数理解析研究所講究録 1212 巻 2001 年 18-31

18

Semiclassical behavior of the scattering phase near acritical value of the potential
Setsuro FUJII\’E
Mathematical Institute, Tohoku University
e-mail : fujiie@math tohoku.ac.jp

藤家雪朗
東北大

・

理

Thierry RAMOND
D\’epartement de Math\’elnatiques, Universite’ Paris Sud(UMR CNRS 8628)
e-mail: thierry.ramond@math.u-psud.fr

0

Introduction

We study the asymptotic behavior in the semiclassical limit of the scattering phase andthe time delay for the one-dimensional Schr\"odinger equation(0.1)$-h^{2} \frac{d^{2}u}{dx^{2}}+V(x)u=Eu$

when the energy is near acritical value of the potential $V(x)$ . We suppose here thatof the potential and consider two cases where $V(x)$the critical value is the maximum
attains the maximum at one point (case I) and two points (case ). We assume moreoverthat these maxima are non-degenerate, i.e. the radii of curvature are finite.
We will show the following facts. Near the critical value , the leading term of theasymptotic expansion as tends to 0of the time delay function (the derivative withrespect to the energy of the scattering phase) is logarithmic in both cases. Moreover its
coefficient is, in case , the radius of curvature of the barrier top and in case , it is theaverage of the radii of curvature of the two barrier tops, plus acontribution frompotential well. This contribution from the potential well is an oscillating functioxi withwavelength $O(h/\log h^{-1})$ near t.he barrier top and it can be continued analytica.lly into
the well, where its behavior is like in the Breit- Wigner formula.
This physical problem is closely related to the purely mathematical problem originatedby H.Weyl [We] on the eigenvalue asymptotics for the Laplace operator in abounded d0is the number of eigenvalues not excedding
with regular boundary.
main
$E$

$V_{0}$$\mathrm{I}\mathrm{I}$$V_{0}$$h$$\mathrm{I}$$\mathrm{I}\mathrm{I}$$\mathrm{t},\mathrm{h}\mathrm{e}$$\Omega\in \mathrm{R}^{\mathrm{n}}$$\mathrm{I}\mathrm{f}N(E)$

19

$E$

(counting multiplicity), then$N(E) \sim\frac{\tau_{n}}{(2\pi)^{n}}\mathrm{v}\mathrm{o}\mathrm{l}(\Omega)E^{n/2}$,$Earrow+\infty$,. L.H\"ormander [H\"o]where $\tau_{n}=\pi^{n/2}/\Gamma(n/2+1)$ is the volume of the unit ball in
generalized this result to compact Riemannian manifolds without boundary with the best
estimate of the error.
for theOn the other hand, the asymptotic behavior of the scattering phaseexterior problem with an obstacle was studied, say, by A.Jensen and T.Kato [Je-Ka],
V.Petkov and G.Popov [Pe-Po] and R.Melrose [Me]. This is defined to be the half of theargument of the determinant of the scattering matrix which is unitary on $L^{2}(S^{n-1})$ . They
also has the Weyl type asymptotic formula when $Earrow+\infty$ :showed that$\mathrm{R}^{n}$$\theta(E)$$\Omega$$\theta(E)$$\theta(E)=-\frac{\tau_{n}}{(2\pi)^{n}}\mathrm{v}\mathrm{o}\mathrm{l}(\Omega)E^{n/2}+O(E^{(n-1)/2})$.

with ascalar potential $V(x)$ , the semiclasIn the case of Schr\"odinger operator on
for simplicity. Then the spectrum
sical problem is also important. Let$P=-h^{2}\Delta+V(x)$onconsists of negative eigenvaluesof the Schr\"odinger operator. We can define the eigenvalue counting function
and the essential spectrum lying on$N(E, h)$ for negative $E$ and the scattering phasefor positive $E$ both depending on
the small parameter . The semiclassical problem is to describe the asymptotic behavior
as tends to 0while $E$ is restricted near afixed energy . Remarkof $N(E, h)$ orthat the high energy regime $(Earrow+\infty)$ is aparticular case of the semiclassical regime
with $h=E^{-1/2}$ (see [Ro]).
The spectral shift function $s(E, h)$ combines these two notions. It is defined as aon modulo areal constant by the following equation:
Schwartz$\mathrm{R}^{n}$$V(x)\in C_{0}^{\infty}(\mathrm{R}^{n})$$L^{2}(\mathrm{R}^{n})$$\mathrm{R}_{+}$$\theta(E, h)$

