数理解析研究所講究録 1208 巻 2001 年 119-127

119

Asymptotic Properties of Solutions to 3-particle
Schr\"odinger Equations
Hiroshi ISOZAKI (磯崎 洋)
Department of Mathematics, Graduate School of Science,
Osaka University, Toyonaka, 560, Japan
Abstract

associated with the 3We construct ageneralized Fourier transformation$H=\triangle+\sum_{a}V_{a}(x^{a})$and characterize all solutions ofbody Schrodinger operator. Theseas the image of$(H-\lambda)u=0$ in the Agmon-Hormander space
waves
spherical
in
of
in
terms
expansions
stationary solutions admit asymptotic
associated with scattering channels.
$\mathcal{F}(\lambda)$$B^{*}$$\mathcal{F}(\lambda)^{*}$$B^{*}$

1Introduction
1.1 Helmholtz equation. Consider the Helmholtz equation in$(-\triangle-\lambda)u=0$,$\lambda>0$$\mathrm{R}^{n}$

(1.1)

.

According to aclassical theorem of Sommerfeld-Rellich, $u=0$ if satisfies (1.1) and$u=O(|x|^{-s})$ as $|x|arrow\infty$ for $s>(n-1)/2$ . Non-trivial solutions arise ffom the decayrate $s\leq(n-1)/2$ , and the border line case $s=(n-1)/2$ was characterized by AgmonH\"ormander [2] : Let be asolution to (1.1). Then satisfies
$u$

$u$

$u$

(1.2)

$\sup_{R>1}\frac{1}{R}\int_{|x|<R}|u(x)|^{2}dx<\infty$

if and only if$u$is written as(1.3)$u(x)= \int_{S^{n-1}}e^{i\sqrt{\lambda}\omega\cdot x}\varphi(\omega)d\omega$. This, combined with the stationary phase method on the sphere,
for some
implies that all the solution of (1.1) satisfying (1.2) admits an asymptotic expansion$\varphi\in L^{2}(S^{n-1})$$u$$u\simeq C(\lambda)r^{-(n-1)/2}e^{i\sqrt{\lambda}r}\varphi(\hat{x})+\overline{C(\lambda)}r^{-(n-1)/2}e^{-i\sqrt{\lambda}r}\varphi(-\hat{x})$where $r=|x|,\hat{x}=x/r$ , and the asymptotic relation$u\simeq v$

,

(1.4)

means that$\lim_{Rarrow\infty}\frac{1}{R}\int_{|x|<R}|u(x)-v(x)|^{2}dx=0$

.

(1.5)

120
The intention of Agmon-Hormander is to characterize the restriction of Fourier transforms on submanifolds inand they introduced the following Besov type space \yen It is
space
8equipped
the Banachwith the norm$\mathrm{R}^{n}$,$||u||_{B}=( \int_{\Omega_{0}}|u(x)|^{2}dx)^{1/2}+\sum_{j=1}^{\infty}(2^{j-1}\int_{\Omega_{j}}|u(x)|^{2}dx)^{1/2}<\infty$

where $\Omega_{0}=\{x\in \mathrm{R}^{n};|x|<1\}$ , $\Omega_{j}=\{x\in \mathrm{R}^{n};2^{j-1}<|x|<2^{j}\}$ for $j\geq 1$ . The norm of
the dual space
is equivalent to the following one$B^{*}$$||u||_{B^{2}}= \sup_{R>0}(\frac{1}{R}\int_{|x|<R}|u(x)|^{2}dx)^{1/2}$

Let for$s\in \mathrm{Y}\mathrm{L}$$u\in L^{2,s}\Leftrightarrow||u||_{s}=||(1+|x|)^{s}u(x)||_{L^{2}(\mathrm{R}^{n})}<\infty$.

For $s>1/2$ we have the following inclusion relations$L^{2,s}\subset B\subset L^{2,1/2}\subset L^{2}\subset L^{2,-1/2}\subset g*\subset L^{2,-s}$We also have$|(u,v)|\leq C||u||_{B}||v||_{B}\cdot$.

.

