E;\hbar)&t(E;\hbar)\end{matrix}\right]=\left[\begin{matrix}\Sigma_{11}(E;\hbar)&\Sigma_{12}(E;\hbar)\\\
\Sigma_{21}(E;\hbar)&\Sigma_{22}(E;\hbar)\end{matrix}\right]$

for the semiclassical operator

$P(x,\hbar D):=-\hbar^{2}\frac{d^{2}}{dx^{2}}+V(x)$

with inverse square $V$ (asymptotically, as $|x|\to\infty$) and obtained the
following result.

######  Theorem III.4.

Let $V\in C^{\infty}({\mathbb{R}})$ with $V>0$ and
$V(x)=\mu_{\pm}^{2}x^{-2}+O(x^{-3})$ as $x\to\pm\infty$ where $\mu_{+}\neq 0$,
$\mu_{-}\neq 0$ and $\partial_{x}^{k}O(x^{-3})=O(x^{-3-k})$ for all $k\geq 0$.
Denote

$V_{0}(x;\hbar):=V(x)+\frac{\hbar^{2}}{4}\langle x\rangle^{-2}$ (36)

and let $E_{0}>0$ be such that for all $0<E<E_{0}$ and $0<\hbar<1$,
$V_{0}(x;\hbar)=E$ has a unique pair of solutions, which we denote by
$x_{2}(E;\hbar)<0<x_{1}(E;\hbar)$. Define

$\begin{split}S(E;\hbar)&:=\int_{x_{2}(E;\hbar)}^{x_{1}(E;\hbar)}\sqrt{V_{0}(y;\hbar)-E}\,dy\\\
T_{+}(E;\hbar)&:=x_{1}(E;\hbar)\sqrt{E}-\int_{x_{1}(E;\hbar)}^{\infty}\big{(}\sqrt{E-V_{0}(y;\hbar)}-\sqrt{E}\big{)}\,dy\\\
T_{-}(E;\hbar)&:=-x_{2}(E;\hbar)\sqrt{E}-\int_{-\infty}^{x_{2}(E;\hbar)}\big{(}\sqrt{E-V_{0}(y;\hbar)}-\sqrt{E}\big{)}\,dy\end{split}$
(37)

as well as $T(E;\hbar):=T_{+}(E;\hbar)+T_{-}(E;\hbar)$. Then for all
$0<\hbar<\hbar_{0}$ where $\hbar_{0}=\hbar_{0}(V)>0$ is small and $0<E<E_{0}$

$\displaystyle\Sigma_{11}(E;\hbar)$
$\displaystyle=e^{-\frac{1}{\hbar}(S(E;\hbar)+iT(E;\hbar))}(1+\hbar\,\sigma_{11}(E;\hbar))$
(38) $\displaystyle\Sigma_{12}(E;\hbar)$
$\displaystyle=-ie^{-\frac{2i}{\hbar}T_{+}(E;\hbar)}(1+\hbar\,\sigma_{12}(E;\hbar))$

where the correction terms satisfy the bounds

$|\partial_{E}^{k}\,\sigma_{11}(E;\hbar)|+|\partial_{E}^{k}\,\sigma_{12}(E;\hbar)|\leq
C_{k}\,E^{-k}\quad\forall\;k\geq 0,$ (39)

with a constant $C_{k}$ that only depends on $k$ and $V$. The same conclusion
holds if instead of (36) we were to define $V_{0}$ as
$V_{0}:=V+\hbar^{2}V_{1}$ with $V_{1}\in C^{\infty}({\mathbb{R}})$,
$V_{1}(x;\hbar)=\frac{1}{4}\langle x\rangle^{-2}+O(x^{-3})$ as $x\to\pm\infty$
with $\partial_{x}^{k}O(x^{-3})=O(x^{-3-k})$ for all $k\geq 0$ and uniformly
in $0<\hbar\ll 1$.

Note the correction of the original potential by $\frac{\hbar^{2}}{4}\langle
x\rangle^{-2}$ in (36). Without this correction the errors $\sigma_{11}$ etc.
diverge as $E\to 0$. The proof of this result of course requires a careful
analysis of the Jost solutions which is then needed in the analysis of the
stationary phase analysis of (35).

The analysis of the Jost solutions is based on the Liouville-Green transform,
which we now recall (see Olver ). Given any second order equation
$f^{\prime\prime}(x)=Q(x)f(x)$ on some interval $I$, and any diffeomorphism
$w:I\to J$ onto some interval $J$, define
$g(w):=(w^{\prime}(x))^{\frac{1}{2}}f(x)$ where $w=w(x)$. Then by the chain
rule, $f^{\prime\prime}=Qf$ is the same as
$g^{\prime\prime}(w)=\tilde{Q}(w)g(w)$ where

$\displaystyle\tilde{Q}(w)$
$\displaystyle:=\frac{Q(x)}{(w^{\prime}(x))^{2}}-(w^{\prime}(x))^{-\frac{3}{2}}\partial_{x}^{2}(w^{\prime}(x))^{-\frac{1}{2}}$
$\displaystyle=\frac{Q(x)}{(w^{\prime}(x))^{2}}-\frac{3}{4}\frac{(w^{\prime\prime}(x))^{2}}{(w^{\prime}(x))^{4}}+\frac{1}{2}\frac{w^{\prime\prime\prime}(x)}{(w^{\prime}(x))^{2}}$

To apply this transformation, one chooses $w$ so that

$\frac{Q(x)}{(w^{\prime}(x))^{2}}=Q_{0}(w)$ (40)

where $Q_{0}$ is some normal form. Then the problem becomes

$\displaystyle g^{\prime\prime}(w)$ $\displaystyle=Q_{0}(w)g(w)-V(w)g(w),$
(41) $\displaystyle V(w)$
$\displaystyle:=\frac{3}{4}\frac{(w^{\prime\prime}(x))^{2}}{(w^{\prime}(x))^{4}}-\frac{1}{2}\frac{w^{\prime\prime\prime}(x)}{(w^{\prime}(x))^{2}}$

where $V$ is treated as a perturbation. This is only admissible if $Q_{0}$ is
in some suitable sense close to $Q$. The determination of $Q_{0}$ is done on a
case by case basis. For example, if $Q$ does not vanish on $I$, then one can
take $Q_{0}={\rm sign}(Q)$ which leads to the classical WKB ansatz, i.e.,

