数理解析研究所講究録
第 1902 巻 2014 年 21-28

21

On multidimensional inverse scattering intime-dependent electric fields$\dagger$

Atsuhide ISHIDA
Faculty of Economics, Otemon Gakuin University
1-15 Nishiai 2-chome, Ibaraki, Osaka 567-8502, Japan-mail: a-ishida@res.otemon.ac.jp$E$1

Introduction

Throughout this paper, we assume the spatial dimension $d\geq 2$ . We report one of the inverse
scattering problems for quantum systems in a time-dependent electric field, whichwas obtained in Adachi-Fujiwara-I [1]. By Enss-Weder time-dependent method [5], we canshow that the high speed limit of the scattering operator determines uniquely the potential
belonging to the wider class than the classes given by the previous work in Adachi-Maehara[3], Adachi-Kamada-Kazuno-Toratani [2], Nicoleau [10] and Fujiwara [6].
$E(t)\in \mathbb{R}^{d}$$V$

The free and full Hamiltonians under the consideration are given by$H_{0}(t)=p^{2}/2-E(t)\cdot x, H(t)=H_{0}(t)+V$(1.1)

acting as the self-adjoint operators on, where $p=-i\nabla_{x}$ is the momentum, $E(t)$ is
the time-dependent electric field and the interaction potential is real-valued multiplicative
operator. $E(t)$ andsatisfy following assumptions.
$L^{2}(\mathbb{R}^{d})$$V$$V=V^{vs}+V^{s}+V^{1}\in \mathscr{V}^{vs}+\mathscr{V}_{\mu,\alpha_{\mu}}^{s}+\mathscr{V}_{\mu,\gamma_{\mu}}^{1}$

Assumption 1.1. The time-dependent electric field$E(t)\in \mathbb{R}^{d}$is represented as$E(t)=E_{0}(1+|t|)^{-\mu}+E_{1}(t)$ ,where $0\leq\mu<1,$$E_{0}\in \mathbb{R}^{d}\backslash \{O\}$and$E_{1}(t)\in C(\mathbb{R}, \mathbb{R}^{d})$such that$| \int_{0}^{t}\int_{0}^{s}E_{1}(\tau)d\tau ds|\leq C\max\{|t|, |t|^{2-\mu_{1}}\}$

with $\mu<\mu_{1}\leq 1.$$\dagger$(1.2)

Joint work with Tadayoshi ADACHI (Kobe University) and Yuko FUJIWARA

(1.3)

22
Roughly speaking about the perturbation partas follows:
for some $\mu_{2}>\mu$ and take$|t|)^{-\mu_{2}}$$E_{1}(t)$, we assume that $|E_{1}(t)|\leq C(1+$$\mu_{1}$(1.4)$\{\begin{array}{ll}\mu_{1}=\mu_{2} \mu<\mu_{2}<1\mu<\mu_{1}<\mu_{2} \mu_{2}=1\mu_{1}=1 \mu_{2}>1.\end{array}$Such $E(t)$ was first dealt with in Adachi-Kamada-Kazuno-Toratani [2]. For brevity’s sake,we suppose that $E_{0}=e_{1}=(1,0, \ldots, 0)\in \mathbb{R}^{d}.$is satisfying
Assumption 1.2.
is the class of real-valued multiplicative operators
and a regular partis decomposed into the sum of a singular partthat.
and satisfies
is compactly supported, belongs to
and
that
in
are
all
boundedits
derivatives
that
and
first
satisfies$V^{vs}$$\mathscr{V}^{vs}$$V^{vs}$$V_{2}^{vs}.$$V_{1}^{vs}$$|\nabla V_{1}^{vs}|\in L^{q_{2}}(\mathbb{R}^{d})$$L^{q_{1}}(\mathbb{R}^{d})$$V_{1}^{vs}$$V_{2}^{vs}\in C^{1}(\mathbb{R}^{d})$$\mathbb{R}^{d}$$V_{2}^{vs}$$\int_{0}^{\infty}\Vert F(|x|\geq R)V_{2}^{vs}(x)\Vert_{\mathscr{D}(L^{2})}dR<\infty$Here$q_{1}$

satisfies that $q_{1}>d/2$ and$q_{1}\geq 2,$$q_{2}$(1.5).

