# Dispersive and Strichartz estimates for 3D wave equation with a class of
many-electric potentials

Haoran Wang  Department of Mathematics, Beijing Institute of Technology,
Beijing 100081, China<EMAIL_ADDRESS>
###### Abstract.

We prove the dispersive and Strichartz estimates for solutions to the wave
equation with a class of many-electric potentials in spatial dimension three.
To obtain the desired dispersive estimate, based on the spectral properties of
the Schrödinger operator involved, we subsequently prove the dispersive
estimate for the corresponding Schrödinger semigroup, obtain a Gaussian type
upper bound, establish Bernstein type inequalities, and finally pass to the
Müller-Seeger’s subordination formula. The desired Strichartz estimates follow
by the established dispersive estimate and the standard argument of Keel-Tao.

Key Words: Dispersive estimate; Strichartz estimate; Wave equation; Hermite
potential; Inverse square potential.
2020 Mathematics Subject Classification: 42B37, 35L05, 35Q40.

##  1\. Introduction

In this paper, we consider the Cauchy problem for the 3D perturbed wave
equation

$\begin{cases}\partial_{tt}u-\Delta
u+Vu=0,\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{3}\\\
u(0,x)=f(x),\quad\partial_{t}u(0,x)=g(x),\end{cases}$ (1.1)

where $V(x):\mathbb{R}^{3}\rightarrow\mathbb{R}$ is the Laguerre potential, a
combination of the inverse square potential and the Hermite potential, i.e.

$V(x)=\frac{a}{|x|^{2}}+\frac{b}{4}|x|^{2},\quad a,b\geq 0,\quad
x\in\mathbb{R}^{3}\setminus\\{0\\}.$ (1.2)

If we write $H_{a,b}$ for the operator in (1.1), i.e.

$H_{a,b}:=-\Delta+\frac{a}{|x|^{2}}+\frac{b}{4}|x|^{2},$ (1.3)

then it is easy to verify that the quadratic form of the operator $H_{a,b}$ is
positive definite, i.e.

$\int_{\mathbb{R}^{3}}\left(|\nabla\varphi|^{2}+\frac{a}{|x|^{2}}|\varphi|^{2}+\frac{b|x|^{2}}{4}|\varphi|^{2}\right)\mathrm{d}x>0,\quad\forall
a,b\geq 0,\quad\varphi\in\mathcal{S}(\mathbb{R}^{3}).$

Hence, the operator $H_{a,b}$ is a symmetric semi-bounded operator on
$L^{2}(\mathbb{R}^{3};\mathbb{C})$, which implies that $H_{a,b}$ admits a
self-adjoint extension, particularly, the Friedrichs’ extension, with a formal
domain $\mathcal{D}$ given by

$\mathcal{D}(H_{a,b})=\left\\{\varphi\in
L^{2}(\mathbb{R}^{3};\mathbb{C}):H_{a,b}\varphi\in
L^{2}(\mathbb{R}^{3};\mathbb{C})\right\\}.$

As a consequence of the Friedrichs’ extension, the unitary propagators like
$e^{itH_{a,b}},e^{it\sqrt{H_{a,b}}}$ are well-defined on the domain of
$H_{a,b}$ via the spectral theorem.

The unique solution $u\in C(\mathbb{R};L^{2}(\mathbb{R}^{3}))$ of the problem
(1.1) can be written as

$u(t,x)=\cos(t\sqrt{H_{a,b}})f(x)+\frac{\sin(t\sqrt{H_{a,b}})}{\sqrt{H_{a,b}}}g(x).$
(1.4)

If the inverse square potential is removed (i.e. $a\equiv 0$ in (1.3)), then
the Hamiltonian $H_{a,b}$ is reduced to the usual harmonic oscillator
$H_{0,b}=-\Delta+\frac{b}{4}|x|^{2}$. If the Hermite potential is removed
(i.e. $b=0$ in (1.3)), then one gets the Schrödinger operator with an inverse
square potential $H_{a,0}=-\Delta+\frac{a}{|x|^{2}}$, which is known to have
the same homogeneity as the free Laplacian $-\Delta$. If the potential $V$ in
(1.1) vanishes, then we recover the classical free wave equation, which is
known to be scaling-invariant. The scaling-invariance of dispersive equations
(including the free Schrödinger and wave equations) is critical for a large
class of dispersive estimates, such as time-decay, Strichartz and local-
smoothing.

In particular, one has the classical dispersive estimate

$\left\|\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}g\right\|_{L^{\infty}(\mathbb{R}^{3})}\leq\frac{C}{|t|}\|g\|_{\dot{B}^{1}_{1,1}(\mathbb{R}^{3})},\quad\forall|t|>0,$
(1.5)

where $\|\cdot\|_{\dot{B}^{1}_{1,1}}$ denotes the standard homogeneous Besov
norm and $C>0$ is a suitable constant independent of $t$ and $g$.

In the past decades, it turns out that such estimates as (1.5) play a
fundamental role in various mathematical fields, including scattering theory,
harmonic analysis and nonlinear dynamics (see e.g. [8, 12, 30, 44]). By the
standard $TT^{\ast}$ argument together with the unitary property of the half-
wave propagator $e^{it\sqrt{-\Delta}}$

$\|e^{it\sqrt{-\Delta}}f\|_{L^{2}(\mathbb{R}^{3})}=\|f\|_{L^{2}(\mathbb{R}^{3})},$

one can derive the Strichartz estimates

$\left\|\cos(t\sqrt{-\Delta})f+\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}g\right\|_{L_{t}^{q}(\mathbb{R};L^{r}(\mathbb{R}^{3}))}\leq
C\left(\|f\|_{\dot{H}^{s}(\mathbb{R}^{3})}+\|g\|_{\dot{H}^{s-1}(\mathbb{R}^{3})}\right)$
(1.6)

for some constant $C>0$, where the pair $(q,r)$ satisfies

$\frac{1}{q}+\frac{1}{r}\leq\frac{1}{2},\quad 2\leq q,r<\infty$ (1.7)

and $s$ is the gap index

$s=3\left(\frac{1}{2}-\frac{1}{r}\right)-\frac{1}{q}.$ (1.8)

We assume $0\leq s<\frac{3}{2}$ to guarantee a room for the pair $(q,r)$. The
Strichartz estimate (1.6) was initially obtained by Segal [41] and then
generalized by Strichartz [42] in connection with Tomas’s restriction theorem
[45]. Later, Ginibre and Velo [19] introduced a new viewpoint, which was
extensively applied by Yajima [48] to obtain a large class of inequalities for
the Schrödinger equation. Finally, Keel and Tao [28] settled the most
challenging endpoint case $q=2$, via bilinear techniques, for an abstract
propagator fulfilling the dispersive estimate as (1.5).

We stress that the above mentioned scaling-invariance is no longer valid for
our model (1.1) due to the appearance of the Hermite potential in (1.3).
Therefore, it is not trivial to verify the dispersive estimate (1.5) (and
hence the Strichartz estimates (1.6)) for the wave equation (1.1), which is
the main purpose of this work. Before stating the main theorems, let us
briefly review some relevant papers to better frame our results. The
dispersive estimate (1.5) has been obtained previously for the Schrödinger and
wave equations in e.g. [4, 5, 10, 11, 13, 14, 20, 21, 22, 23, 24, 25, 27, 37,
40] when the potential $V$ is subcritical (i.e. decaying faster than the
inverse square potential $|x|^{-2}$ near infinity). In the earlier work [4],
Beals and Strauss obtained the $L^{p}-L^{q}$ decay estimates for the solution
to the wave equation (1.1) but not the dispersive estimate in the spirit of
(1.5) for potentials both smooth and small in a suitable sense. Later,
Georgiev and Visciglia [20] proved the dispersive estimate (1.5) for almost
critical potentials, more precisely, for potentials $V\in
C^{\delta}(\mathbb{R}^{3}\setminus\\{0\\}),\delta\in(0,1)$ satisfying

$0\leq V(x)\leq\frac{C}{|x|^{2+\epsilon}+|x|^{2-\epsilon}}\quad\text{for
some}\quad\epsilon>0.$

For the critical inverse square potential $\frac{a}{|x|^{2}}$, it turns out
that the dispersive estimate (1.5) is true when $a\geq 0$ but may be false
when $a<0$ (see [37, 38]). The Strichartz estimate (1.6) was proved by Burq,
Planchon, Stalker, and Tahvildar-Zadeh [6, 7] for both Schrödinger and wave
equations with the potential $\frac{a}{|x|^{2}}$ in two and higher dimensions;
actually, they obtained suitable Morawetz-type estimates for the perturbed
resolvent by the related multiplier results, which, together with the well-
known free counterparts, yields the desired Strichartz estimates for the
perturbed operator $-\Delta+V$. Mizutani [34] studied the Strichartz estimates
for the Schrödinger equation with the critical inverse-square potential
$-\frac{(d-2)^{2}}{4|x|^{2}}$ in dimension $d\geq 3$. It turns out that the
inverse-square potential represents a threshold for the validity of the
dispersive and Strichartz estimates, as shown in [22]. For the Schrödinger
flow $e^{it(\Delta-V)}$, a typical perturbation argument consists of
expressing the action of the flow $e^{it(\Delta-V)}$ by the spectral theorem
and then reducing the matters of proving the desired dispersive estimate to
perform a suitable analysis on the resolvent of the perturbed Laplacian
$-\Delta+V$, in the sense of Agmon-Hörmander. As a standard example of such a
method, we refer the interested reader to the paper of Goldberg and Schlag
[21] (see also Rodnianski and Schlag [40]), in which the dispersive estimate
was obtained for the Schrödinger equation with integrability or decaying
conditions imposed on the potential $V$. Another effective method is studying
the mapping properties of the wave operators of the perturbed Laplacian on
$L^{p}$ and then the dispersive estimate for the Schrödinger flow
$e^{it(\Delta-V)}$ follows from the dispersive estimate for the free Laplacian
(1.5) and the intertwining property. The wave operator argument was introduced
and applied by Yajima in his series of papers [49, 50, 51, 52] to obtain a
large class of inequalities for dispersive equations. It turns out that this
approach leads to a much stronger result since many inequalities including the
resolvent estimate follow as a consequence of the $L^{p}$-boundedness of the
associated wave operators (see e.g. [12, 15, 35, 47]). For growing potentials,
we only mention the interesting paper [16], in which the dispersive estimates
for the wave equation associated to the Hermite operator and the twisted
Laplacian were obtained.

In light of the above considerations, the main purpose of this paper is to
prove the dispersive and Strichartz estimates as (1.5) and (1.6) for the wave
equation (1.1). Different from the convention that the operator $H_{a,b}$ in
(1.3) is often regarded as the Laplacian perturbed by the potential (1.2), we
may regard the operator $H_{a,b}$ as the harmonic oscillator perturbed by an
inverse square potential since the spectral properties of $H_{a,b}$ possess
more similarity to that of the harmonic oscillator $-\Delta+|x|^{2}$. Based on
this observation, we will, without loss of generality, always take $b=1$ in
(1.3) or let $H_{a,1}$ take the place of $H_{a,b}$ throughout the paper.

The main effort will be devoted to prove the dispersive estimate (1.5) since
the Strichartz estimates can be derived from the dispersive estimate via the
standard argument of Keel-Tao [28]. To obtain the desired dispersive estimate,
the steps we shall follow in the sequel consists of proving a dispersive
estimate for the Schrödinger flow $e^{itH_{a,1}}$, obtaining a Gaussian type
upper bound for the heat kernel $e^{-tH_{a,1}}(x,y)$, establishing the
Bernstein’s inequalities associated to Schrödinger operators and finally
passing to the classical subordination formula. In order to carry out these
steps, let us make some preparations. To obtain the dispersive estimate for
the Schrödinger propagator $e^{itH_{a,1}}$, a natural way is to write down the
Schrödinger kernel explicitly, for which a crucial role is played by the
spectrum of the Laplace-Beltrami operator $-\Delta_{\mathbb{S}^{2}}$ on the
unit sphere $\mathbb{S}^{2}$ in $\mathbb{R}^{3}$. It is well-known that
$-\Delta_{\mathbb{S}^{2}}$ admits a purely discrete spectrum, consisting of
real eigenvalues $\mu_{k}=k(k+1),k\in\mathbb{N}:=\\{0,1,2,\cdots\\}$ with
finite multiplicity $2k+1$, and the set of the associated ($L^{2}$-normalised)
eigenfunctions forms a complete orthonormal basis for
$L^{2}(\mathbb{S}^{2};\mathbb{C})$. In view of the symmetry of the wave
equation, we shall always require $t>0$ for simplicity.

The Sobolev space associated to the Schrödinger operator $H_{a,1}$ is defined
by

$\dot{\mathcal{H}}^{s}_{a,1}(\mathbb{R}^{3}):=H_{a,1}^{-s/2}L^{2}(\mathbb{R}^{3})$

and then the usual Sobolev space $\dot{H}^{s}(\mathbb{R}^{3})$ (corresponding
to the free Laplacian $-\Delta$) can be given by

$\dot{H}^{s}(\mathbb{R}^{3}):=\dot{\mathcal{H}}^{s}_{0,0}(\mathbb{R}^{3}).$

The homogeneous Besov norm
$\|\cdot\|_{\dot{\mathcal{B}}^{s}_{p,r}(\mathbb{R}^{3})}$ is defined by (see
also Sect. 5.2)

$\|f\|_{\dot{\mathcal{B}}^{s}_{p,r}(\mathbb{R}^{3})}=\left(\sum_{j\in\mathbb{Z}}2^{sjr}\|\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}^{r}\right)^{1/r},$
(1.9)

where $s\in\mathbb{R},p,r\in[1,\infty]$ and $\psi_{j}$ is a partition of unity

$\sum_{j\in\mathbb{Z}}\psi_{j}(\lambda)=1,\quad\forall\lambda>0.$

In particular, one has

$\|f\|_{\dot{\mathcal{H}}^{s}_{a,1}(\mathbb{R}^{3})}:=\|H_{a,1}^{s/2}f\|_{L^{2}(\mathbb{R}^{3})}=\left\|\left(\sum_{j\in\mathbb{Z}}2^{2sj}|\psi_{j}(\sqrt{H_{a,1}})f|^{2}\right)^{1/2}\right\|_{L^{2}(\mathbb{R}^{3})}=\|f\|_{\dot{\mathcal{B}}^{s}_{2,2}(\mathbb{R}^{3})}.$

Now we state the first result concerning the dispersive estimate for the wave
equation (1.1).