$h$

$h$

$\theta(E, h)$$E_{0}$$(harrow \mathrm{O})$$\mathrm{R}$$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\cdot \mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{T}\mathrm{r}(f^{\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}}(P)-f(P_{0}))=-\int_{-\infty}^{+\infty}f’(E)s(E, h)dE$,

and is an arbitrary function in the Schwartz space. $s(E, h)$ is uniquelydetermined by asking $s(E, h)=\mathrm{O}$ for $E<<0$ . Under this condition, it is easy to see that$s(E, h)=N(E, h)$ for negative $E$ . The remarkable fact discovered by.Birman and
M.G.Krein [Bi-Kr] is the relation between the spectral shift function and the scattering
phase for positive $E$ :$\theta(E, h)=\pi s(E, h)$.
where$P_{0}=-h^{2}\triangle$$f$$\mathrm{M}.\mathrm{S}$$\mathrm{m}\mathrm{o}\mathrm{d} (\pi \mathbb{Z})$The semiclassical asymptotics of the spectral shift function is closely related to its
classical mechanics counterpart. Let $s^{d}(E, h)$ be the classical analogue of the spectral
shift function defined by$\int\int_{\mathrm{R}^{2n}}\{f(p(x, \xi))-f(p_{0}(\xi))\}dxd\xi=-\int_{-\infty}^{+\infty}f’(E)s^{d}(E)dE$,

20

where $p(x, \xi)=\xi^{2}+V(x)$ and$p_{0}(\xi)=\xi^{2}$. We see immediately that$s^{d}(E)= \tau_{n}\int_{\mathrm{R}^{n}}\{(E-V(x))_{+}^{n/2}-E_{+}^{n/2}\}dx$,

where $a_{+}= \max(a, 0)$ . An energy levelis said to be non-trapping for the classicalHamiltonian $p(x, \xi)$ if every Hamiltonian flow on $p^{-1}(E)$ generated by escapes to infinity
as time goes to–infinity.
D.Robert and H.Tamura [RO-Ta] proved that if
is non-trapping in an interval,$s(E, h)$ has auniform complete asymptotic expansioin as
and
$E$

$p$

$\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}+\mathrm{a}\mathrm{n}\mathrm{d}$$E$$harrow \mathrm{O}$$s(E, h)=(2\pi h)^{-n}s^{d}(E)+O(h^{1-n})$as h$arrow \mathrm{O}$

.

(0.2)

If the energy is trapping, however, it is believed that the scattering phase varies very
rapidly because of the presence of poles of the scattering matrix called resonances near
the real axis.
Let us mention here two works on the scattering phase associated with atrapped
trajectory in apotential well. Aresonance generated by such aclosed trajectory is calledshape resonance and is exponentially close to the real axis with respect toC.G\’erard,
A.Martinez and D.Robert [G\’e-Ma-Ro]proved that the scattering phase increases by at
the real part of ashape resonance. In other words, they showed the Breit-Wigner formulaat the bottom of the potential well.
S.Nakamura [Na] considered two Hamiltonians and corresponding to the bounded
and unbounded component of $p^{-1}(E)$ respectively and showed that the spectral shiftfunction is approximated by the sum of that for , the asymptotic behavior of which
we know from the result of [RO-Ta], and the eigenvalue counting function for . Theseeigenvalues are near the shape resonances and cause the rapid variation of the scattering
phase.
The barrier top energy $E=V_{0}$ is trapping because it takes infinite time for classicalparticles to arrive at the barrier top. Remark also that in case $E$ is non-trapping aboveand below and in case , non-trapping above and trapping below because of thepotential well. This change of classical structure at the barrier top is represented as thelogarithmic singularity of $s^{d}(E)$ at $E=V_{0}$ (see Lemma 2.3).
We will see that there is acorrection term for $s^{d}(E)$ in acomplex neighborhood ofof radius $O(h)$ and that it cancels the singularity of $s^{d}(E)$ . This term is written in termsof the Fredholm determinant of the harmonic oscillator. This comes from the fact thatthe operator is reduced to the harmonic oscillator microlocally near the non-degeneratecritical point (see [He-Sj]). Out of this neighborhood, that is where $|V_{0}-E|/h$ tends toinfinity, this term disappears from the leading term and we recover.
Once we prove that the leading term is holomorphic in acomplex domain, we candifferentiate the asymptotic formula term by term. In fact, the scattering phase itself isalso holomorphic in acomplex neighborhood ofand so is the remainder term in the
$h.$

$\pi$

$P_{\mathrm{i}}$$P_{\mathrm{e}}$$P_{\mathrm{e}}$$P_{\mathrm{i}}$$\mathrm{I},$$V_{0}$$\mathrm{I}\mathrm{I}$$V_{0}$$V_{0}$$V_{0}$$P$$s^{\mathrm{c}l}(E)$$V_{0}$