satisfies(1.6)$\{\begin{array}{l}1/q_{2}=1/(2q_{1})+2/d d\geq 51/q_{2}<1/(2q_{1})+1/2 d=41/q_{2}=1/(2q_{1})+1/2 d\leq 3,\end{array}$and $F(|x|\geq R)$ is the characteristic function of $\{x\in \mathbb{R}^{d}||x|\geq R\}.$is the class of real-valued multiplicative operatorswith someand satisfies
belongs tothat$\alpha_{\mu}>0$$\mathscr{V}_{\mu,\alpha_{\mu}}^{s}$$V^{s}$$V^{s}$

is satisfying$C^{1}(\mathbb{R}^{d})$$|V^{s}(x)|\leq C\langle x\rangle^{-\gamma}, |\partial_{x}^{\beta}V^{s}(x)|\leq C_{\beta}\langle x\rangle^{-1-\alpha}, |\beta|=1$(1.7)

with some and such that $1/(2-\mu)<\gamma\leq 1$ andFinally,
with some $\gamma_{\mu}\geq 1/(2(2-\mu))$ is the class of real-valued multiplicative operand satisfies
belongs toators
is satisfying that$\alpha_{\mu}<\alpha\leq\gamma.$$\alpha$$\gamma$$\mathscr{V}_{\mu,\gamma_{\mu}}^{1}$$V^{1}$$V^{1}$$C^{2}(\mathbb{R}^{d})$$|\partial_{x}^{\beta}V^{1}(x)|\leq C\langle x\rangle^{-\gamma_{D}-|\beta|/(2-\mu)}, |\beta|\leq 2$

with some$\gamma_{D}$

,

(1.8)

such that $\gamma_{\mu}<\gamma_{D}\leq 1/(2-\mu)$ .

We note that one can obtain$\int_{0}^{\infty}\Vert F(|x|\geq R)V^{vs}(x)\langle p\rangle^{-2}\Vert_{\mathscr{R}(L^{2})}dR<\infty$(1.9)

by this assumption and it is equivalent to$\int_{0}^{\infty}\Vert V^{vs}(x)\langle p\rangle^{-2}F(|x|\geq R)\Vert_{\mathscr{R}(L^{2})}dR<\infty$(1.10)

23
becauseis a multiplicative operator (see e.g. Reed-Simon [11]).
, we also note that by virtue ofAs for the class, we can treat an oscillation
part. For example, the following function belongs to:$V^{vs}$$\alpha\leq\gamma$$\mathscr{V}_{\mu,\alpha_{\mu}}^{s}$$\mathscr{V}_{\mu,\alpha_{\mu}}^{s}$$V^{s}(x)=\langle x\rangle^{-\gamma}\cos\langle x\rangle^{\gamma-\alpha}$

In fact, we can verify easily thatsome $C>0.$2.

(1.11)$|\nabla_{x}V^{s}(x)|\leq C(\langle x\rangle^{-1-\gamma}+\langle x\rangle^{-1-\alpha})\leq C\langle x\rangle^{-1-\alpha}$

holds with

Results

We first state the case where $V^{1}=0$ . Then we can see the wave operators(2.1)$W^{\pm}= s-\lim_{tarrow\pm\infty}U(t, 0)^{*}U_{0}(t, 0)$

exist as this fact was shown in Adachi-Kamada-Kazuno-Toratani [2], where we denote thepropagators generated by
and $H(t)$ as $U_{0}(t, 0)$ and $U(t, 0)$ . The existence and uniqueness of these propagators are guaranteed by virtue of Yajima [14]. The scattering operator$S=S(V)$ is defined by$S=(W^{+})^{*}W^{-}$(2.2)$H_{0}(t)$The following obtained in [1] is one of those which we would like to report in this paper.