$Q^{-\frac{1}{4}}(x)e^{\pm\int^{x}_{x_{0}}\sqrt{Q}(y)\,dy}\text{\ \ or\ \
}|Q|^{-\frac{1}{4}}(x)e^{\pm i\int^{x}_{x_{0}}\sqrt{|Q|}(y)\,dy}$

depending on whether $Q>0$ or $Q<0$, respectively. If $Q$ does vanish at
$x_{0}\in I$ with $Q^{\prime}(x_{0})\neq 0$, then one maps $x_{0}$ to $w=0$
and chooses $Q_{0}(w)=w$. In other words, the comparison equation is the Airy
equation. The equation for $w$ in that case is $w(x)w^{\prime}(x)^{2}=Q(x)$
which yields

$w(x)={\rm
sign}(x-x_{0})\Big{|}\frac{3}{2}\int_{x_{0}}^{x}\sqrt{|Q(y)|}\,dy\Big{|}^{\frac{2}{3}}$
(42)

which is known as the Langer transform Olver . It is easy to check that $w$ is
(locally around $x_{0}$) smooth (or analytic) provided $Q$ is smooth (or
analytic). It is precisely this Langer transform which is used in CSST , where
it is written as follows for $x\geq 0$

$\displaystyle\zeta$ $\displaystyle=\zeta(x,E;\hbar)$ $\displaystyle:={\rm
sign}(x-x_{1}(E;\hbar))\Big{|}\frac{3}{2}\int_{x_{1}(E;\hbar)}^{x}\sqrt{|V_{0}(x;\hbar)-E|}\,d\eta\Big{|}^{\frac{2}{3}},$

with $x_{1}(E;\hbar)>0$ being the unique turning point (for $E$ small). The
equation transforms as follows.

######  Lemma III.5.

There exists $E_{0}=E_{0}(V)>0$ so that for all $0<E<E_{0}$ one has the
following properties: the equation $V_{0}(x;\hbar)-E=0$ has a unique (simple)
solution on $x>0$ which we denote by $x_{1}=x_{1}(E;\hbar)$. With
$Q_{0}:=V_{0}-E$

$\displaystyle\zeta$ $\displaystyle=\zeta(x,E;\hbar)$ (43)
$\displaystyle:={\rm
sign}(x-x_{1}(E;\hbar))\Big{|}\frac{3}{2}\int_{x_{1}(E;\hbar)}^{x}\sqrt{|Q_{0}(u,E;\hbar)|}\,du\Big{|}^{\frac{2}{3}}$

defines a smooth change of variables $x\mapsto\zeta$ for all $x\geq 0$. Let
$q:=-\frac{Q_{0}}{\zeta}$. Then $q>0$,
$\frac{d\zeta}{dx}=\zeta^{\prime}=\sqrt{q}$, and

$-\hbar^{2}f^{\prime\prime}+(V-E)f=0$

transforms into

$-\hbar^{2}\ddot{w}(\zeta)=(\zeta+\hbar^{2}\tilde{V}(\zeta,E;\hbar))w(\zeta)$
(44)

under $w=\sqrt{\zeta^{\prime}}f=q^{\frac{1}{4}}f$. Here $\dot{\
}=\frac{d}{d\zeta}$ and

$\tilde{V}:=\frac{1}{4}q^{-1}\langle
x\rangle^{-2}-q^{-\frac{1}{4}}\frac{d^{2}q^{\frac{1}{4}}}{d\zeta^{2}}$

The asymptotic description of the Jost solutions is found by matching the Airy
approximations at the turning point $w=0$. A fundamental solution of the
transformed equation (i.e., in the $\zeta$ variable) to the left of the
turning point is described in terms of the Airy function ${\rm Ai},{\rm Bi}$
by the following result from CSST .

######  Proposition III.6.

Let $\hbar_{0}>0$ be small. A fundamental system of solutions to (44) in the
range $\zeta\leq 0$ is given by

$\displaystyle\phi_{1}(\zeta,E,\hbar)$ $\displaystyle={\rm Ai}(\tau)[1+\hbar
a_{1}(\zeta,E,\hbar)]$ $\displaystyle\phi_{2}(\zeta,E,\hbar)$
$\displaystyle={\rm Bi}(\tau)[1+\hbar a_{2}(\zeta,E,\hbar)]$

with $\tau:=-\hbar^{-\frac{2}{3}}\zeta$. Here $a_{1},a_{2}$ are smooth, real-
valued, and they satisfy the bounds, for all $k\geq 0$ and $j=1,2$, and with
$\zeta_{0}:=\zeta(0,E)$,

$\displaystyle|\partial_{E}^{k}a_{j}(\zeta,E,\hbar)|$ $\displaystyle\lesssim
E^{-k}\min\big{[}\hbar^{\frac{1}{3}}\langle\hbar^{-\frac{2}{3}}\zeta\rangle^{\frac{1}{2}},1\big{]}$
(45) $\displaystyle|\partial_{E}^{k}\partial_{\zeta}a_{j}(\zeta,E,\hbar)|$
$\displaystyle\lesssim
E^{-k}\Big{[}\hbar^{-\frac{1}{3}}\langle\hbar^{-\frac{2}{3}}\zeta\rangle^{-\frac{1}{2}}\chi_{[-1\leq\zeta\leq
0]}$
$\displaystyle\qquad\qquad+|\zeta|^{\frac{1}{2}}\chi_{[\zeta_{0}\leq\zeta\leq-1]}\Big{]}$

uniformly in the parameters $0<\hbar<\hbar_{0}$, $0<E<E_{0}$.