21
intersection. We can then estimate the derivative of the remainder term by its supremum
by Cauchy’s formula. Thus we get the asymptotic formula of the time delay function.
The method is based on the exact $WKB$ analysis (see [G\’e-Gr]). In the one-dimesional
matrix and each element is writtencase, the scattering matrix can be defined as aby wronskians of Jost solutions (see Section 2). Therefore the problem is reduced to theconnection problem of Sturm-Liouville equations with small parameter.
In this paper we start from the asymptotic formulas of the scattering matrix obtainedby [Ra] in case Iand by [Fu-Ra 1] in case .
$2\cross 2$$\mathrm{I}\mathrm{I}$

1Results
We consider the one-dimensional Schr\"odinger equation (0.1) where the potential $V(x)$satisfies the following conditions:and dilation analytic, that is, $V(x)$ is holomorphic in asector$S=\{x\in \mathrm{C};|{\rm Im} x|<\tan\theta_{0}|{\rm Re} x|\}\mathrm{U}\{|{\rm Im} x|<\delta\}$ for some $0<\theta_{0}<\pi/2$ and(H1) $V(x)$ is real on$\delta>0$$\mathrm{R}$.

(H2) $V(x)$ is short range, that is, there exists apositive constant$(1+|x|)^{-1-\epsilon}$ in $S$ .

$\epsilon>0$such that $|V(x)|\leq$

be the maximum of the potential on the real axis which we assume to beLetpositive. We consider the two cases:$V_{0}$(Case I)

(Case$\mathrm{I}\mathrm{I}$$V^{-1}(V_{0})=\{\mathit{0}_{1}\}$)$V^{-1}(V_{0})=\{\mathit{0}_{1},\mathit{0}_{2}\}$$(\mathit{0}_{1}<\mathit{0}_{2})$In both cases, we assume furthermore that the radius of curvature
critical points:(H3) $V”(\mathit{0}_{j})=-\rho_{j}^{-1}<0$$\rho_{j}$is finite at each$(j=1,2)$

Put $\lambda=V_{0}-E$ . The energy is under the maximum of the potential when Ais
positive. If Ais positive and sufficiently small, the equation $V(x)-E=\mathrm{O}$ has 2real rootsnear
in both cases and 2other real rootsnear
in case . We then define the action integrals between these turning
as follows. For the simplicity of notations, we use the convention thatpoints andIand 2in case .
1incasestands for$\alpha_{1}(\lambda),$$\beta_{1}(\lambda)$$o_{1}(\alpha_{1}<\mathit{0}_{1}<\beta_{1})$$o_{2}(\alpha_{2}<\mathit{0}_{2}<\beta_{2})$$\alpha_{2}(\lambda),$$\pm\infty$$*$$\mathrm{I}\mathrm{I}$Definition 1.1$\beta_{2}(\lambda)$$\mathrm{I}\mathrm{I}$For $\lambda=V_{0}-E$ positive and sufficiently small, we set$S_{j}( \lambda)=\int_{\alpha_{j}}^{\beta_{j}}\sqrt{V(x)-E}dx$,$(j=1,2)$ ,22$S_{\mathrm{e}}( \lambda)=(\int_{-\infty}^{\alpha_{1}}+\int_{\beta_{\mathrm{r}}}^{\infty})\{\sqrt{E-V(x)}-\sqrt{E}\}dx-\sqrt{E}(\beta_{*}-\alpha_{1})$(in case$S_{\mathrm{i}}( \lambda)=\int_{\beta_{1}}^{\alpha_{2}}\sqrt{E-V(x)}dx$$\mathrm{I}\mathrm{I}$,).

Let us remark here that the classical analogue of the spectral shift function $s^{d}(E)$ isrelated with these actions by(case )
(case $II$ )$I$$2S_{\mathrm{e}}(\lambda)$$s^{d}(E)=\{$$2(S_{\mathrm{e}}(\lambda)+S_{\mathrm{i}}(\lambda))$Avery important fact is thatandhave alogarit.hmic singularity at $\lambda=0$ (seeLemma 2.3). We will see that there is acorrection term which affects the leading term of
the scattering phase and that it regularizes $s^{d}(E)$ by canceling the logarithmic singularity.
This correction term is written by the following interesting function $N(z)$ , which is the
Jost function of the harmonic oscillator (see Remark 2.5). For the properties of thisfunction, see Lemma 2.4.
$S_{\mathrm{e}}$Deflnition 1.2$S_{\mathrm{i}}$We define the function $N(z)$ on$\{z\in \mathbb{C}\backslash ;|\arg z|<\pi\}$$N(z)= \frac{\sqrt{2\pi}}{\Gamma(z+1/2)}e^{z\log(z/e)}$by.