Theorem 2.1. (Adachi-Fujiwara-I [1]) Put$\tilde{\alpha}_{\mu}=\{$

Let$V_{1},V_{2}\in \mathscr{V}^{vs}+\mathscr{V}_{\mu_{)}^{S}\overline{\alpha}_{\mu}}$$\frac{}{}\frac{7-3\mu-\sqrt{(1-\mu)(17-9\mu)}}{2(2-\mu)1+\mu 4(2-\mu)}$(2.3)$0\leq\mu\leq 1/21/2<\mu<1.$. If $S(V_{1})=S(V_{2})$ , then $V_{1}=V_{2}.$In the case where $E(t)\equiv E_{0}$ , that is, the case of the Stark eﬀect, this theorem was
first proved by Weder [12] under the condition
and the additional assumption$\gamma>3/4$ . However, as it is well-known, the short-range condition onunder the Stark eﬀect
is $\gamma>1/2$ . Later Nicoleau [9] proved this theorem for real-valued
satisfying
with $\gamma>1/2$ , under the spatial dimension $d\geq 3$ . After that, thistheorem was obtained by Adachi-Maehara [3] for. In our case where $\mu=0,$
substitute $\mu=0$ in . We have$V^{s}\in \mathscr{V}_{0^{s}0}$$V$$V\in C^{\infty}(\mathbb{R}^{d})$$|\partial_{x}^{\beta}V(x)|\leq C_{\beta}\langle x\rangle^{-\gamma-|\beta|}$$V^{s}\in \mathscr{V}_{0,1/2}^{s}$$\tilde{\alpha}_{\mu}$$\tilde{\alpha}_{0}=\frac{7-3\mu-\sqrt{(1-\mu)(17-9\mu)}}{4(2-\mu)} =\frac{7-\sqrt{17}}{8}<\frac{1}{2}$

.

(2.4)

$\mu=0$If $a<b$ , then$\mathscr{V}_{\mu,b}^{s}\subsetneq \mathscr{V}_{\mu,a}^{s}$. Therefore this implies$\mathscr{V}_{0,1/2}^{s}\subsetneq \mathscr{V}_{0,\tilde{\alpha}_{0}}^{s}$

.

(2.5)

24
, the result corresponding to TheIn the time-dependent case where $0<\mu<1$ andorem 2.1 was also obtained by Adachi-Kamada-Kazuno-Toratani [2] under the assumption
and this implies that above. We can verify that
that
result is finer than the previous one:
$E_{1}(t)\not\equiv O$$\tilde{\alpha}_{\mu}<1/(2-\mu)$$V\in \mathscr{V}^{vs}+\mathscr{V}_{\mu,1/(2-\mu)}^{s}$.

$\mathscr{V}_{\mu,1/(2-\mu)}^{s}\subsetneq \mathscr{V}_{\mu,\overline{\alpha}_{\mu}}^{s}$(2.6), the Dollard-type modified wave operators. If
We next state the case where
due to White [13] (see also Adachi-Tamura [4] and Jensen-Yajima [8])$V^{1}\neq 0$$V^{1}\in \mathscr{V}_{\mu,\gamma_{\mu}}^{1}$(2.7)$W_{D}^{\pm}=tarrow\pm\infty s-hmU(t, 0)^{*}U_{0}(t, 0)M_{D}(t) , M_{D}(t)=e^{-i\int_{0}^{t}V^{1}(p\tau+c(\tau))d\tau}$

can exist by virtue of the condition $\gamma_{D}>1/(2(2-\mu))$ (see [2]), where we put$c(t)= \int_{0}^{t}b(\tau)d\tau, b(t)=\int_{0}^{t}E(\tau)d\tau$

.

(2.8)

Then the Dollard-type modified scattering operator $S_{D}=S_{D}(V^{1}, V^{vs}+V^{s})$ is defined by$S_{D}=(W_{D}^{+})^{*}W_{D}^{-}$

(2.9)

.

Then we also report the following result.
Theorem 2.2. (Adachi-Kjiwara-I [1]) Suppose that a given$\tilde{\gamma}_{\mu}=\frac{1}{2(2-\mu)}+\frac{1-\mu}{4(2-\mu)}$$V^{1}$

satisfies.

$V^{1}\in \mathscr{V}_{\mu,\overline{\gamma}_{\mu}}^{1}$with(2.10)

Put$\tilde{\alpha}_{\mu,D}=\{\begin{array}{ll}\frac{13-5\mu-\sqrt{(1-\mu)(41-25\mu)}}{8(2-\mu)} 0\leq\mu\leq 5/7\frac{1+\mu}{2(2-\mu)} 5/7<\mu<1.\end{array}$(2.11). If $S_{D}(V^{1}, V_{1})=S(V^{1}, V_{2})$ , then $V_{1}=V_{2}$ . Moreover, any one ofLetdetermines uniquely the total potential $V.$
the Dollard-type modified scattering operators$V_{1},V_{2}\in \mathscr{V}^{vs}+\mathscr{V}_{\mu,\overline{\alpha}_{\mu,D}}^{s}$$S_{D}$In the case where $0<\mu<1$ andthis theorem under the condition that