Note that from the standard asymptotic behavior of the Airy functions, viz.,

$\begin{split}{\rm
Bi}(x)&=\pi^{-\frac{1}{2}}x^{-\frac{1}{4}}e^{\frac{2}{3}x^{\frac{3}{2}}}\big{[}1+O(x^{-\frac{3}{2}})\big{]}\text{\
\ as\ \ }x\to\infty\\\ {\rm Bi}(x)&\geq{\rm Bi}(0)>0\quad\forall\;x\geq 0\\\
{\rm
Ai}(x)&=\frac{1}{2}\pi^{-\frac{1}{2}}x^{-\frac{1}{4}}e^{-\frac{2}{3}x^{\frac{3}{2}}}\big{[}1+O(x^{-\frac{3}{2}})\big{]}\text{\
\ as\ \ }x\to\infty\\\ {\rm Ai}(x)&>0\quad\forall\;x\geq 0\end{split}$

the action integral appears naturally in this context, cf. (43). To the right
of the turning point one has the following oscillatory basis.

######  Proposition III.7.

Let $\hbar_{0}>0$ be small. In the range $\zeta\geq 0$ a basis of solutions to
(44) is given by

$\displaystyle\psi_{1}(\zeta,E;\hbar)$ $\displaystyle=({\rm Ai}(\tau)+i{\rm
Bi}(\tau))[1+\hbar b_{1}(\zeta,E;\hbar)]$
$\displaystyle\psi_{2}(\zeta,E;\hbar)$ $\displaystyle=({\rm Ai}(\tau)-i{\rm
Bi}(\tau))[1+\hbar b_{2}(\zeta,E;\hbar)]$

with $\tau:=-\hbar^{-\frac{2}{3}}\zeta$ and where $b_{1},b_{2}$ are smooth,
complex-valued, and satisfy the bounds for all $k\geq 0$, and $j=1,2$

$\displaystyle|\partial_{E}^{k}\,b_{j}(\zeta,E;\hbar)|$ $\displaystyle\leq
C_{k}\,E^{-k}\langle\zeta\rangle^{-\frac{3}{2}}$ (46)
$\displaystyle|\partial_{\zeta}\partial_{E}^{k}b_{j}(\zeta,E)|$
$\displaystyle\leq
C_{k}\,E^{-k}\hbar^{-\frac{1}{3}}\langle\hbar^{-\frac{2}{3}}\zeta\rangle^{-\frac{1}{2}}\langle\zeta\rangle^{-2}$

uniformly in the parameters $0<\hbar<\hbar_{0}$, $0<E<E_{0}$, $\zeta\geq 0$.

We remark that the Langer transform is not the only possibility here. In fact,
in CDST  an alternative approach is used which reduces the potential to a
Bessel normal-form. This is again done by means of a suitable stretching,
i.e., a Liouville-Green transform.

####  III.2.2 Intermediate energies and the top of the barrier

Intermediate energies, including the maximum energy of the potential, can be
treated by means of an approximation of the generalized eigenfunctions. This
was carried out in detail by Costin, Park, and the author by means of a
Liouville Green transformation which reduces the potential near the maximum to
a purely quadratic normal form, see (CPS, , Proposition 2). In this way, one
arrives at a perturbed Weber equation instead of the Airy equation as above.

However, DSS2  follows a different route: a Mourre estimate followed by a
semi-classical version of the propagation bounds in HSS . Mourre Mourre
introduced the powerful idea that the quantum analog, i.e.,

$\chi_{I}(H)i[H,A]\chi_{I}(H)\geq\theta\chi_{I}(H)>0$

where $H=-\Delta+V$, $A=px+xp$, $p=-i\nabla$ and $\chi_{I}(H)$ localizes $H$
to some compact interval $I$ of positive energies, entails a limiting
absorption bound on the resolvent localized to $I$ (which is some form of
scattering). Hunziker, Sigal, Soffer HSS , developed a time-dependent and
abstract approach to Mourre theory by means of propagation estimates in the
spirit of Sigal, Soffer SSof1 . The main result of HSS  is the following
theorem.

######  Theorem III.8.

Let $A,H$ be self adjoint operators on some Hilbert space and assume the
Mourre estimate

$E_{I}i[H,A]E_{I}\geq\theta E_{I}$ (47)

where $\theta>0$ and $I\subset{\mathbb{R}}$ is some compact interval, and
$E_{I}$ is the spectral projector onto $I$ relative to $H$. Assume,
furthermore, that all iterated commutators of $f(H)$ with $A$ are bounded
where $f\in C^{\infty}_{0}({\mathbb{R}})$. Let $\chi^{\pm}$ be the indicator
functions of ${\mathbb{R}}^{\pm}$, respectively. Then for any $m\geq 1$,

$\|\chi^{-}(A-a-\theta^{\prime}t)e^{iHt}g(H)\chi^{+}(A-a)\|\leq
C(m,\theta,\theta^{\prime})\,t^{-m}$

for any $g\in C_{0}^{\infty}(I)$, any $0<\theta^{\prime}<\theta$, uniformly in
$a\in{\mathbb{R}}$.

As simple consequence of this result is the following propagation estimate,
which is clearly most important in the context of Theorem III.2:

$\|\langle A\rangle^{-\alpha}e^{iHt}g(H)\langle A\rangle^{-\alpha}\|\leq
C(\alpha)\,\langle t\rangle^{-\alpha}$ (48)

for any $\alpha>0$. In application one typically takes $A=\frac{1}{2}(px+xp)$,
the generator of dilations, or some variant thereof. Taking $\alpha=1$ shows
that one needs at least $w_{1}$ in the Schrödinger case of Theorem III.2.

One needs to resolve two issues before applying this theory to Theorem III.2:

  * •

We require a semi-classical version of HSS .