######  Theorem 1.1 (Dispersive estimate).

Let $T$ be a constant in $(0,\pi)$, then, for any $t\in(0,T)$, there exists a
positive constant $C$ independent of $t$ such that, for all
$f\in\dot{\mathcal{B}}^{3/2}_{1,1}(\mathbb{R}^{3})$ and
$g\in\dot{\mathcal{B}}^{1}_{1,1}(\mathbb{R}^{3})$

$\begin{split}&\left\|\cos(t\sqrt{H_{a,1}})f\right\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim\frac{1}{\sin
t}\|f\|_{\dot{\mathcal{B}}^{3/2}_{1,1}(\mathbb{R}^{3})},\\\
&\left\|\frac{\sin(t\sqrt{H_{a,1}})}{\sqrt{H_{a,1}}}g\right\|_{L^{\infty}(\mathbb{R}^{3})}\lesssim\frac{1}{\sin
t}\|g\|_{\dot{\mathcal{B}}^{1}_{1,1}(\mathbb{R}^{3})}.\end{split}$ (1.10)

######  Remark 1.2.

Theorem 1.1 is an analog of the main result of [18], in which the dispersive
estimate (1.10) for the Aharonov-Bohm Hamiltonian (a Schrödinger operator with
a scaling-critical magnetic potential)

$\mathcal{L}_{\mathbf{A}}:=\left(i\nabla+\frac{\mathbf{A}(x/|x|)}{|x|}\right)^{2},\quad\mathbf{A}\in
W^{1,\infty}(\mathbb{S}^{1};\mathbb{R}^{2})$

was obtained by using a suitable representation of the fundamental solution of
the wave equation. Thanks to the appearance of the quadratic potential
$|x|^{2}$ in (1.3), we are allowed to obtain an explicit representation
formula for the solution of the Schrödinger and heat equations via the
spectral theorem since the spectral properties of the Hamiltonian $H_{a,1}$
are already well-understood (see [17, Proposition 3.2]). This is not
surprising because one can regard the operator (1.3) as the harmonic
oscillator perturbed by an inverse square potential. In light of the argument
of D’Ancona-Pierfelice [14], the key ingredient to prove the desired
dispersive estimate (1.10) is to obtain a Gaussian type upper bound for the
kernel of the heat semigroup $e^{-tH_{a,1}}$ in the sense of Simon [43].
Similar results based on such a strategy can be found in the previous works
[14, 16, 39].

As a consequence of Theorem 1.1, we can obtain the Strichartz estimates for
the propagator $e^{it\sqrt{H_{a,1}}}$.

######  Theorem 1.3 (Strichartz estimate).

Let $H_{a,1}$ be the Hamiltonian given by (1.3) with $b=1$ and
$a>-\frac{1}{4}$ and let $u$ be given by (1.4). If the pair $(q,r)$ satisfies
the condition (1.7) and $s$ given by (1.8) belongs to the interval $[0,3/2)$,
then there exists some constant $C=C_{a,q,r,s}>0$ independent of $f,g$ such
that

$\|u\|_{L_{t}^{q}\left((0,\pi);L_{x}^{r}(\mathbb{R}^{3})\right)}\leq
C\left(\|f\|_{\dot{\mathcal{H}}^{s}_{a,1}(\mathbb{R}^{3})}+\|g\|_{\dot{\mathcal{H}}^{s-1}_{a,1}(\mathbb{R}^{3})}\right).$
(1.11)

######  Remark 1.4.

Theorem 1.3 is new for the wave equation associated to the perturbed harmonic
oscillator, although the main result of [16] can standardly yield similar
Strichartz estimates for the wave equation associated to the standard harmonic
oscillator $-\Delta+|x|^{2}$. We point out that the Sobolev norm
$\dot{\mathcal{H}}^{s}_{a,1}$ at the right hand side of (1.11) _can_ be
replaced by the Sobolev norm $\dot{H}_{osc}^{s}$ associated with the harmonic
oscillator $-\Delta+|x|^{2}$. Indeed, for dimension $d\geq 3$, the Sobolev
space $\dot{H}^{s}_{a}$ related to the inverse square potential is equivalent
to the usual Sobolev space $\dot{H}^{s}$ associated to the free Laplacian
$-\Delta$ (see [31, Theorem 1.2]). However, the Sobolev norm
$\dot{\mathcal{H}}^{s}_{a,1}$ at the right hand side of (1.11) _cannot_ be
replaced by the classical Sobolev space $\dot{H}^{s}$ since
$\dot{H}_{osc}^{s}$ is _strictly_ contained in $\dot{H}^{s}$ (see [9, Theorem
3]).

##  2\. preliminaries

In this section, we discuss some analytical features of the operator
$H_{a,1}$, based on the spectral properties in [17, Proposition 3.2].
Throughout the paper, we will always use the notation $C$ to denote a
universal positive constant that may vary from line to line.

The operator $H_{a,1}$, if expressed in the polar coordinates $(r,\vartheta)$,
takes the form

$H_{a,1}=-\partial_{r}^{2}-\frac{2}{r}\partial_{r}+\frac{1}{4}r^{2}+\frac{a-\Delta_{\mathbb{S}^{2}}}{r^{2}}.$
(2.1)

In the spherical coordinates $(\theta,\phi)$, we can write
$\vartheta:=(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)$ for
$\vartheta\in\mathbb{S}^{2}$ with $(\theta,\phi)\in[0,2\pi]\times[0,\pi]$ and
then $\Delta_{\mathbb{S}^{2}}$, the Laplace-Beltrami operator on the unit
sphere $\mathbb{S}^{2}$, takes the form

$\Delta_{\mathbb{S}^{2}}=\csc^{2}\phi\partial_{\theta}^{2}+\cot\phi\partial_{\phi}+\partial_{\phi}^{2}.$
(2.2)

Let us consider the eigenvalue problem

$\csc^{2}\phi\partial_{\theta}^{2}Y+\cot\phi\partial_{\phi}Y+\partial_{\phi}^{2}Y=-\gamma
Y,\quad\lambda\geq 0$ (2.3)

and find solutions of the form $Y(\theta,\phi)=\Theta(\theta)\Phi(\phi)$. By
the method of separation of variables, (2.3) reduces to the following two ODEs

$\displaystyle\Theta^{\prime\prime}(\theta)+\beta\Theta(\theta)=0,$ (2.4)
$\displaystyle\Phi^{\prime\prime}(\phi)+\cot\phi\Phi^{\prime}(\phi)+(\mu-\beta\csc^{2}\phi)\Phi(\phi)=0.$
(2.5)

The first equation (2.4) is easy to handle, whose solution $\Theta\in
L^{2}([0,2\pi];\mathbb{C})$ is given by $\Theta(\theta)=e^{im\theta}$ with the
corresponding eigenvalue $\beta=m^{2},m\in\mathbb{Z}$. The second equation
(2.5) is thus reduced to

$\Phi^{\prime\prime}(\phi)+\cot\phi\Phi^{\prime}(\phi)+(\mu-m^{2}\csc^{2}\phi)\Phi(\phi)=0.$
(2.6)

If we make the change of variables

$\xi=\cos\phi,\quad\omega(\xi)=\Phi(\cos\phi),$

then (2.6) is transformed into

$(1-\xi^{2})\omega^{\prime\prime}(\xi)-2\xi\omega^{\prime}(\xi)+\left(\mu-\frac{m^{2}}{1-\xi^{2}}\right)\omega(\xi)=0$
(2.7)

or, equivalently, the self-adjoint form

$[(1-\xi^{2})\omega^{\prime}(\xi)]^{\prime}+\left(\mu-\frac{m^{2}}{1-\xi^{2}}\right)\omega(\xi)=0.$

Note that (2.7) is exactly the generalized Legendre equation, whose solution
is given by $\Phi(\phi)=P_{k}^{m}(\cos\phi)$, the associated Legendre
polynomial of degree $k$ and order $m$ with $|m|\leq k$ and the corresponding
eigenvalue is $\mu_{k}=k(k+1)$ with multiplicity $2k+1$. Therefore, we obtain
the desired ($L^{2}$-normalised) eigenfunction of (2.3)

$Y_{m}^{k}(\vartheta):=Y_{m}^{k}(\theta,\phi)=\left(\frac{2m+1}{4\pi}\frac{(k-m)!}{(k+m)!}\right)^{1/2}P_{k}^{m}(\cos\phi)e^{im\theta},\quad
k,m\in\mathbb{Z},k\geq 0,$ (2.8)

which is the usual spherical harmonics on $\mathbb{S}^{2}$. Associated the
spherical harmonics in (2.8), the zonal function
$Z_{\vartheta}^{(k)}(\vartheta^{\prime})$ is given by

$Z_{\vartheta}^{(k)}(\vartheta^{\prime}):=\sum_{m=-k}^{k}Y_{m}^{k}(\vartheta)\overline{Y_{m}^{k}(\vartheta^{\prime})}.$
(2.9)

It is well-known that the zonal function
$Z_{\vartheta}^{(k)}(\vartheta^{\prime})$ has a uniform upper bound (see e.g.
[32])

$|Z_{\vartheta}^{(k)}(\vartheta^{\prime})|\leq
Z_{\vartheta}^{(k)}(\vartheta)=\frac{2k+1}{4\pi},\quad\forall k\geq
0,\quad\vartheta,\vartheta^{\prime}\in\mathbb{S}^{2}.$ (2.10)

Note that the family of eigenfunctions
$Y_{m}^{k}(\vartheta),k\in\mathbb{N},m\in\mathbb{Z}$ given in (2.8) forms a
complete orthonormal basis for $L^{2}(\mathbb{S}^{2})$. For a function
$\varphi\in L^{2}(\mathbb{R}^{3})$, one can expand $\varphi$ into Fourier
series of the form

$\displaystyle\varphi(x):=\varphi(r,\vartheta)$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{m=-k}^{k}Y_{m}^{k}(\theta,\phi)\int_{0}^{2\pi}\int_{0}^{\pi}\varphi(r,\theta^{\prime},\phi^{\prime})\overline{Y_{m}^{k}(\theta^{\prime},\phi^{\prime})}\mathrm{d}\theta^{\prime}\sin\phi^{\prime}\mathrm{d}\phi^{\prime}$
$\displaystyle=\sum_{k=0}^{\infty}\int_{\mathbb{S}^{2}}\varphi(r,\vartheta^{\prime})\overline{Z^{(k)}_{\vartheta}(\vartheta^{\prime})}\mathrm{d}\sigma(\vartheta^{\prime}),$
(2.11)

where $Y_{m}^{k}(\theta,\phi)$ is given in (2.8) and
$Z^{(k)}_{\vartheta}(\vartheta^{\prime})$ is given in (2.9). One can always
expand an $L^{2}$ function into a Fourier series in terms of the spherical
harmonics (2.8) or the zonal function (2.9) as (2). In view of (2.1), the
action of the operator $H_{a,1}$ on each eigenspace
$\mathcal{H}^{k}:=\mathrm{span}\\{Y_{m}^{k}:m\in\mathbb{Z},|m|\leq k\\}$
(whose dimension is $\dim\mathcal{H}^{k}=2k+1$) can be given via

$H_{a,1}=-\partial_{r}^{2}-\frac{2}{r}\partial_{r}+\frac{1}{4}r^{2}+\frac{\mu_{k}+a}{r^{2}},\quad\mu_{k}=k(k+1).$
(2.12)

We are now in the right position to give the spectrum of $H_{a,1}$. Let us
consider the eigenvalue problem for the operator $H_{a,1}$

$-\Delta e(x)+\frac{a}{|x|^{2}}e(x)+\frac{|x|^{2}}{4}e(x)=\lambda e(x).$
(2.13)

In view of (2) and (2.12), one can obtain the eigenvalues and eigenfunctions
of the ODE (2.16) via a suitable transform (see the transform below (3.6) in
the proof of [17, Proposition 3.2]). Let

$\alpha_{k}=\frac{1}{2}-\sqrt{\left(\frac{1}{2}+k\right)^{2}+a},\quad\text{and}\quad\beta_{k}=\sqrt{\left(\frac{1}{2}+k\right)^{2}+a}$
(2.14)

so that $\alpha_{k}+\beta_{k}=\frac{1}{2}$ for $k\in\mathbb{N}$. Then the
eigenvalue of the operator $H_{a,1}$ in (2.13) can be given in terms of (2.14)
by

$\lambda_{m,k}=2m+\frac{3}{2}-\alpha_{k}=2m+1+\beta_{k},\quad
m,k\in\mathbb{N}:=\\{0,1,2,\cdots\\}$ (2.15)

with the corresponding ($L^{2}$-normalised) eigenfunction

$e_{m,k}(x)=\left(\frac{m!}{2^{\beta_{k}}\Gamma(m+1+\beta_{k})}\right)^{1/2}|x|^{-\alpha_{k}}e^{-\frac{|x|^{2}}{4}}L_{m}^{\beta_{k}}\left(\frac{|x|^{2}}{2}\right)\psi_{k}\left(\frac{x}{|x|}\right),$
(2.16)

where $\psi_{k}$ is given via

$\psi_{k}(\vartheta)\overline{\psi_{k}(\vartheta^{\prime})}=\sum_{n=-k}^{k}Y_{n}^{k}(\vartheta)\overline{Y_{n}^{k}(\vartheta^{\prime})}$
(2.17)

and $L_{m}^{\alpha}(t)$ stands for the generalized Laguerre polynomials

$L_{m}^{\alpha}(t)=\frac{(1+\alpha)_{m}}{m!}\sum_{k=0}^{m}\frac{(-m)_{k}}{(1+\alpha)_{k}}\frac{t^{k}}{k!}.$
(2.18)

Note that the zonal function $Z^{(k)}_{\vartheta}(\vartheta^{\prime})$ in
(2.9) can be expressed in terms of $\psi_{k}$ in (2.17) as

$Z^{(k)}_{\vartheta}(\vartheta^{\prime})=\psi_{k}(\vartheta)\overline{\psi_{k}(\vartheta^{\prime})}.$

In general, for a well-behaved function $F$, one can define the functional
$F(H_{a,1})$ via the spectral theorem

$\left(F(H_{a,1})\varphi\right)(x)=\int_{\mathbb{R}^{3}}\sum_{m,k=0}^{\infty}F(\lambda_{m,k})e_{m,k}(x)\overline{e_{m,k}(y)}\varphi(y)\mathrm{d}y,$

where $\lambda_{m,k}$ is given in (2.15) and $e_{m,k}$ is given in (2.16).
Before closing this section, we shall give a useful lemma concerning the
uniform boundedness of the sum of the product of the zonal function in (2.9)
and the modified Bessel functions of the first kind.

