THE ONE-DIMENSIONAL SCATTERING PROBLEM IN THE QUASI-CLASSICAL APPROXIMATION

M. V. FEDORYUK

Part I

§ 1. FORMULATION OF THE PROBLEM. SUMMARY OF RESULTS

1. Formulation of the problem. Consider the Schrödinger equation

\[ -\frac{h^2}{2m}\,\psi''(x)+(V(x)-E)\psi(x)=0. \tag{1.1} \]

Setting

\[ V(x)-E=q(x),\quad h^{-1}\sqrt{2m}=\lambda, \tag{1.2} \]

we reduce (1.1) to the form

\[ \psi''(x)-\lambda^2 q(x)\psi(x)=0. \tag{1.3} \]

Throughout this paper it is assumed that the function \(q(x)\) is real-valued and that \(\lambda>0\) (with the exception of § 4). The scattering problem is considered under the assumption that \(\lambda\) is a large parameter \((\lambda\to+\infty)\) and that \(q(z)\) is an entire function of \(z\) (with the exception of § 6), satisfying fairly broad assumptions (see § 2). Practically all entire functions \(q(z)\) of physical interest satisfy these conditions.

In quantum mechanics [1] the following problems are usually considered:

\(1^\circ\). The problem of reflection from a barrier of infinite width. In this case

\[ \lim_{x\to\pm\infty} q(x)=q_\pm,\quad q_-<0,\quad q_+>0 \tag{1.4} \]

(the case \(q_+=+\infty\) is possible). Let

\[ \int_{-\infty}^{\hat c}\left|\sqrt{|q(x)|}-\sqrt{|q_-|}\right|\,dx<\infty. \tag{1.5} \]

Denote by \(\psi_1^-\), \(\psi_2^-\) solutions of equation (1.3) such that, as \(x\to-\infty\),

\[ \psi_1^- \sim |q_-|^{-1/4} e^{ik_-x},\quad \psi_2^- \sim |q_-|^{-1/4} e^{-ik_-x},\quad k_-=\lambda\sqrt{|q_-|}. \tag{1.6} \]

The solution \(\psi(x)\) such that

\[ \psi(+\infty)=0 \tag{1.7} \]

has physical meaning. Then

\[ \psi(x)=s_{11}(\lambda)\psi_1^-(x)+s_{12}(\lambda)\psi_2^-(x). \tag{1.8} \]

It is required to find the asymptotics of \(s_{ij}(\lambda)\) as \(\lambda\to+\infty\) (since \(\psi(x)\) is determined up to a constant factor, the same is true for \(s_{ij}\)).

$2^\circ$. The problem of passage through a barrier of finite width. Let $q_\pm<0$, let $q(x)$ have zeros, let (1.5) be satisfied, and

\[ \int_0^{+\infty}\left|\sqrt{|q(x)|}-\sqrt{|q_+|}\right|\,dx<\infty . \tag{1.9} \]

Denote by $\psi_1^+$, $\psi_2^+$ the solutions of (1.3) such that, as $x\to+\infty$,

\[ \psi_1^+\sim |q_+|^{-1/4}e^{ik_+x},\quad \psi_2^+\sim |q_+|^{-1/4}e^{-ik_+x},\quad k_+=\lambda\sqrt{|q_+|}. \tag{1.10} \]

Then

\[ \binom{\psi_1^+}{\psi_2^+} = S\binom{\psi_1^-}{\psi_2^-},\quad S(\lambda)=\|s_{ij}(\lambda)\|. \tag{1.11} \]

It is required to find the asymptotics of the matrix $S(\lambda)$ as $\lambda\to+\infty$.

$3^\circ$. The problem of above-barrier reflection. Let $q_\pm<0$, $q(x)<0$. It is required to find the asymptotics of $S(\lambda)$ as $\lambda\to+\infty$.

In all these problems it is also required to find the asymptotics of $\psi(x,\lambda)$ as $\lambda\to+\infty$, $x$ real.

Since $\psi_1^\pm(x)=\overline{\psi_2^\pm(x)}$, we have

\[ s_{11}=\overline{s}_{22},\quad s_{12}=\overline{s}_{21}. \]

We shall solve all these problems by a unified method, continuing the asymptotics of $\psi(x)$ into the complex plane $z$. This method is presented in detail in [2]. Below, in § 2, we shall give only the information from [2] necessary for reading the article.

2. Summary of results. The classes of potentials $Q_j$ considered below are described in § 2. In particular, throughout this article it is assumed that $q(z)$ has only simple zeros and that $q(x)$ has a finite number of zeros.

Let $x_j$ be the zeros of $q(x)$, $1\le j\le k$, $x_j<x_{j+1}$. Denote

\[ C_-= x_1\sqrt{|q_-|} + \int_{-\infty}^{x_1} \left(\sqrt{|q(x)|}-\sqrt{|q_-|}\right)\,dx,\quad A_-=e^{i\lambda C_-}, \tag{1.12} \]

\[ C_+= -x_k\sqrt{|q_+|} + \int_{x_k}^{+\infty} \left(\sqrt{|q(x)|}-\sqrt{|q_+|}\right)\,dx,\quad A_+=e^{i\lambda C_+}, \tag{1.13} \]

\[ c_j=\int_{x_{2j-1}}^{x_{2j}}\sqrt{|q(x)|}\,dx,\quad a_j=\int_{x_{2j}}^{x_{2j+1}}\sqrt{|q(x)|}\,dx. \tag{1.14} \]

In § 3 the problems $1^\circ$, $2^\circ$ are considered. In particular, the following is proved.