$E_{1}(t)\not\equiv O$, Adachi-Kamada-Kazuno-Toratani [2] proved$V^{1}\in \mathscr{V}_{\mu_{:}\hat{\gamma}_{\mu}}^{1}, V_{1}, V_{2}\in \mathscr{V}^{vs}+\mathscr{V}_{\mu,1/(2-\mu)}^{s}$(2.12)

with $\hat{\mu}=(7-\sqrt{3}-\sqrt{60-22\sqrt{3}})/4$ and$\hat{\gamma}_{\mu}=\{\begin{array}{ll}\frac{-\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{2(1-\mu)^{2}}{1+\mu}}}{2-\mu} 0<\mu\leq\hat{\mu}-\frac{3-\mu}{8}+\sqrt{\frac{(3-\mu)^{2}}{64}+\frac{2\mu^{2}-7\mu+7}{4(2-\mu)^{2}}} \hat{\mu}<\mu<1.\end{array}$(2.13)

25
Computing straightforwardly, we can see$\tilde{\alpha}_{\mu,D}<1/(2-\mu)$and$\mathscr{V}_{\mu,1/(2-\mu)}^{S}\subsetneq \mathscr{V}_{\mu,\overline{\alpha}_{\mu,D}}^{S}, \mathscr{V}_{\mu,\hat{\gamma}_{\mu}}^{1}\subsetneq \mathscr{V}_{\mu,\overline{\gamma}_{\mu}}^{1}$$\tilde{\gamma}_{\mu}<\hat{\gamma}_{\mu}$. We thus obtain.

(2.14)

In particular, there was no result for the case where $\mu=$ O. Here we emphasize that if$5/7\leq\mu<1$ , thenholds, although if $0\leq\mu<5/7$ , thenholds.
$\tilde{\alpha}_{\mu,D}=\tilde{\alpha}_{\mu}$$\tilde{\alpha}_{\mu,\cdot D}>\tilde{\alpha}_{\mu}$Remark 2.3. We assume that, that is,$E(t)\in C(\mathbb{R}, \mathbb{R}^{d})$

is

$T$

-periodic in time with non-zero mean

$E_{0}$$E_{0}= \int_{0}^{T}E(\tau)d\tau/T\neq 0$(2.15),

which was treated by Nicoleau [10] and Fujiwara [6]. In this case, the method in the proofsof Theorems 2.1 and 2.2 does work well also, because we have$|b(t)-tE_{0}| \leq\int_{0}^{T}|E(\tau)-E_{0}|d\tau$(2.16),$|c(t)-t^{2}E_{0}/2| \leq\int_{0}^{|t|}|b(\tau)-\tau E_{0}|d\tau\leq C|t|$,(2.17)

with $C= \int_{0}^{T}|E(\tau)-E_{0}|d\tau$ by the periodicity of $E(t)$ . (2.17) implies $\mu=0$ in (1.2) andin (1.3).

$\mu_{1}=1$

By virtue of this fact, we can obtain an improvement of the results of [10] and [6].
Theorem 2.4. Suppose that
Then the followings hold.

1. Let$V_{1},V_{2}\in \mathscr{V}^{vs}+\mathscr{V}_{0,\overline{\alpha}_{0}}^{s}$$E(t)\in C(\mathbb{R}, \mathbb{R}^{d})$is $T$ -periodic in time with non-zero mean

$E_{0}.$. If $S(V_{1})=S(V_{2})$ , then $V_{1}=V_{2}.$2. Suppose that a given. Let. If $S_{D}(V^{1}, V_{1})=$
satisfies, then $V_{1}=V_{2}$ . Moreover, any one of the Dollard-type modified scattering
operators
determines uniquely the total potential $V.$$V^{1}$$V^{1}\in \mathscr{V}_{0,\overline{\gamma}0}^{1}$$V_{1},V_{2}\in \mathscr{V}^{vs}+\mathscr{V}_{0,\overline{\alpha}_{0,D}}^{s}$$S_{D}(V^{1}, V_{2})$$S_{D}$

Nicoleau [10] proved the uniqueness assuming thatwith $\gamma>1/2$
forand the additional condition $d\geq 3$ . Fujiwara [6] assumed that $V\in$. These two results did not treat the long-range potentials.
$|\partial_{x}^{\beta}V(x)|\leq C_{\beta}\langle x\rangle^{-\gamma-|\beta|}$$V\in C^{\infty}(\mathbb{R})^{d}$$\mathscr{V}^{vs}+\mathscr{V}_{0,1/2}^{s}$3

Short-range CaseBy virtue of Theorem 3.1 below and the Plancherel formula associated with the Radontransform (see Helgason [7]), Theorem 2.1 can be shown in the quite same way as in the
proof of Theorem 1.2 in [12] (see also Enss-Weder [5]).