######  Lemma 2.1.

Let $a>-\frac{1}{4}$ and $\beta_{k}$ be defined in (2.14). Define a function
$K:\mathbb{R}_{+}\times\mathbb{S}^{2}\times\mathbb{S}^{2}\rightarrow\mathbb{C}$
by

$K(\rho,\vartheta,\vartheta^{\prime})=\rho^{-\frac{1}{2}}\sum_{k=0}^{\infty}i^{-\beta_{k}}J_{\beta_{k}}(\rho)Z^{(k)}_{\vartheta}(\vartheta^{\prime}),$
(2.19)

where $Z^{(k)}_{\vartheta}(\vartheta^{\prime})$ denotes the zonal function in
(2.9). Then we have

$\displaystyle\sup_{\rho\geq
0\atop\vartheta,\vartheta^{\prime}\in\mathbb{S}^{2}}|K(\rho,\vartheta,\vartheta^{\prime})|<+\infty$
$\displaystyle\quad\text{for}\quad a>0,$ $\displaystyle\sup_{\rho\geq
0\atop\vartheta,\vartheta^{\prime}\in\mathbb{S}^{2}}\frac{|K(\rho,\vartheta,\vartheta^{\prime})|}{1+\rho^{\beta_{0}-\frac{1}{2}}}<+\infty,$
$\displaystyle\quad\text{for}\quad-\frac{1}{4}<a<0.$

######  Remark 2.2.

We remark that Lemma 2.1 is summarized from [17, Sect. 6]. The series
expression in (2.19) should be understood locally, i.e. in the sense that
there exists some $k_{0}\geq 1$ such that the series

$\rho^{-\frac{1}{2}}\sum_{k=k_{0}+1}^{\infty}i^{-\beta_{k}}J_{\beta_{k}}(\rho)Z^{(k)}_{\vartheta}(\vartheta),\quad\forall\rho<R$

is uniformly convergent for any fixed $R$ and

$K(\rho,\vartheta,\vartheta^{\prime})-\rho^{-\frac{1}{2}}\sum_{k=0}^{k_{0}}i^{-\beta_{k}}J_{\beta_{k}}(\rho)Z^{(k)}_{\vartheta}(\vartheta)\in
L_{loc}^{\infty}(\mathbb{R}_{+}\times\mathbb{S}^{2}\times\mathbb{S}^{2},\mathbb{C}).$

###### Proof of Lemma 2.1.

We give a proof for the benefits of the reader. We only present the proof for
the case $a\geq 0$ since this is enough for the purpose of the present paper.
The proof of the other case $-\frac{1}{4}<a<0$ is similar but a little more
complicated, and we may refer the interested readers to [17, Sect. 6].

Our goal is to show that the function $K$ defined by (2.19) is uniformly
bounded, i.e.

$\left|\rho^{-\frac{1}{2}}\sum_{k=0}^{\infty}i^{-\beta_{k}}J_{\beta_{k}}(\rho)Z^{(k)}_{\vartheta}(\vartheta^{\prime})\right|<+\infty,\quad\forall(\rho,\vartheta,\vartheta^{\prime})\in\mathbb{R}_{+}\times\mathbb{S}^{2}\times\mathbb{S}^{2}.$

To this end, we split $K$ as

$\displaystyle K$
$\displaystyle=\rho^{-\frac{1}{2}}\sum_{k=0}^{\infty}i^{-\beta_{k}}J_{\beta_{k}}(\rho)Z^{(k)}_{\vartheta}(\vartheta^{\prime})$
$\displaystyle=\rho^{-\frac{1}{2}}\sum_{k=0}^{\infty}i^{k+\frac{1}{2}}J_{k+\frac{1}{2}}(\rho)Z^{(k)}_{\vartheta}(\vartheta^{\prime})$
$\displaystyle\qquad+\rho^{-\frac{1}{2}}\sum_{k=0}^{\infty}\left(i^{-\beta_{k}}J_{\beta_{k}}(\rho)-i^{k+\frac{1}{2}}J_{k+\frac{1}{2}}(\rho)\right)Z^{(k)}_{\vartheta}(\vartheta^{\prime})$
$\displaystyle:$ $\displaystyle=\Pi_{1}+\Pi_{2}.$ (2.20)

In view of the Jacobi-Anger expansion for the plane waves (see e.g. [32, 46])

$e^{-ix\cdot
y}=(2\pi)^{3/2}(|x||y|)^{-1/2}\sum_{k=0}^{\infty}i^{k}J_{k+\frac{1}{2}}(|x||y|)Z^{(k)}_{x/|x|}(y/|y|),\quad
x,y\in\mathbb{R}^{3},$

we get

$(2\pi)^{-3/2}e^{-i\rho\vartheta\cdot\vartheta^{\prime}}=\rho^{-1/2}\sum_{k=0}^{\infty}i^{k}J_{k+\frac{1}{2}}(\rho)Z^{(k)}_{\vartheta}(\vartheta^{\prime}).$
(2.21)

Hence, the first term $\Pi_{1}$ in (2) is obviously bounded, in view of
(2.21); actually, one has $\Pi_{1}=(2\pi
i)^{-3/2}e^{-i\rho\vartheta\cdot\vartheta^{\prime}}i$.

The second term $\Pi_{2}$ in (2) is bounded for $\rho\leq\epsilon$ with
$\epsilon$ small enough. Indeed, in view of the bound for the zonal function
(2.10) and the bound for the Bessel function (see e.g. [46])

$|J_{\nu}(r)|\leq\frac{(r/2)^{\nu}}{\Gamma(1+\nu)}e^{r^{2}/4}\quad\forall\nu>0,r\geq
0,$

we have, for $\rho\leq\epsilon$ with $\epsilon$ small enough,

$\displaystyle|\Pi_{2}|$
$\displaystyle=\left|\rho^{-\frac{1}{2}}\sum_{k=0}^{\infty}i^{-\beta_{k}}J_{\beta_{k}}(\rho)Z^{(k)}_{\vartheta}(\vartheta^{\prime})\right|$
$\displaystyle\leq\rho^{-\frac{1}{2}}\sum_{k=0}^{\infty}\frac{2k+1}{4\pi\Gamma(1+\beta_{k})}(\rho/2)^{\beta_{k}}e^{\rho^{2}/4}$
$\displaystyle\leq\frac{e^{\epsilon^{2}/4}}{4\sqrt{2}\pi}\sum_{k=0}^{\infty}\frac{2k+1}{\Gamma(1+\beta_{k})}(\rho/2)^{\beta_{k}-\frac{1}{2}}$
$\displaystyle\leq\frac{(\rho/2)^{\beta_{0}-\frac{1}{2}}e^{\epsilon^{2}/4}}{4\sqrt{2}\pi}\sum_{k=0}^{\infty}\frac{2k+1}{\Gamma(1+\beta_{k})}$
$\displaystyle\leq C\rho^{\beta_{0}-\frac{1}{2}},$

where $C=C(\epsilon)>0$ is a constant independent of
$\rho,\vartheta,\vartheta^{\prime}$.

For $\rho>\epsilon$, we use the contour integral representation for the Bessel
function

$J_{\nu}(r)=\frac{1}{2\pi
i}\int_{\Gamma}e^{\frac{r}{2}(z-\frac{1}{z})}\frac{\mathrm{d}z}{z^{1+\nu}}$

to express the second term $\Pi_{2}$ in (2)

$\displaystyle\Pi_{2}$
$\displaystyle=\rho^{-\frac{1}{2}}\sum_{k=0}^{\infty}\left(i^{-\beta_{k}}J_{\beta_{k}}(\rho)-i^{k+\frac{1}{2}}J_{k+\frac{1}{2}}(\rho)\right)Z^{(k)}_{\vartheta}(\vartheta^{\prime})$
(2.22) $\displaystyle=$ $\displaystyle\frac{\rho^{-1/2}}{2\pi
i}\int_{\Gamma}e^{\frac{\rho}{2}(z-\frac{1}{z})}\left(\sum_{k=0}^{\infty}\left((iz)^{k+\frac{1}{2}-\beta_{k}}-1\right)\frac{Z^{(k)}_{\vartheta}(\vartheta^{\prime})}{(iz)^{k+\frac{1}{2}}}\right)\frac{\mathrm{d}z}{z},$

where $\Gamma\subset\mathbb{C}$ denotes the positive oriented contour running
along the straight line from negative infinity to $z=-1$ below the negative
real axis, the counterclockwise oriented unit circle from $z=-1$ to itself and
the straight line from $z=-1$ to negative infinity above the negative real
axis. For convenience of statements, we denote by $\Gamma_{1}$ the
counterclockwise oriented circumference of radius 1 and by $\Gamma_{2}$ the
two straight lines running between $z=-\infty$ and $z=-1$, and we split the
integral in (2.22) accordingly, i.e.

$\int_{\Gamma}=\int_{\Gamma_{1}}+\int_{\Gamma_{2}}:=I_{1}+I_{2}.$

In view of the analyticity of the integrand of (2.22) outside
$z=\mathbb{R}_{-}+i0\pm$, we can write the integral in (2.22) more rigorously

$\int_{\Gamma}=\int_{\Gamma_{1}}+\int_{\Gamma_{2}}=\lim_{\varepsilon\rightarrow
0+}\left(\int_{\Gamma_{1}^{\varepsilon}}+\int_{\Gamma_{2}^{\varepsilon}}\right),$

where $\Gamma_{1}^{\varepsilon}$ denotes the circumference of radius
$1+\varepsilon$ centered at the origin and $\Gamma_{2}^{\varepsilon}$ denotes
the lines running along $(-\infty,-1-\varepsilon)+i0\pm$. Note that we have
exchanged the order of sum and integral in (2.22), which is permitted for any
$\rho,\vartheta,\vartheta^{\prime}$. Indeed, since $|z|>1$ for all
$z\in\Gamma_{1}^{\varepsilon}\cup\Gamma_{2}^{\varepsilon}$, one has the
absolute convergence of the contour integral along
$\Gamma_{1}^{\varepsilon}\cup\Gamma_{2}^{\varepsilon}$ for any
$\rho,\vartheta,\vartheta^{\prime}$ and hence the exchange of the order of sum
and integral in (2.22) is permitted by Fubini’s theorem.

Let us consider $I_{1}$ (i.e. the integral along $\Gamma_{1}$) firstly. In
view of

$\beta_{k}-(k+\frac{1}{2})=\sqrt{\left(k+\frac{1}{2}\right)^{2}+a}-(k+\frac{1}{2})=\frac{a}{2k+1}+O(k^{-3}),$

we have

$\displaystyle(iz)^{k+\frac{1}{2}-\beta_{k}}-1$
$\displaystyle=-\frac{a}{2k+1}\log(iz)+\frac{a^{2}}{2}\frac{(\log(iz))^{2}}{(2k+1)^{2}}+\frac{O(1)}{k^{3}}$
$\displaystyle:=I_{11}(z,k)+I_{12}(z,k)+I_{13}(z,k)\quad\text{as}\quad
k\rightarrow+\infty$ (2.23)

uniformly in $z\in\Gamma_{1}$. Note that $z^{-\beta_{k}}$ and
$z^{k+\frac{1}{2}}$ admit a branch-cut at the negative real axis of the
complex plane (i.e. at $z\in\mathbb{R}_{-}$), then the function $\log(iz)$
(and $\sqrt{iz}$, etc.) will admit a branch-cut at $z\in\mathbb{R}_{-}$ as
well. Hence, we can, taking (2) into consideration, write $I_{1}$ (i.e. the
integral over $\Gamma_{1}$ in (2.22)) as

$\displaystyle I_{1}$ $\displaystyle=\frac{\rho^{-1/2}}{2\pi
i}\int_{\Gamma_{1}}e^{\frac{\rho}{2}(z-\frac{1}{z})}\left(\sum_{k=0}^{\infty}\left(I_{11}(z,k)+I_{12}(z,k)+I_{13}(z,k)\right)\frac{Z^{(k)}_{\vartheta}(\vartheta^{\prime})}{(iz)^{k+\frac{1}{2}}}\right)\frac{\mathrm{d}z}{z}$
$\displaystyle:=\mathcal{I}_{11}+\mathcal{I}_{12}+\mathcal{I}_{13},$ (2.24)

where every summand $\mathcal{I}_{1j},j=1,2,3$ corresponds to the integrand
with the corresponding $I_{1j},j=1,2,3$. Since $|z|\equiv 1$ for all
$z\in\Gamma_{1}$, we have $|e^{\frac{\rho}{2}(z-\frac{1}{z})}|=1$ and thus, in
view of (2.10) again, we obtain

$|\mathcal{I}_{13}|\lesssim\rho^{-1/2}\sum_{k=0}^{\infty}\frac{2k+1}{k^{3}}\lesssim\epsilon^{-1/2},\quad\text{for}\quad\rho>\epsilon,$

which indicates that $|\mathcal{I}_{13}|$ is bounded uniformly in
$\rho,\vartheta,\vartheta^{\prime}$. From the definitions of the spherical
harmonics (2.8) and the zonal functions (2.9), one can recover the well-known
identity (see [32])

$Z^{(k)}_{\vartheta}(\vartheta^{\prime})=\frac{2k+1}{4\pi}P_{k}(\vartheta\cdot\vartheta^{\prime})$
(2.25)

where $P_{k}$ is the Legendre polynomial of degree $k$ (a solution of the
Legendre equation (2.7)). In view of the identity (2.25) and the generating
function formula for the Legendre polynomials (see e.g. [1, (22.9.12)])

$\sum_{k=0}^{\infty}P_{k}(s)r^{k}=(1-2rs+r^{2})^{-1/2},\quad|r|<1,$ (2.26)

we have, for $\mathcal{I}_{11}$,

$\displaystyle-a\log(iz)\sum_{k=0}^{\infty}\frac{Z^{(k)}_{\vartheta}(\vartheta^{\prime})}{2k+1}(iz)^{-(k+\frac{1}{2})}$
$\displaystyle=-\frac{a\log(iz)}{4\pi}\sum_{k=0}^{\infty}P_{k}(\vartheta\cdot\vartheta^{\prime})(iz)^{-(k+\frac{1}{2})}$
$\displaystyle=-\frac{a(iz)^{-1/2}\log(iz)}{4\pi\sqrt{1+2iz^{-1}\vartheta\cdot\vartheta^{\prime}-z^{-2}}}$
$\displaystyle=-\frac{a(-iz)^{1/2}\log(iz)}{4\pi\sqrt{z^{2}+2iz(\vartheta\cdot\vartheta^{\prime})-1}},\quad|z|>1$
(2.27)

and hence

$\mathcal{I}_{11}=-\frac{a\rho^{-1/2}}{8\pi^{2}i}\lim_{\varepsilon\rightarrow
0+}\int_{\Gamma_{1}^{\varepsilon}}e^{\frac{\rho}{2}(z-\frac{1}{z})}\frac{(-iz)^{1/2}\log(iz)}{\sqrt{z^{2}+2iz(\vartheta\cdot\vartheta^{\prime})-1}}\frac{\mathrm{d}z}{z}.$
(2.28)

Note that $|e^{\frac{\rho}{2}(z-\frac{1}{z})}|\equiv 1$ for all $\rho>0$ and
$z\in\Gamma_{1}$. If $\vartheta\cdot\vartheta^{\prime}$ stays far away from
$\pm 1$, then $1-(\vartheta\cdot\vartheta^{\prime})^{2}>\epsilon$ and the term
$\mathcal{I}_{11}$ in (2.28) is uniformly bounded with respect to
$\varepsilon\rightarrow 0+,\rho>\epsilon$, due to the integrability of the two
singularities of square root type of the integrand at
$z_{\pm}=-i(\vartheta\cdot\vartheta^{\prime})\pm\sqrt{1-(\vartheta\cdot\vartheta^{\prime})^{2}}$.
If $\vartheta\cdot\vartheta^{\prime}=\pm 1$, one sees that these two
singularities at $z_{\pm}$ collapse into a stronger singularity at $z=\pm i$.
Let us assume $\vartheta\cdot\vartheta^{\prime}=-1$ (the other case
$\vartheta\cdot\vartheta^{\prime}=1$ can be similarly treated) for
concreteness, then we obtain