  * •

The top of the barrier energy is trapping in the classical sense.

While the first issue is a routine variant of HSS , the second is not. In the
nontrapping case, Graf Graf  and Hislop, Nakamura HN  showed that the
classical nontrapping condition $\\{a,h\\}>\alpha>0$ on the entire energy
level $\\{h=E_{0}>0\\}$ implies the Mourre estimate (47) for $I$ some small
interval around $E_{0}$ (in the semi-classical case with $\hbar$ sufficiently
small). In the case of surfaces of revolution as in Theorem III.2 this fact,
together with Theorem III.8, implies that one can handle energies in the range
$\epsilon<E<V_{\rm max}(0)-\epsilon$ since they verify a classical nontrapping
condition. On the other hand, for energies near $V_{\rm max}(0)$ this fails
since the top energy is classically trapping. Nevertheless, the Heisenberg
uncertainty principle (or the semiclassical harmonic oscillator) guarantee
(47).

Indeed, with $V(x)=1-\frac{1}{2}\langle Qx,x\rangle+O(|x|^{3})$ with $Q$
positive definite,

$\\{h,a\\}=\xi^{2}-x\cdot\nabla V=\xi^{2}+\langle
Qx,x\rangle+O(|x|^{3})\geq\theta(\xi^{2}+x^{2})$

for small $x$. However, $p^{2}+q^{2}\geq c>0$ by the uncertainty principle,
which indicates that one should expect that (47) continues to hold at a non-
degenerate maximum. For a rigorous rendition of this argument see Briet,
Combes, Duclos BCD , Nakamura Nak , and DSS2 .

Generally speaking, the problem of obtaining a representation of the resolvent
and the spectral measure, and of proving a limiting absorption principle for
energies near a potential barrier has received a lot of attention, see the
monograph by Bony et al. Bony  and the earlier literature cited there such as
the classical work by Helffer, Sjöstrand from the 80s.

This concludes our informal sketch of the proof of (23). As for (24), one
proceeds analogously by dividing energies into three regions, low,
intermediate, and high. In the low and high cases, one obtains pointwise
bounds without weights from the WKB arguments outlined above, followed by
oscillatory integral estimates as in SSS2 . For the intermediate regime one
uses the $L^{2}$ bound (from the Mourre-Hunziker-Sigal-Soffer estimates) which
requires a weight $w_{1}$ followed by the Sobolev embedding theorem. Note that
the latter costs one power of $\ell$, whereas summation over $\ell$ requires
another weight of the form $\ell^{1+\epsilon}$ which explains the loss of
$(1-\partial^{2}_{\theta})^{1+\epsilon}$ on the right-hand side of (24) as
compared to (23).

As a final remark, we would like to emphasize that the sketch of proof of
Theorem III.2 which we just concluded is an adaptation of the argument which
was developed for the Schwarzschild case in DSS2 .

###  III.3 The Schwarzschild case

The results on surfaces of revolution are relevant to another problem namely
the decay of linear waves on a Schwarzschild black hole background. To be more
specific, choose coordinates such that the exterior region of the black hole
can be written as $(t,r,(\theta,\phi))\in\mathbb{R}\times(2M,\infty)\times
S^{2}$ with the metric

$g=-F(r)dt^{2}+F(r)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})$

where $F(r)=1-\frac{2M}{r}$ and, as usual, $M>0$ denotes the mass. We now
introduce the well–known _Regge–Wheeler tortoise coordinate_ $r_{*}$ which (up
to an additive constant) is defined by the relation

$F=\frac{dr}{dr_{*}}.$

In this new coordinate system, the outer region is described by
$(t,r_{*},(\theta,\phi))\in\mathbb{R}\times\mathbb{R}\times S^{2}$,

$g=-F(r)dt^{2}+F(r)dr_{*}^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})$
(49)

with $F$ as above and $r$ is now interpreted as a function of $r_{*}$.
Explicitly, $r_{*}$ is computed as

$r_{*}=r+2M\log\left(\frac{r}{2M}-1\right).$

Generally, the Laplace–Beltrami operator on a manifold with metric $g$ is
given by

$\Box_{g}=\frac{1}{\sqrt{|\det(g_{\mu\nu})|}}\partial_{\mu}\left(\sqrt{|\det(g_{\mu\nu})|}g^{\mu\nu}\partial_{\nu}\right)$

and thus, for the metric $g$ in (49), we obtain

$\Box_{g}=F^{-1}\left(-\partial_{t}^{2}+\frac{1}{r^{2}}\partial_{r_{*}}\left(r^{2}\partial_{r_{*}}\right)\right)+\frac{1}{r^{2}}\Delta_{S^{2}}.$

By setting
$\psi(t,r_{*},\theta,\phi)=r(r_{*})\tilde{\psi}(t,r_{*},\theta,\phi)$ and
writing $x=r_{*}$, the wave equation $\Box_{g}\tilde{\psi}=0$ is equivalent to

$-\partial_{t}^{2}\psi+\partial_{x}^{2}\psi-\frac{F}{r}\frac{dF}{dr}\psi+\frac{F}{r^{2}}\Delta_{S^{2}}\psi=0.$
(50)

The mathematically rigorous analysis of this equation goes back to Wald Wal
and Kay KW , who established uniform boundedness of solutions. In the spirit
of the positive commutator methods outlined above, Dafermos and Rodniansk DR3
found a robust approach based on carefully chosen vector fields and
multipliers. See Luk’s work Luk1 ; Luk2  which is in a similar spirit. As
already noted, Blue and Soffer BSof  proved local decay estimates using
Morawetz estimates. Dafermos and Rodnianski DR2  proved Price’s $t^{-3}$ decay
law for a nonlinear problem but assuming spherical symmetry.