1.22-body Schr\"odinger equation. There are two directions for generalization of the above
facts. One is the extension to Laplacians on non-compact Riemannian manifolds. This isactually aclassical problem and had been studied by Helgason [12], [13] for example. The
general case was studied by Agmon [1], Melrose [24], Melrose-Zworski [25]. Another isthe extension to Schr\"odinger equations. Kato [23] proved that the solution of the 2-b0dy
Schr\"odinger equation$(-\triangle+V(x)-\lambda)u(x)=0$ ,$\lambda>0$

satisfying $u\in L^{2,-}’$ , $\alpha<1/2$ , vanishes identically. Non-trivial solutions to the criticalcase were charac terized by Yafaev [31], G\^atel-Yafaev [6].
For the sake of simplicity, let us mention the 2-body Schr\"odinger operator with shortrange potential $H=-\triangle+V(x)$ , $V(x)=O(|x|^{-1-\epsilon})(\epsilon>0)$ . Suppose is asolution to
the equation$(H-\lambda)u=0$ ,
.
(1.6)
$u$

$\lambda>0$

Then

$u$

satisfies (1.2) if and only if$u$is written as$u=\mathcal{F}(\lambda)^{*}\varphi$

where$\mathcal{F}(\lambda)$,$\varphi\in L^{2}(S^{n-1})$

,

(1.7)

is the operator defined by,$\mathcal{F}(\lambda)f=\int_{\mathrm{R}^{n}}e^{-:\sqrt{\lambda}\omega\cdot x}f(x)dx-\int_{\mathrm{R}^{n}}e^{-i\sqrt{\lambda}\omega\cdot x}V(x)R(\lambda+i0)fdx$

with $R(z)=(H-z)^{-1}$ . The solution
expansion

$u$

(1.8)

of (1.6) satisfying (1.2) admits an asymptotic$u\simeq C(\lambda)r^{-(n-1)/2}e^{i\sqrt{\lambda}}{}^{t}\varphi_{+}(\hat{x})+\overline{C(\lambda)}r^{-(n-1)/2}e^{-i\sqrt{\lambda}r}\varphi_{-}(\hat{x})$

,

(1.6)

121
and

$\varphi\pm \mathrm{a}\mathrm{r}\mathrm{e}$related as follows$\varphi_{+}=\hat{S}(\lambda)J\varphi_{-}$,(1.10)

whereandis the scattering matrix for $H$ . The operator
is aspectral representation (generalized Fourier transformation) for $H$ . In fact there are
two types of generalized Fourier transformation, which are related to the spatial
asymptotics of the resolvent in the following way :$(J\varphi)(\omega)=\varphi(-\omega)$$\hat{S}(\lambda)$$\mathcal{F}(\lambda)$$\mathcal{F}_{\pm}(\lambda)$$\mathcal{F}_{\pm}(\lambda)f=\lim_{rarrow\infty}C_{\pm}(\lambda)r^{(n-1)/2}e^{\mp i\sqrt{\lambda}r}(R(\lambda\pm i0)f)(r\cdot)$.

Moreover$\mathcal{F}(\lambda)f=\mathcal{F}_{+}(\lambda)f=J\overline{\mathcal{F}_{-}(\lambda)\overline{f}}$.

The above facts (1.7), (1.9) and (1.10) are thus closely related each other and arise fromfundamental properties of the generalized Fourier transformation associated with $H$ .
1.33-body Schr\"odinger equation. To extend the above results to many-body Schr\"odinger
equations is avery difficult problem. It was shown in [18] that asolution of the TV-body
Schrodinger equation $(H-\lambda)u=0$ vanishes identically if $u\in L^{2,-}$ ’for some $\alpha<1/2$ andif Ais neither the eigenvalue nor in the set of thresholds of $H$ . This is ageneralizationof Sommerfeld-Rellich’s classical result to the many-body problem. To charcterize thesolutions of the border line case is much harder, since it requires adetailed knowledgeof the $N$-body stationary Schr\"odinger equation, which remains unknown in spite of thesuccess of the proof of asymptotic completeness by the time-dependent method [4], [26],[8], [3].
In this paper, we shall study this problem in the case of 3-particle systems in. We
consider 3particles with mass $m_{i}>0$ and position. To remove the motion of thecenter of mass, our Hamiltonian is defined over the space$u$$\mathrm{R}^{3}$$q_{i}\in \mathrm{R}$.