Let us define, using this function, the regularized actions at the barrier top:

Definition 1.3
Aby

We define the real functions$\sigma_{\mathrm{e}}(\lambda, h)$and$\sigma_{\mathrm{i}}(\lambda, h)$for positive and small$\sigma_{\mathrm{e},\mathrm{i}}(\lambda, h)=S_{\mathrm{e},1}(\lambda)-\frac{h}{2}\{\arg N(i\frac{S_{1}(\lambda)}{\pi h})+\arg N(i\frac{S_{*}(\lambda)}{\pi h})\}$,

where $\arg(iS_{j}(\lambda)/(\pi h))=\pi/2$ for $\lambda>0$ .
can be extended as holomorphic(see Proposition 2.6).
functions to acomplex neighborhood of
We will see (see (2.6)) that far from the barrier top $\lambda=0,$h) coincide with
. More precisely,$\sigma_{\mathrm{e},\mathrm{i}}(\lambda, h)$$\lambda=0$$\sigma_{\mathrm{e},\mathrm{i}}(\lambda,$$S_{\mathrm{e},\mathrm{i}}(\lambda)$$\sigma_{\mathrm{e},\mathrm{i}}(\lambda, h)arrow S_{\mathrm{e},\mathrm{i}}(\lambda)$as$|\lambda|/harrow+\infty$(1.1)

We also define aholomorphic function which will express the width of resonances near
the real axis in case II.
Definition 1.4

We define the positive function$\gamma(\lambda, h)$for positive and small Aby$\gamma(\lambda, h)=\frac{|N(i_{h}^{\lambda}\frac{s_{1}}{\pi}\mathrm{u})N(i_{\pi h}^{s\mathrm{u}}s^{\lambda})|-1}{|N(i_{\pi h}^{\underline{s}}1[perp]\lambda[perp])N(i_{h}^{\lambda}\frac{s_{2}}{\pi}\mathrm{u})|+1}$.

23
As function of(see Lemma 2.7).

$\lambda,$

extends holomorphicaly to acomplex neighborhood of $\lambda=0$

There exists

Theorem 1.5

case$\gamma(\lambda, h)$$C>\mathrm{O}$such that if Ais real and$|\lambda|\leq Ch$, then we have in$I$$h\theta(E, h)=\sigma_{\mathrm{e}}(\lambda, h)+O(h^{2}\log(1/h))$

(1.2)

,

and in case $II$$h \theta(E, h)=\sigma_{\mathrm{e}}(\lambda, h)+h\tan^{-1}\{\gamma(\lambda, h)\tan\frac{\sigma_{\mathrm{i}}(\lambda,h)}{h}\}+O(h^{2}\log(1/h))$

.

(1.3)

The asymptotic formula (1.2) and (1.3) are just the analogues of the results of [RO-Ta]
and [Na] respectively. The second term in the right hand side of (1.3) is related to thepresence of the potential well and causes rapid variations caused by the potential well.
It will be seen more clearly in the next corollary as arapid oscillation of the time delay,
which is the derivative of the scattering phase with respect the energy $E$ .
$\mathrm{o}\mathrm{t}$Corollary 1.6
in caseThere exists$C>\mathrm{O}$such that if Ais real and$|\lambda|\leq Ch$, then we have$I$

$h \frac{d\theta}{dE}=\rho_{1}\log\frac{1}{h}+O(1)$and if Ais real and $|\lambda|\leq Ch/\log(1/h)$ , tlien we htave in case $II$$h \frac{d\theta}{dE}=\frac{\rho_{1}+\rho_{2}}{2}\{1+\frac{\gamma}{(1-\gamma^{2})\cos^{2}(\sigma_{\mathrm{i}}/h)+\gamma^{2}}\}\log\frac{1}{h}+O(1)$The leading term is logarithmic with respect to hence the Weyl law fails in suchsmall neighborhoods of the potential maximum. The function$h$$B(E, h)= \frac{\gamma}{(1-\gamma^{2})\cos^{2}(\sigma_{\mathrm{i}}/h)+\gamma^{2}}$tends to 0for $E<V_{0}$ ,is the contribution from the potentail well. The function
to 1for $E>V_{0}$ and equals 1/3 for $E=V_{0}$ . When is small, $B$ presents spikes at each
and width (see Lemma 2.7). Hence this is an extensionof height
zero ofare givenof the Breit- Wigner fomula to the potential maximum. The zeros ofby the Bohr-Sommerfeld type quantization condition$\gamma(E, h)$$\gamma$$\cos(\sigma_{\mathrm{i}}/h)$$1/\gamma$$\gamma$$\cos(\sigma_{\mathrm{i}}/h)$$\sigma_{\mathrm{i}}(E, h)=(n+\frac{1}{2})\pi h$.