The purpose of this section is to discuss recent work of Donninger and the
authors on pointwise decay for solutions to Eq. (50). Different types of decay
estimates have been proved before. Our results differ from the above in
certain respects: the methods we use are based on constructing the Green’s
function and deriving the needed estimates on it. Previous works in this
direction include mainly the series of papers by Finster, Kamran, Smoller and
Yau, see for example FKSY , where the first pointwise decay result for Kerr
black holes was proved.

As in the case for surfaces of revolution, we freeze the angular momentum
$\ell$ or, in other words, we project onto a spherical harmonic. More
precisely, let $Y_{\ell,m}$ be a spherical harmonic (that is, an eigenfunction
of the Laplacian on $S^{2}$ with eigenvalue $-\ell(\ell+1)$) and insert the
ansatz $\psi(t,x,\theta,\phi)=\psi_{\ell,m}(t,x)Y_{\ell,m}(\theta,\phi)$ in
Eq. (50). This yields the _Regge–Wheeler equation_

$\partial_{t}^{2}\psi_{\ell,m}-\partial_{x}^{2}\psi_{\ell,m}+V_{\ell,\sigma}(x)\psi_{\ell,m}=0$

with $\sigma=1$ where

$V_{\ell,\sigma}(x)=\left(1-\frac{2M}{r(x)}\right)\left(\frac{\ell(\ell+1)}{r^{2}(x)}+\frac{2M\sigma}{r^{3}(x)}\right)$

is known as the _Regge–Wheeler potential_. The other physically relevant
values of the parameter $\sigma$ are $\sigma=-3,0$. For more background we
refer the reader to the introduction of DSS1 , or DR2 .

We immediately note some crucial features of $V_{\ell,\sigma}$: it decays
exponentially as $x\to-\infty$, it decays according to an inverse square law
as $x\to+\infty$ provided $\ell>0$, and like an inverse cube if $\ell=0$.
Moreover, it has a unique nondegenerate maximum which is located at the photon
sphere. It consists of closed light rays and replaces the unique periodic
geodesic which we encountered in Theorem III.2.

So we expect that at least some of the machinery that we described above in
the surface case applies here as well. However, the Regge-Wheeller potential
is considerably more difficult to deal with.

The main result of DSS1  is the following pointwise decay, which captures the
so-called Price law for fixed angular momentum. Strictly speaking, it is still
off by one power of $t$ from the sharpest form of Price’s law which is
$t^{-2\ell-3}$ whereas the following result proves $t^{-2\ell-2}$ (we shall
comment on that issue below). Note how the accelerated decay for higher values
of $\ell$ mirrors what we saw for the surfaces of revolution in Theorem III.1.
Hintz Hintz  recently closed the gap of the missing power of $t$ and thus
finished the proof of Price’s law.

######  Theorem III.9.

Let $(\ell,\sigma)\notin\\{(0,0),(0,-3),(1,-3)\\}$, $\alpha\in\mathbb{N}$ and
$1\leq\alpha\leq 2\ell+3$. Then the solution operators for the Regge–Wheeler
equation satisfy the estimates

$\displaystyle\|w_{\alpha}\cos(t\sqrt{\mathcal{H}_{\ell,\sigma}})f\|_{L^{\infty}(\mathbb{R})}$
$\displaystyle\leq C_{\ell,\alpha}\langle
t\rangle^{-\alpha}\Big{(}\Big{\|}\frac{f^{\prime}}{w_{\alpha}}\Big{\|}_{L^{1}(\mathbb{R})}$
$\displaystyle\qquad\qquad+\Big{\|}\frac{f}{w_{\alpha}}\Big{\|}_{L^{1}(\mathbb{R})}\Big{)}$

and

$\left\|w_{\alpha}\frac{\sin(t\sqrt{\mathcal{H}_{\ell,\sigma}})}{\sqrt{\mathcal{H}_{\ell,\sigma}}}f\right\|_{L^{\infty}(\mathbb{R})}\leq
C_{\ell,\alpha}\langle
t\rangle^{-\alpha+1}\left\|\frac{f}{w_{\alpha}}\right\|_{L^{1}(\mathbb{R})}$

for all $t\geq 0$ where $w_{\alpha}(x):=\langle x\rangle^{-\alpha}$.

The values of $(\sigma,\ell)$ which we exclude here are precisely those where
the Regge-Wheeler potential gives rise to zero energy resonances. Physically
speaking, they correspond to a gauge invariance, such as changing the mass,
and are therefore irrelevant.

The proof of Theorem III.9 is based on representing the solution as an
oscillatory integral in the energy variable $\lambda$, schematically one may
write

$\psi(t,x)=\int
U(t,\lambda)\mathrm{Im}\left[G_{\ell,\sigma}(x,x^{\prime},\lambda)\right]f(x^{\prime})\,dx^{\prime}d\lambda$

where $U(t,\lambda)$ is a combination of $\cos(t\lambda)$ and $\sin(t\lambda)$
terms and $G_{\ell,\sigma}(x,x^{\prime},\lambda)$ is the kernel (Green’s
function) of the resolvent of the operator $\mathcal{H}_{\ell,\sigma}$. In
analogy with Theorem III.1, $G_{\ell,\sigma}(x,x^{\prime},\lambda)$ is
constructed in terms of the Jost solutions and we obtain these functions in
various domains of the $(x,\lambda)$ plane by perturbative arguments: for
$|x\lambda|$ small we perturb in $\lambda$ around $\lambda=0$, whereas for
$|x\lambda|$ large we perturb off of Hankel functions. This is done in such a
way that there remains a small window where the two different perturbative
solutions can be glued together. One of the main technical difficulties of the
proof lies with the fact that we need good estimates for arbitrary derivatives
of the perturbative solutions. This is necessary in order to control the
oscillatory integrals. The most important contributions come from $\lambda\sim
0$ and we therefore need to derive the exact asymptotics of the Green’s
function and its derivatives in the limit $\lambda\to 0$. For instance, we
prove that

$\mathrm{Im}\left[G_{\ell,\sigma}(0,0,\lambda)\right]=\lambda
P_{\ell}(\lambda^{2})+O(\lambda^{2\ell+1})$

as $\lambda\to 0+$ where $P_{\ell}$ is a polynomial of degree $\ell-1$ (we set
$P_{0}\equiv 0$) and the $O$–term satisfies
$O^{(k)}(\lambda^{2\ell+1})=O(\lambda^{2\ell+1-k})$ for all
$k\in\mathbb{N}_{0}$.