$\mathcal{X}=\{-(q_{1}, q_{2}, q_{3});\sum_{i=1}^{3}m_{i}q_{i}=0\}\simeq \mathrm{R}^{6}$

Let $a=(i, j)$ be apair of particles and , and be the 3rd partcle. The reduced masses,
and the Jacobi-coordinates are defined by$i$$m_{a}$$j$$k$$n_{a}$$\frac{1}{m_{a}}=\frac{1}{m_{i}}+\frac{1}{m_{j}}$$x^{a}=\sqrt{2m_{a}}(q_{i}-q_{j})$,,$\frac{1}{n_{a}}=\frac{1}{m_{i}+m_{j}}+\frac{1}{m_{k}}$,$x_{a}= \sqrt{2n_{a}}(q_{k}-\frac{m_{i}q_{i}+m_{j}q_{j}}{m_{i}+m_{j}})$.

The 3-particle Hamiltonians Ho, $H$ are defined by$H_{0}=-\triangle_{x_{a}}-\triangle_{x^{a}}$where$\triangle_{x_{a}}(\triangle_{x^{a}})$,$H=H_{0}+ \sum_{a}V_{a}(x^{a})$,denotes the Laplacian with respect to the variable$T_{a}=-\triangle_{x_{a}}$,$H^{a}=-\triangle_{x^{a}}+V_{a}(x^{a})$,$x_{a}(x^{a})$. We put122$H_{a}=T_{a}+H^{a}=H_{0}+V_{a}(x^{a})$.

Let$R_{a}(z)=(H_{a}-z)^{-1}$ ,$R(z)=(H-z)^{-1}$ ,Let$\mathcal{T}$.

be the set of thresholds of $H$ , namely$\mathcal{T}$where$R^{a}(z)=(H^{a}-z)^{-1}$$\sigma_{p}(H^{a})$$= \{0\}\cup\bigcup_{a}\sigma_{p}(H^{a})$is the set of eigenvalues of$H^{a}$. Let,$\mathcal{T}’=\mathcal{T}\cup\sigma_{p}(H)$$a( \lambda)=\inf\{\lambda-t;t\in \mathcal{T}’, t<\lambda\}$. We define.

.
if
To get the complete result, we assume that each pair potential decays rapidly. Moreand tofunction on
is assumed to be areal
precisely, each pair potential
satisfy$a(\lambda)=\lambda$$\lambda>0$

Assumption$\partial_{y}^{m}V_{ij}(y)=O(|y|^{-m-\rho})$for some $\rho>5$ . Here$\partial_{y}^{m}$$\mathrm{R}^{3}$$C^{\infty}$$V_{ij}(y)$,(1.11)$\forall m\geq 0$

stands for any differentia tion of order $m$ .

Let us stress that this is the only assumption we impose on our 3-particle systems. We
assume no extra assumptions such as nonexistence of zer0-eigenvlue or zer0-resonances.
One can allow Coulombic singularities for . Namely our results below also hold if
is acompactly, whereis asmooth function satisfying (1.11),, and all the multiple commutators of
supported function satisfying
operators.
extend to boundedand$V_{\dot{l}j}$$V_{ij}=V_{\dot{\iota}j}^{(1)}+V_{\dot{l}j}^{(2)}$$V_{ij}^{(2)}$$V_{\dot{l}j}^{(1)}$$|V_{\dot{l}j}^{(2)}(y)|\leq C|y|^{-1}$$V_{ij}^{(2)}$$A= \frac{1}{2_{\dot{l}}}(x\cdot\nabla+\nabla\cdot x)$

1.4 Main results. Let us summarize our main results in this paper. Let $H=H_{0}+ \sum_{a}V_{a}(x^{a})$be the 3-body Schr\"odinger operator with center of mass removed. Each pair potential is. In
assumed to satisfy (1.11). Let be the set of thresholds for $H$ anddenotes the sum ranging over all pairs of particles and overthe following,. The meaning of the notation
the eigenvalues of the subsystem
is similar to this.
$\mathcal{T}’=\mathcal{T}\cup\sigma_{p}(H)$$\mathcal{T}$$\sum_{a,n}=\Sigma_{a}\Sigma_{n}$$H^{a}=-\triangle_{x^{a}}+V_{a}(x^{a})$THEOREM 1. For$\lambda\in\sigma_{cmt}(H)\backslash \mathcal{T}’$$\mathcal{F}(\lambda)$

having the following properties :
diagonalizes $H$ :(1):$\oplus_{a,n}$, there exists a bounded operator$B$ $arrow L^{2}(S^{5})\oplus\oplus_{a,n}L^{2}(S^{2})$$\mathcal{F}(\lambda)$.