$\displaystyle\lim_{\varepsilon\rightarrow 0+}$
$\displaystyle\int_{\Gamma_{1}^{\varepsilon}}e^{\frac{\rho}{2}(z-\frac{1}{z})}\frac{(-iz)^{1/2}\log(iz)}{z-i}\frac{\mathrm{d}z}{z}$
(2.29)
$\displaystyle=i\pi^{2}e^{i\rho}+\mathrm{P.V.}\int_{\Gamma_{1}}e^{\frac{\rho}{2}(z-\frac{1}{z})}\frac{(-iz)^{1/2}\log(iz)}{z-i}\frac{\mathrm{d}z}{z},$
(2.30)

which is simply the Plemelj-Sokhotskyi formula (see e.g. [3]) for the limit of
Cauchy integrals when tending to a singular point. The second term at the
right hand side of (2.29) is a singular integral of the function
$e^{\frac{\rho}{2}(z-\frac{1}{z})}(iz)^{-1/2}\log(iz)$, which is
differentiable with respect to $z=e^{i\theta}$ for $\theta$ varying in some
neighborhood of $\frac{\pi}{2}$ (remember that the discontinuity of the
argument of $z$ is along the negative real axis of the complex plane). Due to
the boundedness of the principal value of a Cauchy integral of a
differentiable function (see [3]), we conclude the boundedness of the term
$\mathcal{I}_{11}$ for all $\rho>\epsilon$. We stress that it is still
possible for the principal value integral to be divergent as
$\rho\rightarrow\infty$ despite its boundedness for any $\rho>\epsilon$. To
exclude this undesired possibility, let us consider the neighborhood of $z=i$
in $\Gamma_{1}$

$\Gamma_{1}^{\eta}:=\\{z=ie^{i\theta}:|\theta|<\eta\ll 1\\}.$

In view of the fact

$\frac{(\pi+\theta)e^{i\theta/2}}{i(e^{i\theta}-1)}=-\frac{\pi}{\theta}+O(1)\quad\text{as}\quad\theta\rightarrow
0,$

we compute the integral over $\Gamma_{1}^{\eta}$ for any $\rho\gg 1$

$\displaystyle\mathrm{P.V.}\int_{-\eta}^{\eta}e^{i\rho\cos\theta}\frac{(\pi+\theta)e^{i\theta/2}}{i(e^{i\theta}-1)}\mathrm{d}\theta$
$\displaystyle=\mathrm{P.V.}\int_{-\eta}^{\eta}e^{i\rho\cos\theta}\left(-\frac{\pi}{\theta}+O(1)\right)\mathrm{d}\theta$
$\displaystyle=O(1)\mathrm{P.V.}\int_{-\eta}^{\eta}e^{i\rho\cos\theta}\mathrm{d}\theta-\pi\mathrm{P.V.}\int_{-\eta}^{\eta}\frac{e^{i\rho\cos\theta}}{\theta}\mathrm{d}\theta.$

The second principal value integral obviously vanishes (due to the parity of
the integrand), i.e.

$\mathrm{P.V.}\int_{-\eta}^{\eta}\frac{e^{i\rho\cos\theta}}{\theta}\mathrm{d}\theta\equiv
0,$

it thus follows (note that the first principal value integral is finite) that

$\mathrm{P.V.}\int_{-\eta}^{\eta}e^{i\rho\cos\theta}\frac{(\pi+\theta)e^{i\theta/2}}{i(e^{i\theta}-1)}\mathrm{d}\theta=O(1).$

Therefore, the integral along $\Gamma_{1}^{\eta}$ is uniformly bounded for all
$\rho$. The integral over $\Gamma_{1}\setminus\Gamma_{1}^{\eta}$ is uniformly
bounded as well since $|e^{\frac{\rho}{2}(z-\frac{1}{z})}|\equiv 1$ for all
$z\in\Gamma_{1}$. Hence, the principal value integral over $\Gamma_{1}$ is
bounded uniformly in $\rho,\vartheta,\vartheta^{\prime}$. If the singularities
$z_{\pm}=-i(\vartheta\cdot\vartheta^{\prime})\pm\sqrt{1-(\vartheta\cdot\vartheta^{\prime})^{2}}$
are very close to each other (i.e.
$|\vartheta\cdot\vartheta^{\prime}|>1-\epsilon$), then, similarly as (2.29),
one can split the integral over $\Gamma_{1}$ into two parts

$\int_{\Gamma_{1}}=\int_{\overbrace{z_{-},z_{+}}}+\int_{\Gamma_{1}\setminus\overbrace{z_{-},z_{+}}},$
(2.31)

where $\overbrace{z_{-},z_{+}}$ denotes the small arc of $\Gamma_{1}$ between
$z_{-}$ and $z_{+}$. The second integral at the right hand side of (2.31) can
be easily bounded just like the principal value integral above and yields the
same uniform boundedness with respect to $\vartheta,\vartheta^{\prime}$. It is
easy to see that there exists some $\hat{\theta}\in\mathbb{R}$ such that

$z_{\pm}=-i(\vartheta\cdot\vartheta^{\prime})\pm\sqrt{1-(\vartheta\cdot\vartheta^{\prime})^{2}}=ie^{\pm
i\hat{\theta}},$

then the first term at the right hand side of (2.31) becomes the integral of

$\displaystyle\frac{(\pi+\theta)e^{i(\rho\cos\theta+\frac{\theta}{2})}}{\sqrt{(ie^{i\theta}-ie^{-i\hat{\theta}})(ie^{i\theta}-ie^{i\hat{\theta}})}}$
$\displaystyle=\frac{\pi
e^{i\rho\cos\theta}}{\sqrt{|\theta+\hat{\theta}||\theta-\hat{\theta}|}}\left(1+O(|\theta-\hat{\theta}|)+O(|\theta+\hat{\theta}|)\right)\times\begin{cases}-1&\quad\text{if}\quad\theta<-\hat{\theta}\\\
1&\quad\text{if}\quad\theta>\hat{\theta}\\\
-i&\quad\text{if}\quad-\hat{\theta}<\theta<\hat{\theta}\end{cases}$
$\displaystyle=\frac{\pi
e^{i\rho\cos\theta}\varphi(\theta)}{\sqrt{|\theta+\hat{\theta}||\theta-\hat{\theta}|}}+O(|\theta-\hat{\theta}|^{-1/2})+O(|\theta+\hat{\theta}|^{-1/2}),$

where $\varphi$ is a suitable function satisfying $|\varphi(\theta)|\equiv 1$.
Hence, we have

$\displaystyle\left|\int_{\overbrace{z_{-},z_{+}}}\right|$
$\displaystyle=\left|\pi\int_{-\hat{\theta}}^{\hat{\theta}}\frac{e^{i\rho\cos\theta}}{\sqrt{\hat{\theta}^{2}-\theta^{2}}}\mathrm{d}\theta+O(1)\right|$
$\displaystyle=\left|\pi\int_{-1}^{1}\frac{e^{i\rho\cos(\hat{\theta}s)}}{\sqrt{1-s^{2}}}\mathrm{d}s+O(1)\right|$
$\displaystyle\lesssim 1\quad\text{uniformly with respect
to}\quad\rho,\hat{\theta}.$

Therefore, the integral over $\Gamma_{1}$ is bounded uniformly in $\rho$ and
$\epsilon$.

Finally, it remains to treat the term $\mathcal{I}_{12}$ in (2)

$\frac{a^{2}\rho^{-1/2}}{4\pi
i}\int_{\Gamma_{1}}e^{\frac{\rho}{2}(z-\frac{1}{z})}\left(\sum_{k=0}^{\infty}\frac{Z^{(k)}_{\vartheta}(\vartheta^{\prime})}{(2k+1)^{2}}(iz)^{-(k+\frac{1}{2})}\right)(\log(iz))^{2}\frac{\mathrm{d}z}{z}.$

Let

$\displaystyle G(z,\vartheta,\vartheta^{\prime}):$
$\displaystyle=\sum_{k=0}^{\infty}\frac{Z^{(k)}_{\vartheta}(\vartheta^{\prime})}{(2k+1)^{2}}(iz)^{-(k+\frac{1}{2})},$
$\displaystyle g(z,\vartheta,\vartheta^{\prime}):$
$\displaystyle=\frac{1}{2i}\sum_{k=0}^{\infty}\frac{Z^{(k)}_{\vartheta}(\vartheta^{\prime})}{2k+1}(iz)^{-(k+\frac{3}{2})},$

then one easily verifies

$G^{\prime}_{z}(z,\vartheta,\vartheta^{\prime})=g(z,\vartheta,\vartheta^{\prime}).$

In view of the formulas (2.25) and (2.26), we have

$g(z,\vartheta,\vartheta^{\prime})=\frac{(iz)^{-3/2}}{8\pi
i\sqrt{1+2iz^{-1}(\vartheta\cdot\vartheta^{\prime})-z^{-2}}}.$

Since $g(z,\vartheta,\vartheta^{\prime})$ has a square root type singularity
when $\vartheta\cdot\vartheta^{\prime}\neq-1$ or has a $(z-i)^{-1}$ type
singularity at $z=i$ when $\vartheta\cdot\vartheta^{\prime}=-1$ (we only
consider this case here, the other case $\vartheta\cdot\vartheta^{\prime}=1$
is similar as before), we conclude that $G(z,\vartheta,\vartheta^{\prime})$
has at most a log-type singularity, which is integrable, and hence the
corresponding integral is uniformly bounded. Therefore, the term $I_{1}$ (i.e.
the integral (2.22) over $\Gamma_{1}$) is uniformly bounded.

We proceed to consider the term $I_{2}$

$I_{2}=\frac{\rho^{-1/2}}{2\pi
i}\int_{\Gamma_{2}}e^{\frac{\rho}{2}(z-\frac{1}{z})}\left(\sum_{k=0}^{\infty}\left((iz)^{k+\frac{1}{2}-\beta_{k}}-1\right)\frac{Z^{(k)}_{\vartheta}(\vartheta^{\prime})}{(iz)^{k+\frac{1}{2}}}\right)\frac{\mathrm{d}z}{z}.$
(2.32)

After changing variables $z=-e^{s}$, exchanging the order of sum and integral
and rearranging terms subsequently, we can rewrite $I_{2}$ of (2.32) (similar
to $I_{1}$) as

$\begin{split}I_{2}&=\frac{\rho^{-1/2}}{2\pi
i}\sum_{k=0}^{\infty}Z^{(k)}_{\vartheta}(\vartheta^{\prime})\left(\Phi_{k}(\rho)+\Psi_{k}(\rho)\right)\\\
&:=\mathcal{I}_{21}+\mathcal{I}_{22},\end{split}$ (2.33)

where

$\displaystyle\Phi_{k}(\rho)$
$\displaystyle=2i^{-(1+\beta_{k})}\sin(\pi\beta_{k})\int_{0}^{\infty}e^{-\rho\sinh
s}(e^{-s\beta_{k}}-e^{-(k+\frac{1}{2})s})\mathrm{d}s,$
$\displaystyle\Psi_{k}(\rho)$ $\displaystyle=-2i\int_{0}^{\infty}e^{-\rho\sinh
s-(k+\frac{1}{2})s}(i^{-\beta_{k}}\sin(\pi\beta_{k})-i^{-(k+\frac{1}{2})}\sin\pi(k+\frac{1}{2}))\mathrm{d}s.$

In view of (2.10) again, we obtain

$\displaystyle|\mathcal{I}_{21}|$ $\displaystyle=\left|\frac{\rho^{-1/2}}{2\pi
i}\sum_{k=0}^{\infty}Z^{(k)}_{\vartheta}(\vartheta^{\prime})\Phi_{k}(\rho)\right|$
$\displaystyle\lesssim\rho^{-1/2}\sum_{k=0}^{\infty}(2k+1)\int_{0}^{\infty}e^{-\rho\sinh
s}\left|e^{-s\beta_{k}}-e^{-(k+\frac{1}{2})s}\right|\mathrm{d}s$
$\displaystyle\lesssim\rho^{-1/2}\sum_{k=0}^{\infty}(2k+1)\left|\frac{1}{\beta_{k}}-\frac{1}{k+\frac{1}{2}}\right|\lesssim\rho^{-1/2}.$

To address $\mathcal{I}_{22}$ in (2.33), we use the fact

$\displaystyle
i^{-\beta_{k}}\sin(\pi\beta_{k})-i^{-(k+\frac{1}{2})}\sin\pi(k+\frac{1}{2})$
$\displaystyle=(i^{-\beta_{k}}-i^{-(k+\frac{1}{2})})\sin(\pi\beta_{k})+i^{-(k+\frac{1}{2})}\left(\sin(\pi\beta_{k})-\sin\pi(k+\frac{1}{2})\right)$
$\displaystyle=i^{-(k+\frac{1}{2})}(-1)^{k+1}\left(\frac{a\pi
i}{2(2k+1)}+\frac{(a\pi)^{2}}{8(2k+1)^{2}}\right)+i^{-(k+\frac{1}{2})}(-1)^{k+1}\frac{(a\pi)^{2}}{2(2k+1)^{2}}+O(k^{-3})$
$\displaystyle=i^{k+\frac{5}{2}}\frac{a\pi}{2(2k+1)}+i^{k+\frac{3}{2}}\frac{5\pi^{2}a^{2}}{8(2k+1)^{2}}+O(k^{-3})\quad\text{as}\quad
k\rightarrow+\infty$

and the formula (2.26) to obtain

$\displaystyle|\mathcal{I}_{22}|$ $\displaystyle=\left|\frac{\rho^{-1/2}}{2\pi
i}\sum_{k=0}^{\infty}Z^{(k)}_{\vartheta}(\vartheta^{\prime})\Psi_{k}(\rho)\right|\leq\Bigg{|}\rho^{-1/2}\int_{0}^{\infty}e^{-\rho\sinh
s-\frac{s}{2}}$
$\displaystyle\times\Bigg{[}\frac{a\pi}{4i^{3/2}\sqrt{1-ie^{-s}(\vartheta\cdot\vartheta^{\prime})-e^{-2s}}}$
$\displaystyle+\frac{5\pi^{2}a^{2}Z^{(k)}_{\vartheta}(\vartheta^{\prime})}{16i^{-(k+\frac{3}{2})}e^{ks}}\left((2k+1)^{-2}+O(k^{-3})\right)\Bigg{]}\Bigg{|}.$

By (2.10) and

$\frac{1}{\sqrt{1-ie^{-s}(\vartheta\cdot\vartheta^{\prime})-e^{-2s}}}\lesssim
1+s^{-1/2},$

we see

$\displaystyle\Bigg{|}\sum_{k=0}^{\infty}$ $\displaystyle
i^{k}e^{-ks}Z^{(k)}_{\vartheta}(\vartheta^{\prime})\left((2k+1)^{-2}+O(k^{-3})\right)\Bigg{|}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}(ke^{ks})^{-1}\lesssim\log|s|\quad\text{uniformly
in}\quad\vartheta,\vartheta^{\prime}.$

Hence, we can bound $\mathcal{I}_{22}$ in (2.33) by

$|\mathcal{I}_{22}|\lesssim\rho^{-1/2}\int_{0}^{\infty}e^{-\rho\sinh
s-\frac{s}{2}}(1+s^{-1/2}+\log|s|)\mathrm{d}s\lesssim\rho^{-1/2}.$

Remember that we have $\rho>\epsilon$ for some $\epsilon>0$ in this case, the
uniform boundedness of $I_{2}$ follows.