As already noted before, for $\ell=0$ the Regge-Wheeler potential decays like
an inverse cube as $x\to\infty$. This case is covered by the following result
of Donninger and the first author DS .

######  Theorem III.10.

Let $V\in C^{[\alpha]+1}(\mathbb{R})$ with
$V(x)=|x|^{-\alpha}[c_{\pm}+O(|x|^{-\beta})]$ as $x\to\pm\infty$ where
$2<\alpha\leq 4$, $\beta=\frac{1}{2}(\alpha-2)^{2}$, $c_{\pm}\in\mathbb{R}$
and $|O^{(k)}(|x|^{-\beta})|\lesssim|x|^{-\beta-k}$ for
$k=1,2,\dots,[\alpha]+1$. Denote by $A$ the self–adjoint Schrödinger operator
$Af:=-f^{\prime\prime}+Vf$ in $L^{2}(\mathbb{R})$ and assume that $A$ has no
bound states and no resonance at zero energy. Then the following decay bounds
hold:

$\displaystyle\|\langle\cdot\rangle^{-\alpha-1}\cos(t\sqrt{A})f\|_{L^{\infty}(\mathbb{R})}$
$\displaystyle\lesssim\langle
t\rangle^{-\alpha}\big{(}\|\langle\cdot\rangle^{\alpha+1}f^{\prime}\|_{L^{1}(\mathbb{R})}$
$\displaystyle\qquad+\|\langle\cdot\rangle^{\alpha+1}f\|_{L^{1}(\mathbb{R})}\big{)}$

and

$\left\|\langle\cdot\rangle^{-\alpha-1}\frac{\sin(t\sqrt{A})}{\sqrt{A}}f\right\|_{L^{\infty}(\mathbb{R})}\lesssim\langle
t\rangle^{-\alpha}\|\langle\cdot\rangle^{\alpha+1}f\|_{L^{1}(\mathbb{R})}$

for all $t\geq 0$.

In particular, this gives $t^{-3}$ for $\alpha=3$ which is the sharp form of
Price’s law for $\ell=0$. It is important to realize that the decay of the
waves in Theorems III.9 and III.10 is really a manifestation of transport
rather than of dispersion. Indeed, d’Alembert’s formula shows that any
solution of

$\partial_{tt}u-\partial_{xx}u=0,\qquad u(0)=f,\;\partial_{t}u(0)=g$

with Schwartz data (say) and $\int g(x)\,dx=0$ satisfies

$\|\langle x\rangle^{-\alpha}u(t)\|_{\infty}\leq C(\alpha)\,t^{-\alpha}$

for any $\alpha\geq 0$. This vanishing mean condition can be attributed to the
zero energy resonance for the free Laplacian in one dimension. Needless to
say, the one-dimensional problem does not exhibit any sort of dispersion but
is governed by linear transport which leads to this arbitrary local decay of
the waves. It is very interesting to note (but perhaps not immediately clear)
that the sharp Huyghens principle in three dimensions is still visible in the
local decay law of Theorem III.9. In fact, we claim that the sharp
$t^{-2\ell-3}$ Price law (at least for $\ell\geq 1$) is a result of the
correction term of the form $\frac{\log x}{x^{3}}$ in the Regge-Wheeler
potential rather than the leading inverse square decay as $x\to+\infty$.

To clarify this point, we now present a simple model case from CDST . With
$a>0$,

$\mathcal{H}:=-\partial_{x}^{2}+V,\quad
V(x)=\left\\{\begin{array}[]{ll}0&\text{\ if\ }x\leq-1\\\
\frac{a^{2}-\frac{1}{4}}{x^{2}}&\text{\ if\ }x\geq 1\end{array}\right.$

Moreover, $V\in C^{\infty}({\mathbb{R}})$ is such that $\mathcal{H}$ has no
zero energy resonance which means that there does not exist a globally
subordinate (or recessive) solution $\mathcal{H}f=0$ other than $f\equiv 0$.
Recall that this refers to solutions of the slowest allowed growth at both
ends, which means here that $f(x)=O(1)$ as $x\to-\infty$ and
$f(x)=O(x^{\frac{1}{2}-a})$ as $x\to+\infty$. Then one has the following local
decay estimates for the wave equation with potential $V$.

######  Proposition III.11.

Under the above assumptions on $\mathcal{H}$,

$\displaystyle\Big{\|}\langle
x\rangle^{-\sigma}\frac{\sin(t\sqrt{\mathcal{H}})}{\sqrt{\mathcal{H}}}P_{(0,\infty)}(\mathcal{H})g\Big{\|}_{\infty}$
$\displaystyle\leq C\langle t\rangle^{-2a-1}\big{\|}\langle
x\rangle^{\sigma}g\big{\|}_{1}$ $\displaystyle\Big{\|}\langle
x\rangle^{-\sigma}\cos(t\sqrt{\mathcal{H}})P_{(0,\infty)}(\mathcal{H})f\Big{\|}_{\infty}$
$\displaystyle\leq C\langle t\rangle^{-2a-2}(\big{\|}\langle
x\rangle^{\sigma}f\big{\|}_{1}$ $\displaystyle\qquad+\big{\|}\langle
x\rangle^{\sigma}f^{\prime}\big{\|}_{1})$

where $\sigma>0$ is sufficiently large depending on $a$. These decay rates are
optimal provided $a\not\in{\mathbb{Z}}_{0}^{+}+\frac{1}{2}$. In the latter
case, one obtains decay $t^{-N}$ for any $N$ (provided $\sigma$ is taken
sufficiently large depending on $N$).