$\mathcal{F}(\lambda)Hf=\lambda \mathcal{F}(\lambda)f$. Then the operator is uniquely extended to a partialby(2) Define
isometry with initial set $H_{ac}(H)=the$ absolutely continuous subspace for $H$ , and final set$(\mathcal{F}f)(\lambda)$$\mathcal{F}(\lambda)f$$\mathcal{F}$$L^{2}((0, \infty);L^{2}(S^{5});\rho_{0}(\lambda)d\lambda)\oplus\oplus_{a,n}L^{2}((\lambda^{a,n}, \infty);L^{2}(S^{2});\rho_{a,n}(\lambda)d\lambda)$,

123$\rho_{0}(\lambda)=\frac{\lambda^{2}}{2}$where,(3) Letthe sense that$\lambda^{a,n}\in\sigma_{p}(H^{a})$$\mathcal{F}_{0}(\lambda)$,$\rho_{a,n}(\lambda)=\frac{1}{2}\sqrt{\lambda-\lambda^{a,n}}$.

,$\mathcal{F}_{a,1}(\lambda)$$\cdots$

be the components of,$\mathcal{F}(\lambda)$,. They are eigenoperators of $H$ in$(H-\lambda)\mathcal{F}_{a,n}(\lambda)^{*}\varphi_{a,n}=0$$(H-\lambda)\mathcal{F}_{0}(\lambda)^{*}\varphi_{0}=0$.
,hold for$f\in
H_{ac}(H)$, the following inversion formula holds:(4) For$\varphi_{0}\in L^{2}(S^{5})$$\varphi_{a,n}\in L^{2}(S^{2})$.

$f= \int_{0}^{\infty}\mathcal{F}_{0}(\lambda)^{*}(\mathcal{F}_{0}f)(\lambda)\rho_{0}(\lambda)d\lambda+\sum_{a,n}\int_{\lambda^{a,n}}^{\infty}\mathcal{F}_{a,n}(\lambda)^{*}(\mathcal{F}_{a,n}f)(\lambda)\rho_{a,n}(\lambda)d\lambda$, the boundary value of the resolvent of
THEOREM 2. For $f\in B$ and A$H$ admits the following asymptotic expansion in the senseof (1.5)$\in\sigma_{cmt}(H)\backslash \mathcal{T}’$$R(\lambda+i0)f$$C( \lambda)\frac{e^{i\sqrt{\lambda}r}}{r^{5/2}}\mathcal{F}_{0}(\lambda)f$$\simeq$$+$,$\sum_{a,n}C_{a,n}(\lambda)\frac{e^{i\sqrt{\lambda-\lambda^{a,n}}r_{a}}}{r_{a}}\mathcal{F}_{a,n}(\lambda)f(\omega_{a})\otimes\varphi^{a,n}(x^{a})$$C(\lambda)=\sqrt{\frac{\pi}{2}}e^{-3\pi i/4}(\lambda_{+})^{3/4}$,$C_{a,n}(\lambda)=\sqrt{\frac{\pi}{2}}h(\lambda-\lambda^{a,n})$where $k_{+}= \max\{k, 0\}$ and $h(t)=1$ if $t\geq 0$ , $h(t)=0$ if $t<0$ , and.
associated with the eigenvalueof$\varphi^{a,n}$,

is the eigenvector$\lambda^{a,n}$$H^{a}$

THEOREM 3. Let
only if is written as$\lambda\in\sigma_{cmt}(H)\backslash \mathcal{T}’$

. Let

$u$

satisfy $(H-\lambda)u=0$ . Then $u\in B^{*}$ if and$u$$u=\mathcal{F}(\lambda)^{*}\varphi$

for some$\varphi\in L^{2}(S^{5})\oplus\oplus_{a,n}L^{2}(S^{2})$THEOREM 4Let
the asymptotic expansion.