It follows from Proposition (2.6) that the distance between two such successive zeros when$|\lambda|\leq Ch/\log(1/h)$ is $2\pi(\rho_{1}+\rho_{2})^{-1}h\log(1/h)$ .

24
2Proofs
2.1

Proof of Theorem 1.6

We calculate the asymptotic formula of the scattering phase starting from the results of[Ra] and [Fu-Ra 1] for the scattering matrix.
Letandbe Jost solutions such that$f_{l}^{\pm}(x)$$f_{f}^{\pm}(x)$$f_{f}^{\pm}(x)\sim e^{\pm:\sqrt{E}x/h}$$f_{l}^{\pm}(x)\sim e^{\pm \mathrm{t}\sqrt{E}x/h}$

as
as

in S,
-oo in S.

${\rm Re} xarrow+\infty$${\rm Re} xarrow$

The existence (and uniqueness) of such solutions is guaranteed by the short range assump. These solutions are holomorphic
tion (H.2) for $E$ in.
in
The two pairs
andare basis of solutions to (0.1) hence they are
related with each other by aconstant:matrix$\Pi_{\theta_{0}}=\{E\in \mathbb{C}\backslash \{0\};|\arg E|<2\theta_{0}\}$$(x, E)\in \mathrm{S}\cross\Pi_{\theta_{0}}$$(f_{l}^{+}, f_{l}^{-})$$(f_{f}^{+}, f_{f}^{-})$$2\cross 2$$\mathrm{T}(E, h)$h)$(\begin{array}{l}f_{l}^{+}f_{l}^{-}\end{array})=\mathrm{T}(E,$.

$(\begin{array}{l}f_{f}^{+}f_{f}^{-}\end{array})$

The determinant of this matrix is 1since. Moreoverandareby definition, hence is of the form

and the wronskians
where$[f_{l}^{+}, f_{l}^{-}]=\det \mathrm{T}[f_{f}^{+}, f_{f}^{-}]$$[f_{l}^{+},f_{l}^{-}]$$\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}-2i\sqrt{E}/h$$[f_{f}^{+}, f_{f}^{-}]$$f_{l,\mathrm{r}}^{-}=(f_{l,\tau}^{+})^{*}$$f^{*}(x, E)=\overline{f(\overline{x},\overline{E})}$$\mathrm{T}$$\tau=(\begin{array}{ll}a bb^{\mathrm{r}} a^{*}\end{array})$

The elements

$a$

and

$b$

,

$aa^{*}-bb^{*}=1$

.

(2.1)

are written in terms of Jost solutions:$a(E, h)= \frac{ih}{2\sqrt{E}}[f_{l}^{+}, f_{f}^{-}]$,$b(E, h)=- \frac{ih}{2\sqrt{E}}[f_{l}^{+}, f_{f}^{+}]$,

and we see that they, as well as $a^{*}(E, h)$ and $b^{*}(E, h)$ , are all holomorphic in.
hand,scattering
On the otherthematrix is defined as the matrix associated with thechange of basis between the outgoing pair of solutions
and the incoming pair of;ifsolutions$E\in\Pi_{\theta_{0}}$$(f_{f}^{+}, f_{l}^{-})$$(f_{l}^{+}, f_{f}^{-})$$p_{+}f_{f}^{+}+p_{-}f_{l}^{-}=q_{+}f_{l}^{+}+q_{-}f_{f}^{-}$,

then$(\begin{array}{l}p_{+}p_{-}\end{array})=\mathrm{S}(E, h)(\begin{array}{l}q_{+}q_{-}\end{array})$

We see by an elementary computation that in terms of$\mathrm{s}=\frac{1}{a}*(\begin{array}{ll}\mathrm{l} -b^{*}b 1\end{array})$

.

$a$

.
and

$b,$

$\mathrm{S}$is written by25

Suppose now that $E$ is positive. Then is unitary by (2.1) and hence its determinant is
is defined as half of theacomplex number with module 1. The scattering phase:argument of.
(2.2)$\mathrm{S}$$\theta(E, h)$$\det \mathrm{S}$$\det \mathrm{S}(E, h)=e^{2\theta(E,h)}|.$

It is real and given in terms of the element$a$of the matrix$\mathrm{T}$by$\theta(E, h)=\arg a(E, h)=-\arg\overline{a(E,h)}$Remark thatby$\theta(E, h)$can also be defined for complex$E\in\Pi_{\theta_{0}}$$\theta(E, h)=\frac{1}{2i}\log\frac{a(E,h)}{a^{*}(E,h)}$

.

(2.3)

as complex valued function.

More precisely we havesuch that
extends holomorphically to the disk
There exists$|E-V_{0}|<Ch$ in case I and $|E-V_{0}|<Ch\log(1/h)$ in case $II$.