###### Proof.

We prove the first bound, the second one being very similar. Thus, let
$\psi(t,x)$ be a solution of the problem

$\partial_{t}^{2}\psi-\partial_{x}^{2}\psi+V\psi=0,\quad\psi(0,x)=0,\;\partial_{t}\psi(0,x)=g$

where $g$ is Schwartz, say, and set for ${\,\rm Re\,}(p)>0$

$\hat{\psi}(p,x):=\int_{0}^{\infty}e^{-tp}\psi(t,x)\,dt$

Then

$(\mathcal{H}+p^{2})\hat{\psi}(p,\cdot)=g$

which has a unique bounded solution

$\displaystyle\hat{\psi}(p,x)$
$\displaystyle=\int_{-\infty}^{\infty}G(p;x,y)\,g(y)\,dy$
$\displaystyle=\int_{-\infty}^{x}\frac{f_{+}(x,p)f_{-}(y,p)}{W(p)}\,g(y)\,dy$
$\displaystyle\qquad-\int_{x}^{\infty}\frac{f_{+}(y,p)f_{-}(x,p)}{W(p)}\,g(y)\,dy$

with constant Wronskian
$W(p):=f_{+}(x,p)f_{-}^{\prime}(x,p)-f_{+}^{\prime}(x,p)f_{-}(x,p)$. Here
$f_{\pm}(x,p)$ are the Jost solutions

$(\mathcal{H}+p^{2})f_{\pm}(\cdot,p)=0,\qquad f_{\pm}(x,p)\sim e^{\mp
xp}\text{\ as\ }x\to\pm\infty$

The goal is now to obtain the expansion of $f_{\pm}(x,p)$ in small $p$, as
this then yields the large time asymptotics of, with arbitrary $p_{0}>0$,

$\psi(t,x)=\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\int_{p_{0}-i\infty}^{p_{0}+i\infty}e^{tp}\;G(p;x,y)\,dp\;g(y)\,dy$
(51)

via contour deformation and Watson’s lemma. By choice of potential $V$,

$\displaystyle f_{-}(x,p)$ $\displaystyle=e^{px}\text{\ for\ }x\leq-1$
$\displaystyle f_{+}(x,p)$ $\displaystyle=\frac{\pi i}{2}e^{a\pi
i/2}H_{a}^{(1)}(ipx)\big{(}\frac{2px}{\pi}\big{)}^{\frac{1}{2}}\text{\ for\
}x\geq 1$

One can continue $f_{-}(x,p)$ to the right of $x=-1$ which yields an entire
function in $p$ for each fixed $x$. The nonresonance condition for $p=0$ means
that $f_{-}(\cdot,0)$ and $x^{\frac{1}{2}-a}$ are linearly independent at
$x=1$. Since $H_{a}^{(1)}=J_{a}+iY_{a}$ and – up to constant factors –

$J_{a}(u)\sim u^{a}(1+O(u^{2})),\quad Y_{a}(u)\sim u^{-a}(1+O(u^{2}))$

as $u\to 0$ with analytic $O(u^{2})$ (at least provided $a$ is not an
integer), we conclude that

$W(p)=c(V)\,p^{\frac{1}{2}-a}\big{[}1+O(p^{2})+\tilde{c}(V)\,p^{2a}(1+O(p^{2}))\big{]}\text{\
\ as\ }p\to 0$

with $O(p^{2})$ analytic in a neighborhood of $p=0$ and with $c(V)\neq 0$.
This is obtained by computing $W(p)$ at $x=1$, say, and by noting that the
most singular contribution to $W(p)$ around $p=0$ is
$c(V)\,p^{\frac{1}{2}-a}$. By inspection, $c(V)=0$ is the same as a zero
energy resonance which is excluded. If $a$ is a positive integer, then as
$p\to 0$

$W(p)=c(V)\,p^{\frac{1}{2}-a}\big{[}1+O(p^{2})+\tilde{c}(V)\,p^{2a}\log(p)(1+O(p^{2}))\big{]}$

For simplicity, let us first freeze $x,y$, say $x=y=1$. Then one concludes
from the preceding that

$G(p;1,1)=C(V)\,p^{2a}\big{[}1+O(p^{2})+\tilde{c}(V)\,p^{2a}(1+O(p^{2}))\big{]}$
(52)

for small $p\in{\mathbb{C}}\setminus(-\infty,0]$, and analytic $O(p^{2})$
around $p=0$, whereas for the case of $a\in{\mathbb{Z}}$,

$\displaystyle G(p;1,1)$ $\displaystyle=C(V)\,p^{2a}\log(p)\big{[}1+O(p^{2})$
$\displaystyle\qquad+\tilde{c}(V)\,p^{2a}\log(p)(1+O(p^{2}))\big{]}$

The stated decay law now follows via Watson’s lemma in a standard fashion.
Note the special role of integer but odd $2a$ (which is the exceptional case
in the statement of the proposition): in that case (52) is analytic in small
$p$ whence one can push the contour in (51) through $p=0$ leading to
exponential decay (at least as far as the contribution of small $p$ is
concerned).