26
be given such that
Theorem 3.1. (Reconstruction Formula [1]) Let$v=|v|\hat{v}$Put. Let $\eta>0$ be given, andbe such that
. Put
. Let
supp
with supp, whereis the same as in Theorem 2.1 andandis the Fouriertransformation. Then$\hat{v}\in \mathbb{R}^{d}$$\Phi_{0},$$\mathscr{F}\Psi_{0}\subset\{\xi\in \mathbb{R}^{d}||\xi|\leq\eta\}$$\mathscr{F}\Phi_{0},$$V^{vs}\in \mathscr{V}^{vs}$$\Psi_{0}\in \mathscr{S}(\mathbb{R}^{d})$$|\hat{v}\cdot e_{1}|<1.$$\mathscr{F}\Phi_{0},$$\mathscr{F}\Psi_{0}\in C_{0}^{\infty}(\mathbb{R}^{d})$$\Phi_{v}=e^{iv\cdot x}\Phi_{0},$ $\Psi_{v}=e^{iv\cdot x}\Psi_{0}$$\mathscr{F}$$\tilde{\alpha}_{\mu}$$V^{s}\in \mathscr{V}_{\mu,\overline{\alpha}_{\mu}}^{s}$$|v|(i[S,p_{j}] \Phi_{v}, \Psi_{v})=\int_{-\infty}^{\infty}((V^{vs}(x+\hat{v}t)p_{j}\Phi_{0}, \Psi_{0})-(V^{vs}(x+\hat{v}t)\Phi_{0},p_{j}\Psi_{0})$(3.1)$+(i(\partial_{x_{j}}V^{s})(x+\hat{v}t)\Phi_{0}, \Psi_{0}))dt+o(1)$holds as$|v|arrow\infty$for$1\leq j\leq d.$

To prove Theorem 3.1, the following estimate is the key.

Proposition 3.2. Let

$v$

and

$\Phi_{v}$be as in Theorem 3.1 and$\epsilon>0$. Put$\Theta(\alpha)=\{\begin{array}{ll}\alpha+\frac{(\alpha-\mu)(1-\alpha)}{(1-\mu)(2-\alpha)} \alpha>\mu\alpha-\frac{\mu-\alpha}{1-\mu} \mu/(2-\mu)<\alpha\leq\mu.\end{array}$(3.2)$\int_{-\infty}^{\infty}\Vert(V^{s}(x)-V^{s}(vt+c(t)))U_{0}(t, 0)\Phi_{v}\Vert dt=O(|v|^{-\Theta(\alpha)+\epsilon})$

(3.3)

Then

holds as$|v|arrow\infty$for$V^{s}\in \mathscr{V}_{\mu,\mu/(2-\mu)}^{s}.$In Adachi-Maehara [3], the corresponding estimate to this proposition was(3.4)$\int_{-\infty}^{\infty}\Vert(V^{s}(x)-V^{s}(vt+c(t)))U_{0}(t, 0)\Phi_{v}\Vert dt=O(|v|^{-\alpha})$(see Lemma 2.2 in [3]). When we denote the error term of (3.1) by $R(v)$ , $\lim_{|v|arrow\infty}R(v)=0$
is equivalent to $2(-\alpha)+1<$ O. Therefore $\alpha>1/2$ was required. On the other hand, in
Adachi-Kamada-Kazuno-Toratani [2], the corresponding one was$\int_{-\infty}^{\infty}\Vert(V^{s}(x)-V^{s}(vt+c(t)))U_{0}(t, 0)\Phi_{v}\Vert dt=O(|v|^{-\rho})$

,

(3.5)

where
(3.6)

$\rho=\frac{(2-\mu)\alpha-\mu}{2(1-\mu)}$(see Lemma 3.4 in [2]) and $\lim_{|v|arrow\infty}R(v)=0$ is equivalent to $2(-\rho)+1<$ O. Solving thisinequality for , we see that $\alpha>1/(2-\mu)$ was required. In our estimate,comesfrom the inequality $2(-\Theta(\alpha))+1<0$ which is equivalent to $\lim_{|v|arrow\infty}R(v)=0.$$\alpha$$\alpha>\tilde{\alpha}_{\mu}$27
4Long-range CaseIn the case where $V^{1}\neq 0$ , the reconstruction formula is represented as follows, which also
yields the proof of Theorem 2.2.