Therefore, we obtain the desired bound

$\sup_{\rho\geq
0\atop\vartheta,\vartheta^{\prime}\in\mathbb{S}^{2}}|K(\rho,\vartheta,\vartheta^{\prime})|<+\infty$

and the proof of Lemma 2.1 is finished. ∎

##  3\. dispersive estimate for Schrödinger flow

In this section, we will construct a representation formula for the kernel of
the Schrödinger propagator $e^{-itH_{a,1}}$ and then prove a dispersive
inequality for this kernel. Such a dispersive inequality will be used to prove
the main theorem (Theorem 1.1) of this paper, as mentioned in the
introduction.

######  Proposition 3.1 (Schrödinger kernel).

Let $H_{a,1}$ be the operator given by (1.3) and let $u$ be the unique
solution to the Schrödinger equation

$\begin{cases}i\partial_{t}u(t,x)=H_{a,1}u(t,x),\\\ u(0,x)=f(x)\in
L^{2}(\mathbb{R}^{3}),\end{cases}$

then the solution $u$ can be represented as

$u(t,x)=\left(e^{-itH_{a,1}}f\right)(x):=\int_{\mathbb{R}^{3}}K^{S}(x,y)f(y)\mathrm{d}y,$
(3.1)

where $K^{S}(x,y)$ corresponds to the kernel of the Schrödinger propagator
$e^{-itH_{a,1}}$.

Moreover, if we write $K_{t}^{S}(r_{1},\vartheta_{1};r_{2},\vartheta_{2})$ for
the kernel $K^{S}(x,y)$ in polar coordinates $(r,\vartheta)$, then we have

$K_{t}^{S}(r_{1},\vartheta_{1};r_{2},\vartheta_{2})=\frac{e^{-\frac{r_{1}^{2}+r_{2}^{2}}{4i\tan
t}}}{2i\sqrt{r_{1}r_{2}}\sin
t}\sum_{k=0}^{\infty}I_{\beta_{k}}\Bigg{(}\frac{r_{1}r_{2}}{2i\sin
t}\Bigg{)}Z_{\vartheta_{1}}^{(k)}(\vartheta_{2}),$ (3.2)

where $Z_{\vartheta}^{(k)}(\vartheta^{\prime})$ is given in (2.9) and
$\beta_{k}$ in (2.14).

######  Remark 3.2.

The integral at the right hand side of (3.1) should be understood in the sense
of improper integrals, i.e.

$u(t,x)=\lim_{R\rightarrow+\infty}\int_{B_{R}(0)}K^{S}(x,y)f(y)\mathrm{d}y,\quad\text{where}\quad
B_{R}(0):=\\{y\in\mathbb{R}^{3}:|y|<R\\},$

which is consistent with the statement of Remark 2.2.

###### Proof of Proposition 3.1.

Since the spectrum of the operator $H_{a,1}$ consists of pure points, we have,
by the spectral theorem,

$e^{-itH_{a,1}}(x,y)=\sum_{m,k\in\mathbb{N}}e^{-it\lambda_{m,k}}e_{m,k}(x)\overline{e_{m,k}(y)}.$
(3.3)

Substituting the eigenvalues (2.15) and eigenfunctions (2.16) into (3.3), one
gets

$\begin{split}K_{t}^{S}(r_{1},\vartheta_{1};r_{2},\vartheta_{2})&=(r_{1}r_{2})^{-1/2}e^{-\frac{r_{1}^{2}+r_{2}^{2}}{4}}\sum_{k=0}^{\infty}\frac{(r_{1}r_{2})^{\beta_{k}}e^{-it(1+\beta_{k})}}{2^{\beta_{k}}}\psi_{k}(\vartheta_{1})\overline{\psi_{k}(\vartheta_{2})}\\\
&\qquad\qquad\times\sum_{m=0}^{\infty}e^{-2imt}\frac{m!L_{m}^{\beta_{k}}\left(\frac{r_{1}^{2}}{2}\right)L_{m}^{\beta_{k}}\left(\frac{r_{2}^{2}}{2}\right)}{\Gamma(m+1+\beta_{k})},\end{split}$
(3.4)

where $\psi_{k}$ is defined in (2.17). From the Poisson kernel formula for the
generalized Laguerre polynomials (see e.g. [2, (6.2.25)]): for
$|r|<1,\alpha>-1$

$\sum_{m=0}^{\infty}\frac{m!L_{m}^{\alpha}(x)L_{m}^{\alpha}(y)r^{m}}{\Gamma(1+\alpha+m)}=\frac{e^{-\frac{r}{1-r}(x+y)}}{(1-r)(xyr)^{\alpha/2}}I_{\alpha}\left(\frac{2\sqrt{xyr}}{1-r}\right),$
(3.5)

we obtain

$\begin{split}K_{t}^{S}(r_{1},\vartheta_{1};r_{2},\vartheta_{2})=\frac{(r_{1}r_{2})^{-1/2}e^{-\frac{r_{1}^{2}+r_{2}^{2}}{4i\tan
t}}}{2i\sin
t}\sum_{k=0}^{\infty}\psi_{k}(\vartheta_{1})\overline{\psi_{k}(\vartheta_{2})}I_{\beta_{k}}\Bigg{(}\frac{r_{1}r_{2}}{2i\sin
t}\Bigg{)}.\end{split}$ (3.6)

Finally, in view of (2.17) and (2.9), the desired formula (3.2) follows. ∎

######  Remark 3.3.

Note that in the representation formula of the Schrödinger kernel (3.2) in
Proposition 3.1, there appears a sine factor $\sin t$, which implies that the
time points $t=k\pi$ with $k\in\mathbb{Z}$ have to be excluded. This
phenomenon is natural due to the introduction of the Hermite potential
$|x|^{2}$, which implies that the effect of the Hermite potential on the
Laplacian is stronger than that of the inverse square potential $|x|^{-2}$, or
in other words, the effect of the inverse square potential $|x|^{-2}$ can be
viewed as a ”small” perturbation of the harmonic oscillator (in the sense that
the discreteness of the spectrum is preserved). Hence, the dispersive bound
for the kernel (3.2) will be of the form $|\sin t|^{-\frac{3}{2}}$ (see
Proposition 3.4 below) with $t\neq k\pi,k\in\mathbb{Z}$, which is different
from that for the free Schrödinger kernel case $|t|^{-\frac{3}{2}}$, but
consistent with that for the (unperturbed) harmonic oscillator (see e.g.
[29]).

In the following, we prove a dispersive bound for the Schrödinger kernel
obtained in Proposition 3.1.

######  Proposition 3.4.

Let $H_{a,1}$ be the operator in (1.3), then there exists some constant $C>0$
such that

$\|e^{-itH_{a,1}}\|_{L^{1}(\mathbb{R}^{3})\rightarrow
L^{\infty}(\mathbb{R}^{3})}\leq C|\sin t|^{-\frac{3}{2}},\quad\forall t\neq
k\pi,\quad k\in\mathbb{Z}.$ (3.7)

######  Remark 3.5.

Interpolating between the dispersive estimate (3.7) and the unitary property
of the Schrödinger propagator $e^{-itH_{a,1}}$

$\|e^{-itH_{a,1}}f\|_{L^{2}(\mathbb{R}^{3})}=\|f\|_{L^{2}(\mathbb{R}^{3})},$

one immediately gets the time decay estimate

$\|e^{-itH_{a,1}}f\|_{L^{p^{\prime}}(\mathbb{R}^{3})}\lesssim|\sin
t|^{-3(\frac{1}{2}-\frac{1}{p^{\prime}})}\|f\|_{L^{p}(\mathbb{R}^{3})},\quad
p\in[1,2],\quad\frac{1}{p}+\frac{1}{p^{\prime}}=1,$

which is an analog of [17, Theorem 1.11 (i)] for $e^{-itH_{a,0}}$.

###### Proof of Proposition 3.4.

The proof follows immediately from Lemma 2.1 and we sketch it here for
completeness. Since the sine function is periodic, we may, without loss of
generality, assume $t\in(0,\pi)$ for simplicity. Note that the Schrödinger
kernel $e^{-itH_{a,1}}(x,y)$ in (3.2) can be rewritten as

$e^{-itH_{a,1}}(x,y)=-i(2\sin
t)^{-\frac{3}{2}}e^{-\frac{|x|^{2}+|y|^{2}}{4i\tan t}}K\left(\frac{|xy|}{2\sin
t},\frac{x}{|x|},\frac{y}{|y|}\right),$

where $K(\cdot,\cdot,\cdot)$ is the function defined in Lemma 2.1. Then, the
desired dispersive bound (3.7) follows by Lemma 2.1. Indeed, let us verify
(3.7) according to $a=0$ or $a>0$. If $a=0$, then we are done since in this
case the result reduces to the dispersive estimate for the standard harmonic
oscillator (see [29]). If $a>0$, we have

$\displaystyle\|e^{-itH_{a,1}}\|_{L^{1}(\mathbb{R}^{3})\rightarrow
L^{\infty}(\mathbb{R}^{3})}$ $\displaystyle\leq|e^{-itH_{a,1}}(x,y)|$
$\displaystyle\leq|\sin t|^{-\frac{3}{2}}\left|K\left(\frac{|xy|}{2\sin
t},\frac{x}{|x|},\frac{y}{|y|}\right)\right|$ $\displaystyle\leq C|\sin
t|^{-\frac{3}{2}},$

where we use Lemma 2.1 in the last inequality. Hence, the proof of the
dispersive inequality (3.7) is completed. ∎

######  Remark 3.6.

The parameter $a$ in the inverse square potential $\frac{a}{|x|^{2}}$ is
usually assumed to satisfy $a\geq-\frac{1}{4}$ ($a\geq-\frac{(d-2)^{2}}{4}$
for general dimension $d\geq 2$) due to the sharp Hardy inequality (see e.g.
[26])

$\int_{\mathbb{R}^{d}}\frac{|u(x)|^{2}}{|x|^{2}}\mathrm{d}x\leq\frac{4}{(d-2)^{2}}\int_{\mathbb{R}^{d}}|\nabla
u(x)|^{2}\mathrm{d}x,\quad d\geq 3.$

However, we require $a\geq 0$ throughout the paper. We do not consider the
case $-\frac{1}{4}\leq a<0$ in the present paper, in light of the result of
[17, Theorem 1.11], which says that, for
$\frac{1}{p}+\frac{1}{p^{\prime}}=1,p\in[1,2]$, we have

$\|e^{it(\Delta-a|x|^{-2})}f\|_{L^{p^{\prime}}(\mathbb{R}^{3})}\lesssim|t|^{-3(\frac{1}{2}-\frac{1}{p^{\prime}})}\|f\|_{L^{p}(\mathbb{R}^{3})},\quad\text{when}\quad
a\geq 0$ (3.8)

and

$\|e^{it(\Delta-a|x|^{-2})}f\|_{L^{p^{\prime},\alpha_{1}}(\mathbb{R}^{3})}\lesssim\frac{(1+|t|^{\alpha_{1}})^{1-\frac{2}{p^{\prime}}}}{|t|^{3(\frac{1}{2}-\frac{1}{p^{\prime}})}}\|f\|_{L^{p,\alpha_{1}}(\mathbb{R}^{3})},\text{when}-\frac{1}{4}<a<0,$
(3.9)

where $\alpha_{1}=\frac{1}{2}-\sqrt{\frac{1}{4}+a}$ is a strictly positive
number and $\|\cdot\|_{L^{p,\alpha_{1}}(\mathbb{R}^{3})}$ denotes the weighted
$L^{p}$ norm defined by

$\|u\|_{L^{p,\alpha_{1}}(\mathbb{R}^{3})}:=\left(\int_{\mathbb{R}^{3}}(1+|x|^{-\alpha_{1}})^{2-p}|u(x)|^{p}\mathrm{d}x\right)^{1/p},\quad\forall
p\geq 1.$

It is easy to see that the (short time) dispersive estimate (3.7) for the
Schrödinger propagator $e^{-itH_{a,1}}$ is an analog of the time-decay
estimate (3.8) for $a\geq 0$ (note that $H_{a,0}=-\Delta+a|x|^{-2}$).
Analogous to (3.9), it should hold the following weighted decay estimate for
$-\frac{1}{4}<a<0$

$\|e^{-itH_{a,1}}\|_{L^{p,\alpha_{1}}(\mathbb{R}^{3})\rightarrow
L^{p^{\prime},\alpha_{1}}(\mathbb{R}^{3})}\lesssim\frac{(1+|\sin
t|^{\alpha_{1}})^{1-\frac{2}{p^{\prime}}}}{|\sin
t|^{3(\frac{1}{2}-\frac{1}{p^{\prime}})}},\quad t\neq k\pi,k\in\mathbb{Z}.$
(3.10)

Since $\alpha_{1}>0$ when $-\frac{1}{4}<a<0$, the decay estimate (3.9) is
weaker than the usual one (3.8) and so is (3.10). We cannot recover the
dispersive estimate (3.7) for the regime $-\frac{1}{4}<a<0$ since the weight
function in (3.9) seems to be indispensable (actually, the weighted estimate
(3.9) seems to be best possible, in view of [17, Remark 1.12]) and this is
exactly the reason why we restrict to consider the case $a\geq 0$ in the
present paper. In addition, it is not known whether the weighted estimate
(3.9) (or (3.10)) holds for the critical case $a=-\frac{1}{4}$. Nevertheless,
these unsolved problems may be left as the main objects of further
investigations.

##  4\. heat kernel

In this section, we construct a representation formula for the kernel of the
heat flow $e^{-tH_{a,1}}$ and then prove the Gaussian boundedness. The study
of heat kernels for Schrödinger operators with potentials on Euclidean spaces
$\mathbb{R}^{d}$ or manifolds has its own interest.

######  Proposition 4.1 (Heat kernel).

Let $H_{a,1}$ be the operator (1.3) and let $u$ be the unique solution to the
Schrödinger equation

$\begin{cases}\partial_{t}u(t,x)+H_{a,1}u(t,x)=0,\\\ u(0,x)=f(x)\in
L^{2}(\mathbb{R}^{3}),\end{cases}$

then the solution $u$ can be represented as

$u(t,x)=\left(e^{-tH_{a,1}}f\right)(x):=\int_{\mathbb{R}^{3}}K^{H}(x,y)f(y)\mathrm{d}y,$
(4.1)

where $K^{H}(x,y)$ corresponds to the kernel of the heat flow $e^{-tH_{a,1}}$.