We now discuss the Watson lemma in more detail. First, we move the contour in
(51) onto the imaginary axis:

$\psi(t,1)=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{tp}\;G(p;1,1)\,dp$

The contribution due to $1-\chi(p)$ is shown via integration by parts to decay
faster than any power of $t$ (use that $G(iE;x,y)=O(E^{-1})$ for large $E$,
uniformly in $x,y$). On the other hand, for the contribution of $\chi$ we
retain only finitely many terms from $G(p;1,1)$ with a remainder that is
smooth enough around $p=0$ so as to yield the desired decay again by
integration by parts. Finally, the first remaining term is of the form (up to
a constant factor $C(V)$)

$\int_{-i{\varepsilon}}^{i{\varepsilon}}e^{tp}\;p^{2a}\chi(p)\,dp$

We also have a $\log p$ factor if $a\in{\mathbb{Z}}$. One now extends this to

$\int_{\gamma}e^{tp}\;p^{2a}\,dp$ (53)

where $\gamma$ is an curve which contains $[-i{\varepsilon},i{\varepsilon}]$
and is asymptotic to $[0,e^{i\theta}\infty]$ and $[-e^{i\theta}\infty,0]$,
respectively, and the ends. Noting that the integrals we inserted here
decrease like $t^{-N}$ for any $N$ by integration by parts. By Cauchy’s
theorem this is the same as

$2\sin(2a\pi)\int_{0}^{\infty}e^{-tp}\;p^{2a}\,dp=2\sin(2a\pi)t^{-2a-1}\Gamma(2a+1)$

which is the decay rate stated in the proposition. Note that if
$a=\ell+\frac{1}{2}$ then this term vanishes leading to the exceptional
behavior stated above. On the other hand, if $a\in{\mathbb{Z}}$, then this
contribution does not vanish due to the $\log(p)$ factor. Finally, we need to
remove the restriction $x=y=1$. However, we have set up our argument in such a
way that this modification is easy. First, the contribution of
$|p|>{\varepsilon}$ is again shown to decay at an arbitrary rate via
integration by parts. Now this procedure brings down as many powers of $x,y$
as given by the desired power of $t^{-1}$. Next, the contribution of the
finitely many terms involving $p^{2a}$ etc. is similar to before, and each one
of these terms comes with a corresponding weight in $x$ and $y$. Finally, the
remainder in $G(p;x,y)$ after subtracting that initial segment is again
sufficiently smooth in $p$ and therefore integration by parts yields the
desired decay leading to another instance of requiring large $\sigma$. ∎

The significance of this proposition lies with proximity of $V$ to the Regge-
Wheeler potential. Indeed, we replaced the exponential tails on the left by
zero, and retained the inverse square tails on the right (ignoring the higher-
order corrections). In case of the Regge-Wheeler potential one has
$a^{2}-\frac{1}{4}=\ell(\ell+1)$ which implies that $a=\ell+\frac{1}{2}$ which
is the exceptional case of Proposition III.11. Formally speaking
$2a+1=2\ell+2$ corresponds exactly to the decay rate of Theorem III.9, whereas
the Price law $t^{-2\ell-3}$ is therefore seen to be a result of the
$\frac{\log x}{x^{3}}$ correction to the far field in $V_{\ell,\sigma}$. In
fact, it is shown in CDST  that the Price law is due to the nonanalytic term
$p^{2a+1}\log p$ instead of $p^{2a}$ in (53). To accomplish this, one derives
an expansion of $f_{+}(x,p)$ in small $p$ taking into account as many terms
from $V_{\ell,\sigma}$ as required for obtaining Price’s law and the next few
corrections to it. The route taken in CDST  consists of a reduction of the
Regge-Wheeler potential to a normal form by means of a Liouville-Green
transform. The normal form here consists of the potential without any
corrections to the leading $\frac{\ell(\ell+1)}{x^{2}}$ decay. The branching
around $p=0$ then results from the change of independent variable. Arguing as
in the previous proof then yields the sharp $t^{-2\ell-3}$ Price law.

To conclude this survey, let us state the main local decay result from DSS2 .

######  Theorem III.12.

The following decay estimates hold for solutions $\psi$ of (50) with data
$\psi[0]=(\psi_{0},\psi_{1})$:

$\displaystyle\|\langle x\rangle^{-\frac{9}{2}-}\psi(t)\|_{L^{2}}$
$\displaystyle\lesssim\langle t\rangle^{-3}\|\langle
x\rangle^{\frac{9}{2}+}({/\\!\\!\\!\nabla}^{5}\partial_{x}\psi_{0},{/\\!\\!\\!\nabla}^{5}\psi_{0},{/\\!\\!\\!\nabla}^{4}\psi_{1})\|_{L^{2}}$
(54) $\displaystyle\|\langle x\rangle^{-4}\psi(t)\|_{L^{\infty}}$
$\displaystyle\lesssim\langle t\rangle^{-3}\|\langle
x\rangle^{4}({/\\!\\!\\!\nabla}^{10}\partial_{x}\psi_{0},{/\\!\\!\\!\nabla}^{10}\psi_{0},{/\\!\\!\\!\nabla}^{9}\psi_{1})\|_{L^{1}}$
(55)

where ${/\\!\\!\\!\nabla}$ stands for the angular derivatives. The notation
$a\pm$ stands for $a\pm{\varepsilon}$ where ${\varepsilon}>0$ is arbitrary
(the choice determines the constants involved). Also, instead of
$({/\\!\\!\\!\nabla}^{10},{/\\!\\!\\!\nabla}^{9})$ in (55) one needs less,
namely $({/\\!\\!\\!\nabla}^{\sigma+1},{/\\!\\!\\!\nabla}^{\sigma})$ where
$\sigma>8$ is arbitrary. Here $L^{2}:=L^{2}_{x}({\mathbb{R}};L^{2}(S^{2}))$,
$L^{1}:=L^{1}_{x}({\mathbb{R}};L^{1}(S^{2}))$, and
$L^{\infty}:=L^{\infty}_{x}({\mathbb{R}};L^{\infty}(S^{2}))$.

It is obtained by summation in $\ell$ following the same line of reasoning
that lead to Theorem III.2 above. The most significant complication is due to
the asymmetry of the Regge-Wheeler potential: while the inverse square
potential for $x\to\infty$ is covered by CSST  as before, the exponentially
decaying part on the left requires another WKB analysis. We refer the reader
to DSS2  for the details.

###### Acknowledgements.

This article is dedicated to the memory of Jean Bourgain. The author was
partly supported by the National Science Foundation grant DMS-1902691. The
author thanks an anonymous referee for numerous helpful comments which
improved the presentation.

## Data Availability Statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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