Theorem 4.1. (Reconstruction Formula [1]) Let
be given such that
Put $v=|v|\hat{v}$ . Let $\eta>0$ be given, andbe such that
suppwith supp
. Let
. Put
, whereare the same as in Theorem 2.2.
and
and
Then
$\hat{v}\in \mathbb{R}^{d}$$\Phi_{0},$$V^{vs}\in \mathscr{V}^{vs},$$\mathscr{F}\Phi_{0},$$V^{1}\in \mathscr{V}_{\mu,\overline{\gamma}_{\mu}}^{1}$$V^{S}\in \mathscr{V}_{\mu,\overline{\alpha}_{\mu,D}}^{S}$$\tilde{\alpha}_{\mu,D}$$\mathscr{F}\Psi_{0}\in C_{0}^{\infty}(\mathbb{R}^{d})$$\Phi_{v}=e^{ivx}\Phi_{0},$ $\Psi_{v}=e^{ivx}\Psi_{0}$$\mathscr{F}\Psi_{0}\subset\{\xi\in \mathbb{R}^{d}||\xi|\leq\eta\}$$\mathscr{F}\Phi_{0},$$|\hat{v}\cdot e_{1}|<1.$$\Psi_{0}\in \mathscr{S}(\mathbb{R}^{d})$$\tilde{\gamma}_{\mu}$$|v|(i[S_{D},p_{j}] \Phi_{v}, \Psi_{v})=\int_{-\infty}^{\infty}((V^{vs}(x+\hat{v}t)p_{j}\Phi_{0}, \Psi_{0})-(V^{vs}(x+\hat{v}t)\Phi_{0},p_{j}\Psi_{0})$(4.1)$+(i(\partial_{x_{j}}V^{s})(x+\hat{v}t)\Phi_{0}, \Psi_{0})+(i(\partial_{x_{j}}V^{1})(x+\hat{v}t)\Phi_{0}, \Psi_{0}))dt+o(1)$holds as$|v|arrow\infty$for$1\leq j\leq d.$

To prove Theorem 4.1, the following estimate is the key.
Proposition 4.2. Let

$v$

and

$\Phi_{v}$be as in Theorem 4.1,$\epsilon>0$

and$V^{1}\in \mathscr{V}_{\mu,1/(2(2-\mu))}^{1}$. Put$\Theta_{D}(\gamma_{D})=\{\begin{array}{ll}1 \gamma_{D}>1/2\frac{2\gamma_{D}(2-\mu)-1}{1-\mu} \gamma_{D}\leq 1/2.\end{array}$(4.2)$\int_{-\infty}^{\infty}\Vert(V^{1}(x)-V^{1}(t(p-b(t))+c(t)))U_{D}(t)\Phi_{v}\Vert dt=O(|v|^{-\Theta_{D}(\gamma_{D})+\epsilon})$

(4.3)

Then

holds as$|v|arrow\infty$, where $U_{D}(t)=U_{0}(t, 0)M_{D}(t)$ and $M_{D}(t)$ is the same as in (2.7).

In Adachi-Kamada-Kazuno-Toratani [2], the corresponding estimate to this proposition

was$\int_{-\infty}^{\infty}\Vert(V^{1}(x)-V^{1}(t(p-b(t))+c(t)))U_{D}(t)\Phi_{v}\Vert dt=O(|v|^{-\rho\iota})$

,

(4.4)

where
$\rho_{1}=\frac{(1-\sigma_{\kappa})(\tilde{\gamma}_{D}+2-\mu)}{(1-\mu)(\sigma_{\kappa}(\tilde{\gamma}_{D}+2-\mu)-1+\tilde{\gamma}_{D})}$

(4.5)

with
and $\sigma_{\kappa}=1-\kappa(1-\mu)/(2-\mu)$ for $0<\kappa<1$ (see Lema 4.5 in[2]). When we denote the error term of (4.1) by $R_{D}(v)$ , $\lim_{|v|arrow\infty}R_{D}(v)=0$ is equivalent
to $2(-\rho_{1})+1<$ O. Therefore
was required. In our case, $\lim_{|v|arrow\infty}R_{D}(v)=0$ isequivalent to $2(-\Theta_{D}(\gamma_{D}))+1<0$ andcomes from this inequality.
$\tilde{\gamma}_{D}=(2-\mu)\gamma_{D}$$\gamma_{D}>\hat{\gamma}_{\mu}$$\gamma_{D}>\tilde{\gamma}_{\mu}$28
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