Moreover, if we write $K_{t}^{H}(r_{1},\vartheta_{1};r_{2},\vartheta_{2})$ for
the kernel $K^{H}(x,y)$ in polar coordinates $(r,\vartheta)$, then we have

$\displaystyle
K_{t}^{H}(r_{1},\vartheta_{1};r_{2},\vartheta_{2})=\frac{e^{-\frac{r_{1}^{2}+r_{2}^{2}}{4}\coth
t}}{2\sqrt{r_{1}r_{2}}\sinh
t}\sum_{k=0}^{\infty}I_{\beta_{k}}\Bigg{(}\frac{r_{1}r_{2}}{2\sinh
t}\Bigg{)}Z^{(k)}_{\vartheta}(\vartheta^{\prime}),$ (4.2)

where $\beta_{k}$ is defined in (2.14) and
$Z^{(k)}_{\vartheta}(\vartheta^{\prime})$ is the zonal function in (2.9).

######  Remark 4.2.

The integral at the right hand side of (4.1) should be understood in the sense
of improper integrals as the Schrödinger counterpart, i.e.

$u(t,x)=\lim_{R\rightarrow+\infty}\int_{B_{R}(0)}K^{H}(x,y)f(y)\mathrm{d}y,\quad\text{where}\quad
B_{R}(0):=\\{y\in\mathbb{R}^{3}:|y|<R\\}.$

###### Proof of Proposition 4.1.

The proof is similar to Proposition 3.1 for the Schrödinger case. Since the
spectrum of the operator $H_{a,1}$ consists of pure points, we have, by the
spectral theorem,

$e^{-tH_{a,1}}(x,y)=\sum_{m,k\in\mathbb{N}}e^{-t\lambda_{m,k}}e_{m,k}(x)\overline{e_{m,k}(y)}.$
(4.3)

Substituting the eigenvalues (2.15) and eigenfunctions (2.16) into (4.3), one
gets

$\displaystyle
K_{t}^{H}(r_{1},\vartheta_{1};r_{2},\vartheta_{2})=(r_{1}r_{2})^{-1/2}$
$\displaystyle
e^{-\frac{r_{1}^{2}+r_{2}^{2}}{4}}\sum_{k=0}^{\infty}\frac{(r_{1}r_{2})^{\beta_{k}}e^{-t(1+\beta_{k})}}{2^{\beta_{k}}}\psi_{k}(\vartheta_{1})\overline{\psi_{k}(\vartheta_{2})}$
$\displaystyle\qquad\times\sum_{m=0}^{\infty}e^{-2mt}\frac{m!L_{m}^{\beta_{k}}\left(\frac{r_{1}^{2}}{2}\right)L_{m}^{\beta_{k}}\left(\frac{r_{2}^{2}}{2}\right)}{\Gamma(m+1+\beta_{k})}.$
(4.4)

By the formula (3.5), we obtain

$\begin{split}K_{t}^{H}(r_{1},\vartheta_{1};r_{2},\vartheta_{2})=\frac{(r_{1}r_{2})^{-1/2}e^{-\frac{r_{1}^{2}+r_{2}^{2}}{4\tan
t}}}{2\sinh
t}\sum_{k=0}^{\infty}\psi_{k}(\vartheta_{1})\overline{\psi_{k}(\vartheta_{2})}I_{\beta_{k}}\Bigg{(}\frac{r_{1}r_{2}}{2\sinh
t}\Bigg{)}.\end{split}$ (4.5)

Finally, in view of (2.17) and (2.9), the desired formula (4.2) follows. ∎

In the following, we prove a Gaussian type upper bound for the heat kernel in
Proposition 4.1.

######  Proposition 4.3.

Let $H_{a,1}$ be the operator (1.3), then there exists some constant $C>0$
such that

$|e^{-tH_{a,1}}(x,y)|\leq C|\sinh t|^{-\frac{3}{2}}e^{-\frac{|x-y|^{2}}{4\tanh
t}},\quad\forall t>0.$ (4.6)

###### Proof.

By using the function $K(\cdot,\cdot,\cdot)$ defined in Lemma 2.1, the heat
kernel (4.2) can be rewritten as

$e^{-tH_{a,1}}(x,y)=e^{-\frac{|x|^{2}+|y|^{2}}{4\tanh t}}(2\sinh
t)^{-\frac{3}{2}}K\left(\frac{\rho}{2\sinh
t},\frac{x}{|x|},\frac{y}{|y|}\right),\quad t>0.$

Then the desired bound (4.6) follows by Lemma 2.1. ∎

######  Remark 4.4.

Note that for small time $t$ (for example $t\in(0,1)$), the bound (4.6) can be
regarded as the usual Gaussian upper bound $t^{-3/2}e^{-\frac{|x-y|^{2}}{4t}}$
for heat kernels due to the simple facts $\sinh t\sim t\sim\tanh t$ as
$t\rightarrow 0$. However, for large time $t\gg 1$, one can see a exponential
decay $e^{-\frac{3}{2}t}$ for the heat kernel, which is faster than the usual
polynomial decay $t^{-3/2}$, due to the fact $\sinh t\sim e^{t}$ as
$t\rightarrow+\infty$. The Gaussian bound in (4.6) is known to play an
essential role in proving the corresponding Bernstein type inequalities, which
will be used to obtain the dispersive estimate for the half-wave propagator.

##  5\. Bernstein inequalities, Besov spaces & subordination formula

In this section, we prove the Bernstein inequalities, define the Besov spaces
and give the subordination formula adapted to the present setting.

###  5.1. Bernstein inequalities

In this subsection, we prove the $L^{p}$ Bernstein inequalities concerning the
frequency truncations of the Schrödinger operator $H_{a,1}$ given by (1.3). In
the following lemma, we prove the Bernstein inequalities for more general
cases $1\leq p\leq q\leq\infty,s\in\mathbb{R}$, although only the case
$(p,q,s)=(1,\infty,0)$ is used to prove the main theorem (Theorem 1.1).

######  Lemma 5.1 (Bernstein inequalities).

Let $\psi\in C_{c}^{\infty}(\mathbb{R})$ with $\psi\in[0,1]$ and
$\mathrm{supp}\psi\subset[1/2,2]$. Then, for any $f\in
L^{p}(\mathbb{R}^{3}),s\in\mathbb{R}$ and $j\in\mathbb{Z}$, it holds

$\|\psi(2^{-j}\sqrt{H_{a,1}})H_{a,1}^{s}f\|_{L^{q}(\mathbb{R}^{3})}\leq
C2^{2sj+3j\left(\frac{1}{p}-\frac{1}{q}\right)}\|f\|_{L^{p}(\mathbb{R}^{3})},\quad
1\leq p\leq q\leq\infty$ (5.1)

for some constant $C>0$.

###### Proof.

The proof is rather standard as in the theory of function spaces associated to
Schrödinger operators with potentials and we sketch it here for completeness.

From Proposition 4.3, it is known that the heat kernel of the operator
$H_{a,1}$ has a Gaussian upper bound, i.e.

$|e^{-tH_{a,1}}(x,y)|\leq
Ct^{-\frac{3}{2}}e^{-\frac{|x-y|^{2}}{ct}},\quad\text{for some}\quad c,C>0.$

By [36, Proposition 5.1], we conclude that the integral kernel of the
truncated operator $\psi(2^{-j}\sqrt{H_{a,1}})$ satisfies

$|\psi(2^{-j}\sqrt{H_{a,1}})(x,y)|\leq\frac{C_{N}2^{3j}}{(1+2^{j}|x-y|)^{N}},$
(5.2)

for all $N>0$, where $C_{N}>0$ is a constant independent of $j$. The bound can
be extended by scaling to a slightly more general case, i.e.

$|H_{a,1}^{s}\psi(2^{-j}\sqrt{H_{a,1}})(x,y)|\leq\frac{C_{N}2^{2sj+3j}}{(1+2^{j}|x-y|)^{N}}.$

Then the desired inequality (5.1) follows by applying Young’s inequality:

$\displaystyle\|H_{a,1}^{s}\psi(2^{-j}\sqrt{H_{a,1}})f\|_{L^{q}(\mathbb{R}^{3})}$
$\displaystyle=\left\|\int_{\mathbb{R}^{3}}H_{a,1}^{s}\psi(2^{-j}\sqrt{H_{a,1}})(x,y)f(y)\mathrm{d}y\right\|_{L^{q}(\mathbb{R}^{3})}$
$\displaystyle\leq
C_{N}2^{2sj+3j}\left\|\int_{\mathbb{R}^{3}}\frac{|f(y)|\mathrm{d}y}{(1+2^{j}|x-y|)^{N}}\right\|_{L^{q}(\mathbb{R}^{3})}$
$\displaystyle\leq
C_{N}2^{2sj+3j}\|(1+2^{j}|\cdot|)^{-N}\|_{L^{r}(\mathbb{R}^{3})}\|f\|_{L^{p}(\mathbb{R}^{3})}$
$\displaystyle\leq
C_{N}2^{2sj+3j\left(\frac{1}{p}-\frac{1}{q}\right)}\|f\|_{L^{p}(\mathbb{R}^{3})},$

where $1+\frac{1}{q}=\frac{1}{r}+\frac{1}{p}$. ∎

###  5.2. Besov spaces

In this subsection, we define the Besov spaces in the present setting for
general cases $1\leq p\leq q\leq\infty,s\in\mathbb{R}$, although only the case
$(p,q,s)=(1,1,1)$ is needed in the main theorem (Theorem 1.1). Take a
partition of unity

$1=\sum_{j\in\mathbb{Z}}\psi(2^{-j}\lambda):=\sum_{j\in\mathbb{Z}}\psi_{j}(\lambda),\quad\lambda\in(0,+\infty),$

where $\psi\in C_{c}^{\infty}$ with $0\leq\psi\leq 1$ and
$\mathrm{supp}\psi\subset[1/2,2]$. For $1\leq p,q<\infty$ and
$s\in\mathbb{R}$, the Besov space $\dot{\mathcal{B}}^{s,\psi}_{p,q}$
associated to the operator $H_{a,1}$ is defined by

$\dot{\mathcal{B}}^{s,\psi}_{p,q}(\mathbb{R}^{3}):=\left\\{f\in
L^{2}(\mathbb{R}^{3}):\|f\|^{q}_{\dot{\mathcal{B}}^{s,\psi}_{p,q}}=\sum_{j\in\mathbb{Z}}2^{jqs}\|\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}^{q}<+\infty\right\\}.$
(5.3)

As the classical Besov norm, the distorted Besov norm in (5.3) is independent
of the choice of a partition of unity in the following sense.

######  Proposition 5.2.

Let $\varphi,\psi\in C_{c}^{\infty}(\mathbb{R})$ be two different partitions
of unity with supports in $[1/2,2]$. Then, for $1\leq p,q<\infty$ and
$s\in\mathbb{R}$, we have

$\|f\|_{\dot{\mathcal{B}}^{s,\psi}_{p,q}(\mathbb{R}^{3})}\sim\|f\|_{\dot{\mathcal{B}}^{s,\varphi}_{p,q}(\mathbb{R}^{3})}.$
(5.4)

###### Proof.

To prove the equivalence (5.4), we claim that for any $m\in\mathbb{N}$, there
exists a constant $C_{m}>0$ such that

$\|\psi_{j}(\sqrt{H_{a,1}})\varphi_{k}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}\leq
C_{m}2^{-2m|j-k|}\|f\|_{L^{p}(\mathbb{R}^{3})},\quad\forall j,k\in\mathbb{Z}.$
(5.5)

We only consider the case $j\geq k$ in (5.5) and the other case can be treated
similarly. For $m\in\mathbb{N}$, we write

$\psi_{j}(\sqrt{H_{a,1}})\varphi_{k}(\sqrt{H_{a,1}})f=H_{a,1}^{-m}\psi_{j}(\sqrt{H_{a,1}})H_{a,1}^{m}\varphi_{k}(\sqrt{H_{a,1}})f$

which, together with Bernstein inequalities (5.1), implies that

$\begin{split}\|\psi_{j}(\sqrt{H_{a,1}})\varphi_{k}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}&\leq
C_{m}2^{-2jm}\|H_{a,1}^{m}\varphi_{k}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}\\\
&\leq C_{m}2^{-2(j-k)m}\|f\|_{L^{p}(\mathbb{R}^{3})}.\end{split}$

Then the claimed estimate (5.5) follows. To prove (5.4), since the roles of
$\varphi$ and $\psi$ are equivalent, it is sufficient to show that

$\|f\|_{\dot{\mathcal{B}}^{s,\psi}_{p,q}(\mathbb{R}^{3})}\leq
C\|f\|_{\dot{\mathcal{B}}^{s,\varphi}_{p,q}(\mathbb{R}^{3})},\quad
f\in\dot{\mathcal{B}}^{s,\varphi}_{p,q}(\mathbb{R}^{3}).$

Since $\varphi$ is a partition of unity, it holds

$\psi_{j}(\sqrt{H_{a,1}})f=\sum_{k\in\mathbb{Z}}\psi_{j}(\sqrt{H_{a,1}})\varphi_{k}(\sqrt{H_{a,1}})f,\quad
j\in\mathbb{Z}.$

On the other hand, one has
$\varphi_{k}=(\varphi_{k-1}+\varphi_{k}+\varphi_{k+1})\varphi_{k}$ in view of
the support property of $\varphi$. Hence, we have

$\psi_{j}(\sqrt{H_{a,1}})f=\sum_{k\in\mathbb{Z}}\psi_{j}(\sqrt{H_{a,1}})(\varphi_{k-1}(\sqrt{H_{a,1}})+\varphi_{k}(\sqrt{H_{a,1}})+\varphi_{k+1}(\sqrt{H_{a,1}}))\varphi_{k}(\sqrt{H_{a,1}})f,$

which implies

$\|\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}\leq\sum_{k\in\mathbb{Z}}\|\psi_{j}(\sqrt{H_{a,1}})(\varphi_{k-1}(\sqrt{H_{a,1}})+\varphi_{k}(\sqrt{H_{a,1}})+\varphi_{k+1}(\sqrt{H_{a,1}}))\varphi_{k}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}.$

We now apply (5.5) to deduce that for $m>|s|/2$

$\|\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}\leq\sum_{k\in\mathbb{Z}}2^{-2|j-k|m}\|\varphi_{k}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}.$
(5.6)

Inserting (5.6) into the expression of the Besov norm in (5.3) gives

$\displaystyle\|f\|^{q}_{\dot{\mathcal{B}}^{s,\psi}_{p,q}(\mathbb{R}^{3})}$
$\displaystyle=\sum_{j\in\mathbb{Z}}2^{jsq}\|\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}^{q}$
$\displaystyle\leq
C\sum_{j\in\mathbb{Z}}2^{jsq}\left(\sum_{k\in\mathbb{Z}}2^{-2|j-k|m}\|\varphi_{k}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}\right)^{q}$
$\displaystyle\leq
C\sum_{j\in\mathbb{Z}}\left(\sum_{k\in\mathbb{Z}}2^{-2|j-k|m+js}\|\varphi_{k}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}\right)^{q}$

which, together with Young’s inequality, implies

$\|f\|_{\dot{\mathcal{B}}^{s,\psi}_{p,q}(\mathbb{R}^{3})}\leq
C\left(\sum_{k\in\mathbb{Z}}2^{sqk}\|\varphi_{k}(\sqrt{H_{a,1}})f\|_{L^{p}(\mathbb{R}^{3})}^{q}\right)^{1/q}=\|f\|_{\dot{\mathcal{B}}^{s,\varphi}_{p,q}(\mathbb{R}^{3})}$

as desired. ∎

In view of Proposition 5.2, for $1\leq p,q<\infty$ and $s\in\mathbb{R}$, we
may briefly write $\|\cdot\|_{\dot{\mathcal{B}}^{s}_{p,q}}$ for
$\|\cdot\|_{\dot{\mathcal{B}}^{s,\psi}_{p,q}(\mathbb{R}^{3})}$ if there is no
confusion, where $\psi\in C_{c}^{\infty}(\mathbb{R})$ is a partition of unity
with $0\leq\psi\leq 1$ and $\mathrm{supp}\psi\subset[1/2,2]$.