$\lambda\in\sigma_{cmt}(H)\backslash \mathcal{T}’$. Let $u\in B^{*}$ satisfy $(H-\lambda)u=0$ . Then tz admits$u\simeq C(\lambda)r^{-5/2}e^{i\sqrt{\lambda}r}\varphi_{0}^{(+)}(\hat{x})+\overline{C(\lambda)}r^{-5/2}e^{-i\sqrt{\lambda}r}\varphi_{0}^{(-)}(\hat{x})$$+ \sum_{a,n}[C_{a,n}(\lambda)r_{a}^{-1}e^{i\sqrt{\lambda-\lambda^{a,n}}r_{a}}\varphi_{a,n}^{(+)}(\omega_{a})\otimes\varphi^{a,n}(.x^{a})$$+\overline{C_{a,n}(\lambda)}r_{a}^{-1}e^{-i\sqrt{\lambda-\lambda^{a,n}}r_{a}}\varphi_{a,n}^{(-)}(\omega_{a})\otimes\varphi^{a,n}(x^{a})]$in the sense of (1.5), where$C(\lambda)=(2\pi)^{-1/2}e^{-5\pi i/4}(\lambda_{+})^{-5/4}$,$C_{a,n}(\lambda)=(2\pi)^{-1/2}e^{-\pi i/2}((\lambda-\lambda^{a,n})_{+})^{-1/2}$.

(1.12)

124
Let$\varphi^{(\pm)}={}^{t}(\varphi_{0}^{(\pm)}, \varphi_{a,1}^{(\pm)}, \cdots)$. Then$\varphi^{(+)}=\hat{\mathrm{S}}(\lambda)\mathrm{J}\varphi^{(-)}$

where$\hat{\mathrm{S}}(\lambda)$

is the -matrix and$S$$\mathrm{J}$: (to$(\theta)$,$\mathrm{J}$(1.13),

is the reflection$\varphi_{a,1}(\omega_{a})$,$\cdots$)$arrow(\varphi_{0}(-\theta), \varphi_{a,1}(-\omega_{a}),$ $\cdots)$, there exist a unique solution
For any
expansion (1.12) is valid.
$\varphi^{(-)}$$u$.

of $(H-\lambda)u=0$ and(1.14)$\varphi^{(+)}$

for which the

Theorem 2is afundamental result concerning the behavior at infinity of solutions tostationary Schr\"odinger equations. We shall discuss its other applications elsewhere.

1.5 Related works. Let us consider the $N$ -body Schr\"odinger operator $H=H_{0}+\Sigma_{1\leq i<j\leq N}$. As is inferred from the 2-body case, the above problem boils down to theasymptotic expansion of the resolvent $(H-\lambda\mp i0)^{-1}$ at infinity, which leads to theconstruction of generalized eigenfunctions for $H$ and to the properties of -matrices. All
the difficulties of the $N$-body problem arise ffom the directions $\{q_{i}=q_{j}\}$ , along which
does not decay. Let us call them singular directions in thisthe pair potential
paper.
Most of the study of the stationary $N$-particle Schr\"odinger equation has been done
outside the singular directions. The asymptotic expansion of the resolvent outside thesingular directions (free region) was obtained by Herbst-Skibsted [14].
In aseries of papers [9], [10], [11], [27], [28] Hassell and Vasy continued the study ofgeneralized eigenfunction for $H$ and -matrices in the free region. As for the behavioraround the singular directions, Vasy [29] studied it by projecting the solution of $(H-\lambda)u=$
onto the bound states of subsystems and investigating the spatial asymptotics in the free
region for the subsystem. Let us also mention the work of Vasy [30] on the propagation of
singularirties for the -matrix, the underlying idea of which is to look at the micr0-local
behavior of the resolvent at infinity.
The works [15], [16] studied directly the scattering matrix and generalized eigenfunctions around the singular directions. Since they lean heavily upon the spectral property
near the zero energy of subsystems, they are restricted to the 3-body case.
$V_{ij}(q_{i}-q_{j})$$\mathrm{S}$$V_{j}\dot,(q_{\dot{l}} -q_{j})$$\mathrm{S}$$0$$\mathrm{S}$

1.6 Methods. The present paper is essentially acontinuation of our previous works [15],[16], which are based on the micr0-local resolvent estimates for $N$ -body Schr\"odingcr operators. The germ of the idea of the generalized Fourier transformation for 3-particle
system has already been given in [16]. However we need two optimal results to overcome
new difficulties in the many-body problem.
The first one is the Agmon-H\"ormander space , , which is not only optimal for the
restriction of the Fourier transform on the sphere, but also appropriate to deal with themulti-channel property of the many-body problem. We encounter two types of spherical. The former is dominant in the free region,andscattering waves,enableswhile the latter is dominant near the singular direction $\{x^{a}=0\}$ . The space
us to show the orthogonality of these two waves and hence the expansion of the resolvent$B$$r^{-5/2}e^{i\sqrt{\lambda}r}$$B^{*}$$r_{a}^{-1}e^{i\sqrt{\lambda-\lambda^{a,n}}r_{a}}$$B^{*}$