Lemma 2.1

$C>\mathrm{O}$$\theta(E, h)$

are holomorphic in
is singular only at zeros of
Proof: Since andand . The zeros of are complex conjugates of those ofand hence it is enough tostudy the asymptotic distribution of zeros of , that is, resonances. This was done in[Ra] and [Fu-Ra 1] in cases Iand respectively. They calculated the leading term of the(see Theorem 2.2) and majorated by Rouche”s theorem theasymptotic expansion of
distance between its zeros and those of .
Recall here the asymptotic formula of $a^{*}(E, h)$ obtained in [Ra] and [Fu-Ra 1]. See.
andalso [Fu] where$a$$a^{*}$$a^{*}$$\Pi_{\theta_{0}},$$\theta(E, h)$$a$$a^{*}$$a$$a^{*}$$\mathrm{I}\mathrm{I}$$a^{*}$$a^{*}$$S_{l}+S_{f}=-S_{\mathrm{e}}$Theorem 2.2$\square$$S_{12}=S_{\mathrm{i}}$The asymptotic formula of $a^{*}(E, h)$ is given by the following formulas:(Case )
$I$$a^{*}(E, h)=e^{(S_{1}-iS_{\mathrm{e}})/h}N(i \frac{S_{1}}{\pi h})(1+O(h\log h))$

,

(2.4)

(Case $II$)$a^{*}(E, h)=e^{(S_{1}+S_{2}-iS_{\mathrm{e}})/h}(e^{iS_{\mathrm{i}}/h}+N(i \frac{S_{1}}{\pi h})N(i\frac{S_{2}}{\pi h})e^{-iS_{\mathrm{i}}/h})$$\cross(1+O(h\log h))$Let us calculate the argument of$a^{*}$

through (2.4) and (2.5). For simplicity, put$r_{j}( \lambda, h)=|N(i\frac{S_{j}(\lambda)}{\pi h})|$,$\phi_{j}(\lambda, h)=\arg N(i\frac{S_{j}(\lambda)}{\pi h})$

.

(2.5)

26
In case , we get immediately$\mathrm{I}$.

$\pi h\theta(E, h)=S_{\mathrm{e}}-h\phi_{1}+O(h^{2}\log h)=\sigma_{\mathrm{e}}+O(h^{2}\log h)$

In case$\mathrm{I}\mathrm{I}$, we have$\pi h\theta(E, h)=S_{\mathrm{e}}-h\arg(e^{iS_{\mathrm{I}}/h}+r_{1}r_{2}e^{i(\phi_{1}+\phi_{2})}e^{-iS_{\mathrm{i}}/h})+O(h^{2}\log h)$$=\sigma_{\mathrm{e}}-h\arg(e^{i\sigma_{\mathrm{I}}/h}+r_{1}r_{2}e^{-i\sigma_{1}/h})+O(h^{2}\log h)$,and since$\arg(e^{:\sigma_{\mathrm{I}}/h}+r_{1}r_{2}e^{-:\sigma 1/h})=\arg\{(1+r_{1}r_{2})\cos\frac{\sigma_{\mathrm{i}}}{h}+i(1-r_{1}r_{2})\sin\frac{\sigma_{\mathrm{i}}}{h}\}$$= \tan^{-1}(\frac{1-r_{1}r_{2}}{1+r_{1}r_{2}}\tan\frac{\sigma_{1}}{h})$,we get (1.3).

2.2

Regularized actlons

We first recall the analytic property in acomplex neighborhood ofof the action
integralsandwhich were defined for small positive Ain Definition 1.1.
Let $D(R)$ denote the disk {A}. See [Fu-Ra 1] for the proof of the following
lemma.
$\lambda=\mathrm{O}$$S_{j}(\lambda),$$S_{\mathrm{e}}(\lambda)$$S_{\mathrm{i}}(\lambda)$$\in \mathrm{c};|\lambda|<R$

Lemma 2.3 There exist apositive constantholomorphic in $D(R)$ such thatand$g_{\mathrm{i}}(\lambda)$$S_{j}(\lambda),$$R$

and functions $g_{j}(\lambda)(j=1,2),$andand
are all real for $0<\lambda<R$$g_{\mathrm{e}}(\lambda)$$S_{\mathrm{e}}(\lambda)$$S_{\mathrm{i}}(\lambda)$$(j=1,2)$ .