###  5.3. Subordination formula

In this subsection, we give a simplified version of the classical
subordination formula introduced by Müller and Seeger [33].

######  Proposition 5.3 (Subordination formula).

Let $\eta\in C_{c}^{\infty}(\mathbb{R})$ with
$\mathrm{supp}\eta\subset[1/2,2]$. Then there exist a symbol function $a\in
C^{\infty}(\mathbb{R}\times\mathbb{R})$ with
$\mathrm{supp}a(\cdot,\sigma)\subset[1/16,4]$ for every $\sigma\in\mathbb{R}$
fulfilling

$|\partial_{s}^{j}\partial_{\sigma}^{k}a(s,\sigma)|\leq
C_{j,k}(1+|s|)^{-k},\quad\forall j,k\geq 0$

and a Schwartz function
$\rho(s,\sigma)\in\mathcal{S}(\mathbb{R}\times\mathbb{R})$ such that

$\eta(\sqrt{x}/2^{j})e^{it\sqrt{x}}=\rho(t\sqrt{x}/2^{j},2^{j}t)+\eta(\sqrt{x}/2^{j})2^{j/2}\sqrt{t}\int_{0}^{\infty}a(s,2^{j}t)e^{i2^{j-2}t/s}e^{istx/2^{j}}\mathrm{d}s,$
(5.7)

for every $x\geq 0$ and $t\geq 2^{-j}$ with $j\in\mathbb{Z}$.

The proof of Proposition 5.3 can be found in the paper of Müller and Seeger
[33] (see also D’Ancona, Pierfelice and Ricci [16] in Appendix); we omit the
details here.

##  6\. proof of Theorem 1.1

With all the required ingredients in hand, we are now in the right position to
prove the dispersive estimate for the wave equation (Theorem 1.1). For the
convenience of the reader, we list these ingredients again: dispersive
estimate for Schrödinger flow (3.7) in Proposition 3.4 of Section 3, Gaussian
bound for heat kernel (4.6) in Proposition 4.3 of Section 4, Bernstein
inequalities (5.1) in Proposition 5.1 and subordination formula (5.7) in
Proposition 5.3 of Section 5.

To prove Theorem 1.1, a crucial ingredient is the following frequency-
localized decay estimate for the half-wave propagator $e^{it\sqrt{H_{a,1}}}$
since the desired dispersive estimate (1.10) then follows from the definition
of the distorted Besov space $\dot{\mathcal{B}}^{1}_{1,1}(\mathbb{R}^{3})$ by
summing all the frequencies.

######  Lemma 6.1.

Let $0<T<\pi$ and $\psi\in C_{c}^{\infty}(\mathbb{R})$ with
$0\leq\psi(\lambda)\leq 1$ and $\mathrm{supp}\psi\subset[1/2,2]$. Then, for
all $j\in\mathbb{Z},|t|<T$ and $f\in L^{1}(\mathbb{R}^{3})$, there exists a
constant $C>0$ independent of $t$ such that

$\|\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}f\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C2^{-3j}\langle 2^{j}t\rangle^{-1}\|f\|_{L^{1}(\mathbb{R}^{3})},$ (6.1)

where $\psi_{j}(\lambda):=\psi(2^{-j}\lambda)$ and
$\langle\cdot\rangle:=(1+|\cdot|^{2})^{1/2}$.

###### Proof.

It suffices to consider the case $t>0$ since the case $t<0$ can be treated
similarly. We follow the proof of the main result of [16]. Note that
$\psi_{j}=\psi_{j}(\psi_{j-1}+\psi_{j}+\psi_{j+1})$, then we have for
$t<2^{-j}$

$\|\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}f\|_{L^{\infty}(\mathbb{R}^{3})}\leq\sum_{k=0}^{2}\|\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}\psi_{j+k-1}(\sqrt{H_{a,1}})f\|_{L^{\infty}(\mathbb{R}^{3})}.$

By Lemma 5.1, we obtain

$\displaystyle\|\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}f\|_{L^{\infty}(\mathbb{R}^{3})}$
$\displaystyle\leq
C\sum_{k=0}^{2}\|\psi_{j}(\sqrt{H_{a,1}})\|_{L^{2}(\mathbb{R}^{3})\rightarrow
L^{\infty}(\mathbb{R}^{3})}\|e^{it\sqrt{H_{a,1}}}\|_{L^{2}(\mathbb{R}^{3})\rightarrow
L^{2}(\mathbb{R}^{3})}$
$\displaystyle\qquad\qquad\times\|\psi_{j+k-1}(\sqrt{H_{a,1}})\|_{L^{1}(\mathbb{R}^{3})\rightarrow
L^{2}(\mathbb{R}^{3})}\|f\|_{L^{1}(\mathbb{R}^{3})}$ $\displaystyle\leq
C2^{3j}\|f\|_{L^{1}(\mathbb{R}^{3})}\sim\frac{2^{3j}}{1+2^{j}|t|}\|f\|_{L^{1}(\mathbb{R}^{3})}.$

For $t\geq 2^{-j}$, we apply Proposition 5.3 to write

$\displaystyle\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}f$
$\displaystyle=\rho(2^{-j}t\sqrt{H_{a,1}},2^{j}t)f$
$\displaystyle\quad+\psi_{j}(\sqrt{H_{a,1}})2^{\frac{j}{2}}\sqrt{t}\int_{0}^{\infty}a(s,2^{j}t)e^{i2^{j-2}t/s}e^{i2^{-j}stH_{a,1}}f\mathrm{d}s$
$\displaystyle:=S_{1}+S_{2}.$

For the first term $S_{1}$, by applying Lemma 5.1 and noting the fact
$\rho\in\mathcal{S}(\mathbb{R}\times\mathbb{R})$, we have

$\displaystyle|S_{1}|$ $\displaystyle\leq
C\|\tilde{\psi}_{j}(\sqrt{H_{a,1}})\|_{L^{2}(\mathbb{R}^{3})\rightarrow
L^{\infty}(\mathbb{R}^{3})}\|\rho(2^{-j}t\sqrt{H_{a,1}},2^{j}t)\|_{L^{2}(\mathbb{R}^{3})\rightarrow
L^{2}(\mathbb{R}^{3})}$
$\displaystyle\qquad\quad\times\|\tilde{\psi}_{j}(\sqrt{H_{a,1}})\|_{L^{1}(\mathbb{R}^{3})\rightarrow
L^{2}(\mathbb{R}^{3})}\|f\|_{L^{1}(\mathbb{R}^{3})}$ $\displaystyle\leq
C2^{3j}\|\rho(2^{-j}t\sqrt{x},2^{j}t)\|_{L_{x}^{\infty}(\mathbb{R})}\|f\|_{L^{1}(\mathbb{R}^{3})}$
$\displaystyle\leq C2^{3j}(2^{j}t)^{-1}\|f\|_{L^{1}(\mathbb{R}^{3})}$
$\displaystyle\leq C2^{2j}t^{-1}\|f\|_{L^{1}(\mathbb{R}^{3})}$
$\displaystyle\sim\frac{2^{3j}}{1+2^{j}|t|}\|f\|_{L^{1}(\mathbb{R}^{3})},$

where $\tilde{\psi}_{j}=\psi_{j-1}+\psi_{j}+\psi_{j+1}$.

For the second term $S_{2}$, by (3.7) (dispersive estimate for
$e^{-itH_{a,1}}$) and Lemma 5.1, we obtain

$\displaystyle|S_{2}|$ $\displaystyle\leq
C2^{\frac{j}{2}}\sqrt{t}\int_{0}^{\infty}|a(s,2^{j}t)|(2^{-j}st)^{-3/2}\|f\|_{L^{1}(\mathbb{R}^{3})}\mathrm{d}s$
$\displaystyle\leq
C2^{2j}t^{-1}\|f\|_{L^{1}(\mathbb{R}^{3})}\sim\frac{2^{3j}}{1+2^{j}|t|}\|f\|_{L^{1}(\mathbb{R}^{3})}.$

Therefore, the proof of (6.1) is completed. ∎

######  Remark 6.2.

Theorem 1.1 can be derived from Lemma 6.1 by the definition of the distorted
Besov space $\dot{\mathcal{B}}^{1}_{1,1}$ in (1.9). Indeed, let
$\tilde{\psi}_{j}=\tilde{\psi}(2^{-j}\sqrt{H_{a,1}})$ and
$\tilde{\psi}(\lambda)=\lambda^{-1}\psi(\lambda)$ with $\psi$ as in Lemma 6.1.
Note that from (6.1), we have

$\|e^{it\sqrt{H_{a,1}}}\tilde{\psi}_{j}f\|_{L^{\infty}(\mathbb{R}^{3})}\leq
C2^{2j}|t|^{-1}\|f\|_{L^{1}(\mathbb{R}^{3})},$

which implies

$\displaystyle\left\|\frac{e^{it\sqrt{H_{a,1}}}}{\sqrt{H_{a,1}}}f\right\|_{L^{\infty}(\mathbb{R}^{3})}$
$\displaystyle\leq\sum_{j\in\mathbb{Z}}\left\|\frac{e^{it\sqrt{H_{a,1}}}}{\sqrt{H_{a,1}}}\psi_{j}(\sqrt{H_{a,1}})f\right\|_{L^{\infty}(\mathbb{R}^{3})}$
$\displaystyle\leq
C|t|^{-1}\sum_{j\in\mathbb{Z}}2^{2j}\|H^{-1/2}_{a,1}\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{1}(\mathbb{R}^{3})}$
$\displaystyle=C|t|^{-1}\sum_{j\in\mathbb{Z}}2^{j}\|\tilde{\psi}_{j}(\sqrt{H_{a,1}})f\|_{L^{1}(\mathbb{R}^{3})}$
$\displaystyle=C|t|^{-1}\|f\|_{\dot{\mathcal{B}}^{1,\tilde{\psi}}_{1,1}(\mathbb{R}^{3})}$
$\displaystyle\sim|t|^{-1}\|f\|_{\dot{\mathcal{B}}^{1}_{1,1}(\mathbb{R}^{3})}.$

######  Remark 6.3.

In view of Remark 3.6, we point out that the argument of obtaining the
dispersive estimate for the wave equation in Theorem 1.1 fails if
$-\frac{1}{4}\leq a<0$. A weighted version of the dispersive estimate (1.10)
should be valid for this case $-\frac{1}{4}\leq a<0$, but this is open
question. Nevertheless, motivated by [18, Remark 1.2], it is interesting to
conjecture that the corresponding Strichartz estimates as in Theorem 1.3
should be true for the case $-\frac{1}{4}<a<0$ or even the critical case
$a=-\frac{1}{4}$ as Mizutani [34] for the Schrödinger equation; these problems
may be the main objects of future investigations.

##  7\. proof of Theorem 1.3

In this section, we prove Strichartz estimates in Theorem 1.3. It is
sufficient to show that for admissible $q,r$ in (1.7) and $s$ in (1.8), it
holds

$\|e^{it\sqrt{H_{a,1}}}f\|_{L_{t}^{q}(I;L_{x}^{r}(\mathbb{R}^{3}))}\leq
C\|f\|_{\dot{H}_{osc}^{s}(\mathbb{R}^{3})},$ (7.1)

where $C>0$ is a constant independent of
$f\in\dot{H}_{osc}^{s}(\mathbb{R}^{3})$ and $I=[a,b]\subset(0,\pi)$.

######  Proposition 7.1.

Let $\psi\in C_{c}^{\infty}(\mathbb{R})$ and
$\psi_{j}(\lambda)=\psi(2^{-j}\lambda)$ with $0\leq\psi(\lambda)\leq 1$ and
$\mathrm{supp}\psi\subset[1/2,2]$. Then, for all $j\in\mathbb{Z}$ and $f\in
L^{2}(\mathbb{R}^{3})$, there exists some constant $C>0$ such that

$\|\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}f\|_{L_{t}^{q}(I;L_{x}^{r}(\mathbb{R}^{3}))}\leq
C2^{2sj}\|f\|_{L^{2}(\mathbb{R}^{3})},$ (7.2)

with $I=[a,b]\subset(0,\pi)$ and $(q,r),s$ in (1.7), (1.8), respectively.

###### Proof.

First, we prove that

$\displaystyle\left\|\left(\sum_{j\in\mathbb{Z}}|\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}f|^{2}\right)^{1/2}\right\|_{L_{t}^{q}(I;L_{x}^{r}(\mathbb{R}^{3}))}$
$\displaystyle\leq
C\left\|\left(\sum_{j\in\mathbb{Z}}|H_{a,1}^{\frac{s}{2}}\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}f|^{2}\right)^{1/2}\right\|_{L^{2}(\mathbb{R}^{3})}.$
(7.3)

Indeed, by Lemma 6.1, we obtain

$\displaystyle\|\psi_{j}(\sqrt{H_{a,1}})e^{it\sqrt{H_{a,1}}}f\|_{L^{\infty}(\mathbb{R}^{3})}$
$\displaystyle=\|e^{it\sqrt{H_{a,1}}}\tilde{\psi}_{j}(\sqrt{H_{a,1}})\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{\infty}(\mathbb{R}^{3})}$
$\displaystyle\leq
C\frac{2^{3j}}{1+2^{j}|t|}\|\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{1}(\mathbb{R}^{3})},$

where $\tilde{\psi}_{j}=\psi_{j-1}+\psi_{j}+\psi_{j+1}$. Moreover, by the
unitary property of the propagator $e^{it\sqrt{H_{a,1}}}$

$\|e^{it\sqrt{H_{a,1}}}\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{2}(\mathbb{R}^{3})}=\|\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{2}(\mathbb{R}^{3})},$

we can conclude, by following the argument of Keel-Tao [28, Corollary 1.3],
that

$\|e^{it\sqrt{H_{a,1}}}\psi_{j}(\sqrt{H_{a,1}})f\|_{L_{t}^{q}(I;L_{x}^{r}(\mathbb{R}^{3}))}\leq
C\|H_{a,1}^{\frac{s}{2}}\psi_{j}(\sqrt{H_{a,1}})f\|_{L^{2}(\mathbb{R}^{3})}.$

Thus, the desired estimate (7) follows by Minkowski’s inequality since
$q,r\geq 2$.