125
ocal version of Yafaev’s resolThe second new tool employed in this paper is avent estimates concerning the spherical part of the radiation condition [32]. Our Theorem
3.5 is amany-body counter part of Agmon-Hormander’s result [2], Theorem 7.4, and is
crucial to construct the generalized Fourier transformation near the singular directions.
$\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}\ovalbox{\tt\small REJECT}$

1.6 Generalized eigenfunctions with 2-cluster incoming state. Finally, we shall discuss theasymptotic expansion of generalized eigenfunction for $H$ . From the practical point ofview, in the real scattering experiment, the most important case is the one in which theinitial state is of 2-cluster. Suppose in the remote past the pair $a=(i,j)$ forms abound. Then the generalized eigenfunctionand eigenstate
state with energy
is written as$\lambda^{a,n}<0$$\varphi^{a,n}(x^{a})$$\Psi(x, \lambda, \omega_{a})$$\Psi(x, \lambda,\omega_{a})=e^{i\sqrt{\lambda-\lambda^{an}|}\omega_{a}\cdot x_{a}}\varphi^{a,n}(x^{a})-v$(1.16),$v=R(\lambda+i0)f$ ,(1.16)$f= \sum_{c\neq a}V_{c}(x^{c})\varphi^{a,n}(x^{a})e^{i\sqrt{\lambda-\lambda^{a,n}}\omega_{a}\cdot x_{a}}$.

(1.17)

In our previous work [16], we derived asymptotic expansions of at infinity. However,
the results were not satisfactory in that we have separated 3-cluster scattering and 2cluster scattering. Moreover in Theorem 1.3 of [16], we multiplied atechnical localization), although it was removed in [17], Theorem
to (we wrote it asfactorpaper, we can derive amore transparent
present
analysisof the6.6. By virtue of theasymptotic expansion.
$v$$\psi_{b}(D_{x_{b}})$$\psi_{\beta}(D_{x_{\beta}})$$v$

THEOREM 5Let
$v$

$v$

$\simeq$$+$be as in (1.16) and$\alpha=(a, \lambda^{a,n}, \varphi^{a,n})$. Then in the sense of (1.5)$C_{0}(\lambda)r^{-5/2}e^{i\sqrt{\lambda}r}\hat{S}_{0\alpha}(\lambda;\hat{x},\omega_{a})$,$\sum_{\beta}C_{\beta}(\lambda)r_{b}^{-1}e^{i\sqrt{\lambda-\lambda^{b,m}}r_{b}}A_{\beta\alpha}(\lambda;\theta_{b}, \omega_{a})\otimes\varphi^{a,n}(x^{a})$where $r=|x|,\hat{x}=x/r$ , $r_{b}=|x_{b}|$ ,$\theta_{b}=x_{b}/r_{b}$,$C_{0}(\lambda)=e^{-\pi i/4}2\pi\lambda^{-1/4}(\lambda-\lambda^{a,n})^{-1/4}$,$C_{\beta}(\lambda)=2\pi i(\lambda-\lambda^{a,n})^{-1/4}(\lambda-\lambda^{b,m})^{-1/4}$,

is the scattering amplitude associated with the scatter $.ng$ process, inand.
with eigenvalue
which after the collision the pair takes the bound state$A_{\beta\alpha}(\lambda, \theta_{b}, \omega_{a})$$b$$\varphi^{b,m}(x^{b})$$\lambda^{b,m}$

If the initial state is of 3-cluster, the behavior of the generalized eigenfunction is much
more complicated. In fact, Hassell [9] derived an asymptotic expansion ofaway from singular directions, which contains in addition to the expected termwith $0<p<5/2$ .
extra terms like$\Psi(x, \lambda, \theta)-$$e^{i\sqrt{\lambda}\theta x}$$r^{-5/2}e^{i\sqrt{\lambda}r}a_{0}(\hat{x})$$r^{-p}e^{i\phi(\hat{x})r}a(\hat{x})$126
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