$S_{j}(\lambda)=\pi\rho_{j}\lambda(1+\lambda g_{j}(\lambda))$$S_{\mathrm{e},\mathrm{i}}( \lambda)=S_{\mathrm{e},\mathrm{i}}(0)+\frac{1}{2\pi}(S_{1}(\lambda)+S_{*}(\lambda))\log\lambda+\lambda g_{\mathrm{e},\mathrm{i}}(\lambda)$,

where $\log\lambda>0$ when $\arg\lambda=0$ .
We also recall some properties of the function $N(z)$ .
Lemma 2.4

$N(z)$ is holomorphic in $\{z\in \mathrm{C}\backslash \{0\};|\arg z|<\pi\}$$\lim_{|z|arrow\infty}N(z)=1$

and in this domain,

.

(2.6)

In particular, on the positive imaginary axis $z=it$ , $t>0$ , we have$|N(it)|^{2}=1+e^{-2\pi t}$

,

(2.7)

27

$\arg N(it)=t\log t+tg(t)$ ,(2.8)

where is areal and analytic function and extends holomorphically to acomplex neighborhood of the origin.
$g$

Proof: The formula (2.6) is nothing but the Stirling formula and (2.7) follows easily fromthe product formula of the Gamma function:$| \Gamma(\frac{1}{2}+\cdot t)|^{2}=\Gamma(\frac{1}{2}+\dot{\iota}t)\Gamma(\frac{1}{2}-i.t)=\frac{\pi}{\cosh\pi t}$

.

(2.9)

For (2.8), we have$\arg N(i\mathrm{t})=t\log t-t-\arg\Gamma(1/2+it)$.

Using (2.9), we can rewrite the last term of the right hand side as.

$\arg\Gamma(\frac{1}{2}+it)--\frac{i}{2}\log\pi-i\log\Gamma(\frac{1}{2}+it)-\frac{i}{2}\log(\cosh\pi t)$

This function can be extended analyticallyHence we can write $\arg N$ (it) in the form (2.8).

$.\mathrm{t}\mathrm{o}.\mathrm{C}\backslash i(\mathrm{Z}+1/2)$

and equals 0when.

$t=0\square$

Remark 2.5 The function $N(z)$ can be characterized as Jost function of the harmonic
oscillator (see [Vo]). Let
be the solutions to (0.1) with $V(x)=x^{2}$ and $h=1$ whose
asymptotic behavior is$\psi_{\pm}(x)$$\psi_{\pm}(x)\sim(x^{2}-E)^{-1/4}\exp(\pm\int_{x_{0}}^{x}(y^{2}-E)^{1/2}dy)$

as x$arrow\mp\infty$.

It is possible to define these solutions for arbitrarywhen
is negative. Thenthe Jost function of the harmonic oscillator, which is defined as the wronskian of thesesolutions, is written by$E$$x_{0}\in \mathbb{R}$$\frac{1}{2}[\psi_{+}, \psi_{-}]=N(-\frac{E}{2})$.

Proposition 2.6 There exists
such that the functions
andcan be extended as holomorphic functions with respect to Ain $D(Ch)$ and the following
asymptotic formula holds in this domain:$C>\mathrm{O}$$\sigma_{\mathrm{e},\mathrm{i}}(\lambda, h)=S_{\mathrm{e},\mathrm{i}}(0)-\frac{\rho_{1}+\rho_{*}}{2}\lambda\log\frac{1}{h}+O(\lambda)$$\sigma_{\mathrm{e}}(\lambda, h)$as$harrow \mathrm{O}$

Proof: From Lemmas 2.3 and 2.4, we get$\sigma_{\mathrm{e},\mathrm{i}}(\lambda, h)=S_{\mathrm{e},\mathrm{i}}(0)-\frac{\rho_{1}+\rho_{*}}{2}\lambda\log\frac{1}{h}+G(\lambda, h)$,.

$\sigma_{\mathrm{i}}(\lambda, h)$28$\mathrm{w}1\iota 11$G$= \lambda g_{\mathrm{e},\mathrm{i}}-\sum_{j=1,*}[\frac{S_{j}}{2\pi}\{g(\frac{S_{j}}{\pi h})+\log(\rho_{j}(1+\lambda g_{j}))\}+\frac{\rho_{j}}{2}\lambda^{2}g_{j}\log\frac{1}{h}]$.
$\square$

2.3

Proof of Corollary 1.7

Here we deduce Corollary 1.6 from Theorem 1.5 by making use of the analyticity of theberemainder terms in the asymptotic formulas (1.2) and (1.3). Let(1.3)
respectively;(1.2),
the the remainder terms of$R_{I}(\lambda, h),$$R_{I}(\lambda, h)=\pi h\theta(E, h)-\sigma_{\mathrm{e}}(\lambda, h)$,$R_{II}( \lambda, h)=\pi h\theta(E, h)-\sigma_{\mathrm{e}}(\lambda, h)-h\tan^{-1}\{\gamma(\lambda, h)\tan\frac{\sigma_{\mathrm{i}}(\lambda,h)}{h}\}$First let us observe some properties of the function$R_{II}(\lambda, h)$$\gamma(\lambda, h)$.