Next, by the almost orthogonality

$\left\|\left(\sum_{j\in\mathbb{Z}}|\psi_{j}(\sqrt{H_{a,1}})f|^{2}\right)^{1/2}\right\|_{L^{r}(\mathbb{R}^{3})}\sim\|f\|_{L^{r}(\mathbb{R}^{3})},\quad\forall
1<r<\infty,$ (7.4)

we obtain

$\|e^{it\sqrt{H_{a,1}}}f\|_{L_{t}^{q}(I;L_{x}^{r}(\mathbb{R}^{3}))}\leq
C\|H_{a,1}^{\frac{s}{2}}f\|_{L^{2}(\mathbb{R}^{3})}.$

Therefore, we have

$\left\|\cos(t\sqrt{H_{a,1}})f+\frac{\sin(t\sqrt{H_{a},1})}{\sqrt{H_{a,1}}}g\right\|_{L_{t}^{q}(I;L_{x}^{r}(\mathbb{R}^{3}))}\leq
C\|f\|_{\dot{H}_{osc}^{s}(\mathbb{R}^{3})}+\|g\|_{\dot{H}_{osc}^{s-1}(\mathbb{R}^{3})}$

as desired. ∎

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Relationships Between Six Circumcircles Sangaku Journal of Mathematics (SJM)
©SJM
ISSN 2534-9562
Volume 3 (2019), pp. 98-105
Received 19 September 2019. Published on-line 28 October 2019
web: http://www.sangaku-journal.eu/
©The Author(s) This article is published with open access111This article is
distributed under the terms of the Creative Commons Attribution License which
permits any use, distribution, and reproduction in any medium, provided the
original author(s) and the source are credited..

Relationships Between Six Circumcircles

Stanley Rabinowitz

545 Elm St Unit 1, Milford, New Hampshire 03055, USA

e-mail<EMAIL_ADDRESS>
web: http://www.StanleyRabinowitz.com/

Abstract. If $P$ is a point inside $\triangle ABC$, then the cevians through
$P$ divide $\triangle ABC$ into six small triangles. We give theorems about
the relationships between the radii of the circumcircles of these triangles.
We also state some results about the relationships between the circumcenters
of these triangles.

Keywords. Circumcircles, triangle geometry, circumcenters, cevians, cevasix
configuration.

Mathematics Subject Classification (2010). 51M04.

##  1\. Introduction

Let $P$ be any point inside a triangle $ABC$. The cevians through $P$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1.

Figure 1. numbering of the six triangles

The relationships between the radii of the circles inscribed in these
triangles was investigated in [8]. The relationships between the radii of
certain excircles associated with these triangles was investigated in [7]. In
this paper, we will investigate the relationships between the radii of the
circles circumscribed about these triangles.

We will make use of The Extended Law of Sines which states that if $a$, $b$,
and $c$ are the lengths of the sides of a triangle opposite angles $A$, $B$,
and $C$, then

${a\over\sin A}={b\over\sin B}={c\over\sin C}=2R$

where $R$ is the circumradius of $\triangle ABC$.

##  2\. Radii

We begin with some relationships between the radii of the six circumcircles.

######  Theorem 2.1.

Let $P$ be any point inside $\triangle ABC$. The cevians through $P$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1. Let $R_{i}$ be the circumradius of $T_{i}$. Then
$R_{1}R_{3}R_{5}=R_{2}R_{4}R_{6}$.

###### Proof.

By The Extended Law of Sines in $\triangle PBD$, we have

$R_{1}={BD\over 2\sin\angle BPD},$

with similar expressions for the other $R_{i}$. Thus,

$R_{1}R_{3}R_{5}={BD\over 2\sin\angle BPD}\cdot{CE\over 2\sin\angle
CPE}\cdot{AF\over 2\sin\angle APF}\phantom{.}$

and

$R_{2}R_{4}R_{6}={DC\over 2\sin\angle DPC}\cdot{EA\over 2\sin\angle
EPA}\cdot{FB\over 2\sin\angle FPB}.$

But $BD\cdot CE\cdot AF=DC\cdot EA\cdot FB$ by Ceva’s Theorem. Also, angles
$\angle BPD$ and $\angle EPA$ are vertical angles, so they are congruent and
their sines are equal. Similarly, $\sin\angle CPE=\sin\angle FPB$ and
$\sin\angle APF=\sin\angle DPC$. Therefore, we conclude that
$R_{1}R_{3}R_{5}=R_{2}R_{4}R_{6}$. ∎

######  Corollary 2.2.

Let $P$ be any point inside $\triangle ABC$. The cevians through $P$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1. Let $r_{i}$ be the radius of the nine-point circle of
$T_{i}$ (the circle through the midpoints of the sides). Then
$r_{1}r_{3}r_{5}=r_{2}r_{4}r_{6}$.

###### Proof.

This follows immediately from the fact that $R_{i}=2r_{i}$. ∎

We have some additional results for specific locations of point $P$.

######  Theorem 2.3.

Let $O$ be the circumcenter of $\triangle ABC$ and assume that $O$ lies inside
$\triangle ABC$. The cevians through $O$ divide $\triangle ABC$ into six
smaller triangles, labeled $T_{1}$ through $T_{6}$ as shown in Figure 1. Let
$R_{i}$ be the circumradius of $T_{i}$. Then $R_{1}=R_{2}$, $R_{3}=R_{4}$, and
$R_{5}=R_{6}$.

Figure 2. Circumcenter: $R_{1}=R_{2}$

###### Proof.

By symmetry, it suffices to show that $R_{1}=R_{2}$ (Figure 2). Since $O$ is
the circumcenter of $\triangle ABC$, $OB=OC$. Angles $\angle ODB$ and $\angle
ODC$ are supplementary, so their sines are equal. Thus, by The Extended Law of
Sines, we have

$R_{1}={OB\over 2\sin\angle ODB}={OC\over 2\sin\angle ODC}=R_{2}$

as required. ∎

######  Theorem 2.4.

Let $N$ be the Nagel Point of $\triangle ABC$. The cevians through $N$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1. Let $R_{i}$ be the circumradius of $T_{i}$. Then
$R_{1}=R_{4}$, $R_{2}=R_{5}$, and $R_{3}=R_{6}$.

Figure 3. Nagel Point: $R_{1}=R_{4}$

Note: The Nagel Point of a triangle is the point of concurrence of $AD$, $BE$,
and $CF$, where $D$, $E$, and $F$ are the points where the excircles of
$\triangle ABC$ touch the sides $BC$, $CA$, and $AB$, respectively [1, p.
160]. The Nagel point is usually denoted $Na$ or $N_{a}$, but here we will
refer to it as $N$, for simplicity.

###### Proof.

First note that by symmetry, it suffices to show that $R_{1}=R_{4}$ (Figure
3). If $BC=a$, $CA=b$, $AB=c$, and $s=(a+b+c)/2$, then it is known that
$BD=AE=s-c$ [1, p. 88]. Thus, by The Extended Law of Sines and the fact that
$\angle BND=\angle ENA$, we have

$R_{1}={BD\over 2\sin\angle BND}={AE\over 2\sin\angle ENA}=R_{4}$

as required. ∎

######  Theorem 2.5.

Let $H$ be the orthocenter of $\triangle ABC$. The cevians through $H$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1. Let $C_{i}$ be the circumcircle of $T_{i}$. Let $R_{i}$ be
the radius of $C_{i}$. Then $R_{1}=R_{6}$, $R_{2}=R_{3}$, and $R_{4}=R_{5}$.

Figure 4. Orthocenter: $R_{1}=R_{6}$

###### Proof.

By symmetry, it suffices to show that $R_{1}=R_{4}$ (Figure 4), i.e., that
$C_{1}$ and $C_{6}$ coincide. Since $\angle BDH+\angle HFB=180^{\circ}$,
quadrilateral $BDHF$ is cyclic. Thus, the circle through points $B$, $D$, and
$H$ is the same as the circle through points $B$, $F$, and $H$. ∎

##  3\. Circumcenters

Now we collect together some interesting results concerning the centers of the
six circumcircles. Most, but not all, of these results are scattered about in
the literature. We will use the notation [XYZ] to denote the area of
$\triangle XYZ$.

######  Theorem 3.1.

Let $P$ be any point inside $\triangle ABC$. The cevians through $P$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1. Let $O_{i}$ be the circumcenter of $T_{i}$. Then
$[O_{1}O_{3}O_{5}]=[O_{2}O_{4}O_{6}]$ (Figure 5).

Figure 5. Two triangles have same area

The following proof is due to Dubrovsky [2].

###### Proof.

Since $O_{1}$ is the circumcenter of $\triangle BPD$, it must lie on the
perpendicular bisector of $BP$. The same remark holds true for $O_{6}$.
Therefore, $O_{1}O_{6}\perp BP$. In the same way, $O_{6}O_{5}\perp PF$,
$O_{5}O_{4}\perp AP$, $O_{4}O_{3}\perp PE$, $O_{3}O_{2}\perp CP$, and
$O_{2}O_{1}\perp PD$. Hence $O_{1}O_{6}\parallel O_{3}O_{4}$,
$O_{6}O_{5}\parallel O_{2}O_{3}$, and $O_{5}O_{4}\parallel O_{1}O_{2}$.
Therefore, hexagon $O_{1}O_{2}O_{3}O_{4}O_{5}O_{6}$ has its opposite sides
parallel. But it is known [3] that if $ABCDEF$ is a hexagon with its opposite
sides parallel, then $[ACE]=[BDF]$. Thus
$[O_{1}O_{3}O_{5}]=[O_{2}O_{4}O_{6}]$. ∎

######  Theorem 3.2.

Let $M$ be the centroid of $\triangle ABC$. The medians through $M$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1. Let $O_{i}$ be the circumcenter of $T_{i}$. Then
$O_{1}O_{4}=O_{2}O_{5}=O_{3}O_{6}$. (Figure 6).

Figure 6. Red segments are congruent

###### Proof.

This follows from Proposition 4 of [5]. ∎

The following result comes from [5].

######  Theorem 3.3.

Let $P$ be any point inside $\triangle ABC$. The cevians through $P$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1. Let $O_{i}$ be the circumcenter of $T_{i}$. Then the points
$O_{i}$ lie on a circle if and only if either $P$ is the centroid of
$\triangle ABC$ (Figure 7) or $P$ is the orthocenter of $\triangle ABC$ (in
which case $O_{6}=O_{1}$, $O_{2}=O_{3}$, and $O_{4}=O_{5}$).

Figure 7. $O_{i}$ lie on a circle when $P=M$

The “if” portion of this theorem is the well-known Van Lamoen’s Theorem, [4].

######  Theorem 3.4.

Let $P$ be any point inside $\triangle ABC$. The cevians through $P$ divide
$\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through $T_{6}$ as
shown in Figure 1. Let $O_{i}$ be the circumcenter of $T_{i}$. Then the points
$O_{i}$ lie on a conic (Figure 8).

Figure 8. $O_{i}$ lie on a conic

The following proof comes from [5].

###### Proof.

Since $O_{1}O_{6}\parallel O_{3}O_{4}$, $O_{6}O_{5}\parallel O_{2}O_{3}$, and
$O_{5}O_{4}\parallel O_{1}O_{2}$, the result follows from the converse of
Pascal’s Theorem. ∎

Before stating our next result, we will need the following lemma which comes
from [6].

######  Lemma 3.5.

Let P be any point on the median $AD$ of $\triangle ABC$. Let $BE$ and $CF$ be
the cevians through $P$. Suppose the circumcircles of triangles $BPF$ and
$CEP$ meet at points $P$ and $Q$. Then $\angle BPQ=\angle CPD$ (Figure 9).

Figure 9.

######  Theorem 3.6.

Let P be any point on the median $AD$ of $\triangle ABC$. The cevians through
$P$ divide $\triangle ABC$ into six smaller triangles, labeled $T_{1}$ through
$T_{6}$ as shown in Figure 1. Let $O_{i}$ be the circumcenter of $T_{i}$. Then
(Figure 10)
(a) the points $O_{1}$, $O_{2}$, $O_{3}$, and $O_{6}$ lie on a circle,
(b) the points $O_{3}$, $O_{4}$, $O_{5}$, and $O_{6}$ lie on a circle.

Figure 10. The situation when $AD$ is a median

###### Proof.

Part (a). Since $O_{1}$ is the circumcenter of $\triangle BPD$, $O_{1}$ must
lie on the perpendicular bisector of $PB$. Similarly, $O_{6}$ lies on the
perpendicular bisector of $BP$. In the same way, $O_{1}O_{2}\perp PD$ and
$O_{2}O_{3}\perp PC$. Draw the circumcircles of triangles $PEC$ and $PFB$.
These circles meet at $P$ and $Q$ (Figure 11). The line joining the centers of
two intersecting circles is perpendicular to the common chord, so
$O_{3}O_{6}\perp PQ$. Since $\angle PNO_{2}=\angle PKO_{2}$, quadrilateral
$PNKO_{2}$ is cyclic and hence $\angle NPK=\angle NO_{2}K$. By the same
reasoning, quadrilateral $PLMO_{6}$ is cyclic and so $\angle LPM=\angle
LO_{6}M$. But by Lemma 3.5, $\angle NPK=\angle LPM$. Therefore $\angle
NO_{2}K=\angle LO_{6}M$. This makes quadrilateral $O_{6}O_{1}O_{3}O_{2}$
cyclic and $O_{1}$, $O_{2}$, $O_{3}$, and $O_{6}$ lie on a circle as claimed.
∎

Figure 11.

Part (b). This result is due to Suppa [9]. The proof is similar to the proof
of part (a) and details can be found in [6].

## References

  * [1] Nathan Altshiller-Court, College Geometry, 2nd edition, Dover Publications, New York, 2007.
  * [2] Vladimir Dubrovsky, Solution to Problem 3425. Romantics of Geometry, August, 2019.
https://www.facebook.com/groups/parmenides52/permalink/2384338005013225/

  * [3] Mark Kantrowich, Solution to Problem from the 1984 Annual Greek High School Competition. Crux Mathematicorum, 11(1985)184.
https://cms.math.ca/crux/backfile/Crux_v11n06_Jun.pdf

  * [4] Floor van Lamoen, Problem 10830, American Mathematical Monthly, 2000(107)863, 2002(109)396–397.
  * [5] Alexei Myakishev and Peter Y. Woo, On the circumcenters of cevasix configurations, Forum Geometricorum, 3(2003)57–63.
http://forumgeom.fau.edu/FG2003volume3/FG200305.pdf

  * [6] Aris Pavlákis, Comment on Problem 3473. Romantics of Geometry, Sept. 2019.
https://www.facebook.com/groups/parmenides52/permalink/2410533599060332/

  * [7] Stanley Rabinowitz, Relationships Between Six Excircles. Sangaku Journal of Mathematics, 3(2019)73-90. http://www.sangaku-journal.eu/2019/SJM_2019_73-90_Rabinowitz.pdf
  * [8] Stanley Rabinowitz, Relationships Between Six Incircles. Sangaku Journal of Mathematics, 3(2019)51-66. http://www.sangaku-journal.eu/2019/SJM_2019_51-66_Rabinowitz.pdf
  * [9] Ercole Suppa, Problem 3473. Romantics of Geometry, Aug. 2019.
https://www.facebook.com/groups/parmenides52/permalink/2410533599060332/