.

h) is holomorphic
Lemma 2.7 There exist positive C and R such that the functionin (-R,$R)\cross i(-Ch,$Ch). Moreover, on (-R, R) in particular, $0<\gamma<1$ and$\gamma(\lambda,$(i) if $\lambda=O(h)$ , there exist $0<\gamma_{0}<\gamma_{1}<1$ independent of Aand of h such thatand in particular $\gamma(0, h)=1/3$ ,$\gamma_{0}<\gamma(\lambda, h)<\gamma_{1}$(ii)$\mathrm{i}f|\lambda/h|arrow\infty$,$O(e^{-2S_{1}(\lambda)/\hslash}+e^{-2S_{2}(\lambda)/h})$$(\lambda>0)$$1-O(e^{(S_{1}(\lambda)+S_{2}(\lambda))/h})$$(\lambda<0)$$\gamma(\lambda, h)=\{$,.

Proof: With (2.7), one obtains$\gamma(\lambda, h)=$and the lemma follows easily. In particular this function has singularities at the points
satisfying $S_{j}(\lambda)=(n+1/2)\pi ih(j=1,.2)$ .
$\square$and
Proposition 2.8
There exists apositive constant such thatare holomorphic vvith respect to Ain $D(Ch)$ and in $D(Ch/\log(1/h))$ respectively forsufficiently small .
$C$

$h$

$R_{I}(\lambda, h)$$R_{II}(\lambda, h)$29

Proof: The functions
andare holomorphic in the required domain byLemma 2.1 and Proposition 2.6. It remains toshow that the last term ofis alsoholomorphic in $D(Ch/\log(1/h))$ . Let us calculate the derivative:$\theta(E, h)$$\sigma_{\mathrm{e}}(\lambda, h)$$R_{II}$$h \{\tan^{-1}(\gamma\tan\frac{\sigma_{\mathrm{i}}}{h})\}’=\frac{\gamma\sigma_{\mathrm{i}}’+h\gamma’\cos(\sigma_{\mathrm{i}}/h)\sin(\sigma_{\mathrm{i}}/h)}{(1-\gamma^{2})\cos^{2}(\sigma_{\mathrm{i}}/h)+\gamma^{2}}$.

(2.10)

being holomorphic, it suffices to see that the denominator $D(\lambda, h)=$does not vanish in $D(Ch/\log(1/h))$ . First we see that for real Ain
this domain,is real and bounded from below by apositive constant independentof both Aand . Next for complex , we see
Both$\gamma$and$\sigma_{\mathrm{i}}$$(1-\gamma^{2})\cos^{2}(\sigma_{\mathrm{i}}/h)+\gamma^{2}$$d(\lambda, h)$$h$$\lambda$$\gamma(\lambda, h)arrow\frac{1}{3}$,$|{\rm Im} \frac{\sigma_{\mathrm{i}}}{h}|\leq C\frac{\rho_{1}+\rho_{2}}{2}+O(\frac{1}{\log(1/h)})$,stays away from 0for sufficiently small
as tends to 0. Hence, by continuity,and .
Proposition 2.8 enables us to estimate the derivatives ofandin terms ofthemselves by Cauchy’s integral formula; if afunction
is holomorphic in, thenits derivative is bounded from above in $D(r/2)$ by 2 $\sup_{D(r)}|R(\lambda)|/r$ . Recalling that
$R_{I}=O(h^{2}\log(1/h))$ and $R_{II}=O(h^{2}\log(1/h))$ , we obtain
$h$

$C$

$d(\lambda, h)$$h$$\square$$R_{I}$$R(\lambda)$$\frac{dR_{I}}{d\lambda}=O(h\log h^{-1})$,$\frac{dR_{II}}{d\lambda}=O(h(\log h)^{2})$$R_{II}$$\overline{D(r)}$.

On the other hand, we know from Proposition 2.6 that$\frac{d\sigma_{\mathrm{e},\mathrm{i}}}{dE}=\frac{\rho_{1}+\rho_{2}}{2}\log\frac{1}{h}+O(1)$and since $hd\gamma/dE=O(1)$$h \frac{d}{dE}\{\tan^{-1}(\gamma\tan\frac{\sigma_{\mathrm{i}}}{h})\}=\frac{\rho_{1}+\rho_{*}}{2}\frac{\gamma}{(1-\gamma^{2})\cos^{2}(\sigma_{\mathrm{i}}/h)+\gamma^{2}}\log\frac{1}{h}+O(1)$This completes the proof of Corollary 1.6.

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