Theorem 3.2. Let the conditions of problem $2^\circ$ be satisfied, let $q(z)$ be an entire function, $q(z)\in Q_1$, and let $q(x)$ have $2m$ zeros. Then, as $\lambda\to+\infty$, $m>1$

\[ s_{11}=-i(A_+A_-)^{-1}s_{11}^0,\quad s_{21}=A_-A_+^{-1}s_{21}^0, \tag{1.15} \]

\[ s_{11}^0= 2^{m-1}\exp\left(\lambda\sum_{j=1}^m c_j\right) \left(\prod_{j=1}^{m-1}\cos\lambda a_j+O(\lambda^{-1})\right), \tag{1.16} \]

where $a_j$, $c_j$ are defined by formula (1.14), and $s_{21}^0$ has the same form as $s_{11}^0$. For $m=1$ the last factor in (1.16) should be replaced by $1+O(\lambda^{-1})$.

For \(m=1, 2\), and in some particular cases for \(m>2\), these formulas were obtained earlier (see [1], [3–6]). Analogous formulas have been obtained by us for problem \(1^\circ\) (Theorem 3.1). Asymptotic expansions in powers of \(\lambda^{-1}\) have also been found for the reflection coefficient in the simplest case.

In § 4, problem \(2^\circ\) is studied in more detail: \(q(x)\) has 4 zeros. It is well known (see [3]) that in this case the transmission coefficient \(D_+(\lambda)\) has maxima for certain values \(\lambda=\lambda_n\) (the resonance phenomenon). In the first approximation (see [3])

\[ \lambda_n=a_1^{-1}\left(n\pi+\frac{\pi}{2}+O(n^{-1})\right). \tag{1.17} \]

In § 4 an asymptotic expansion of \(\lambda_n\) in powers of \(n^{-1}\) is found (see (4.1.13)), and the resonance values \(D_+(\lambda_n)\) are computed.

1. Let \(c_1\ne c_2\) (\(c_j\), see (1.14)). Then, as \(n\to\infty\),

\[ D_+(\lambda_n)=4\left|\sqrt{\frac{q_+}{q_-}}\exp[-2\lambda_n|c_1-c_2|]\right|(1+O(n^{-1})). \tag{1.18} \]

1. Let \(c_1=c_2\); then

\[ D_+(\lambda_n)=1+O(n^{-2}). \tag{1.19} \]

For rectangular potentials, as is well known, at the resonance values \(\lambda_n\) there is complete transmission; but for analytic potentials, as we see, this is not so. The only exception is even \(q(x)\). In § 4 it is also shown that there exists a sequence of complex values \(\lambda_n^*\), close to \(\lambda_n\), with exponentially small imaginary part and such that \(D_+(\lambda_n^*)=1\). The \(\lambda_n^*\) are determined by formula (4.1.12). The problem of quasistationary levels is also investigated (see Theorem 4.2). Less precise results are obtained in the case where \(q(x)\) has \(m>4\) zeros.

In §§ 5, 6 problem \(3^\circ\) is considered (above-barrier reflection). A result known to physicists is rigorously proved (Theorem 5.1, item \(1^\circ\); see [7–9]), and new cases are investigated (Theorem 5.1, item \(2^\circ\), and Theorem 6.1). In § 6 some results proved in § 3 for entire functions \(q(z)\) are transferred to meromorphic \(q(z)\).

In § 2 we give information on the Schrödinger equation needed for reading the article, describe the classes of potentials \(Q_j\), and prove a number of auxiliary propositions (Lemma 2.10 is apparently of independent interest). The method used in the present article is set forth in detail in the proof of the theorem of § 1.

§ 2. SOME FACTS ABOUT THE SCHRÖDINGER EQUATION

Classes of potentials \(q(z)\). Auxiliary propositions

Many of the definitions given below are contained in [2]. A reader familiar with article [2] may omit items 1–4 of this section.

Throughout § 2 it is assumed that \(\lambda>0\).

1. Stokes lines. Let \(q(z)\) be an entire function. The zeros of \(q(z)\) are called turning points. Denote

\[ \xi(z)=\int \sqrt{q(z)}\,dz,\qquad \xi(z_0,z)=\int_{z_0}^{z}\sqrt{q(t)}\,dt. \tag{2.1.1} \]

A Stokes line (S. L.) is a regular connected component of the curve \(\operatorname{Re}\xi(z)=\mathrm{const}\), passing through a turning point. Let \(\Phi\) denote the set of all S. L., and \(R_z\) the Riemann sphere. We shall say that \(q(z)\in Q_0\) if \(q(z)\) is an entire function and the following conditions are satisfied:

1) \(\overline{\Phi}=\Phi\);

2) \(\Phi\setminus l=\overline{\Phi}\setminus l\);

3) \(\displaystyle \lim_{\substack{z\to\infty\\ z\in l}}|\operatorname{Im}\xi(z_0,z)|=+\infty\)

for every S. L. \(l\), and the same is true for \(-q(z)\);

1. all zeros of \(q(z)\) are simple.

It is proved in [2] that if conditions 1) and 3) are fulfilled, then every connected component of the set \(R_z\setminus\Phi\) is a domain of one of the following two types:

1. A domain \(D_1\) of half-plane type. \(\partial D_1\) (the boundary of \(D_1\)) consists of one connected component, \(\xi(z)\) is single-valued in \(D_1\), and \(\xi(D_1)\) is the half-plane \(\operatorname{Re}\xi>a\) (or \(\operatorname{Re}\xi<a\)).

2. A domain \(D_2\) of strip type. \(\partial D_2\) consists of two connected components, \(\xi(z)\) is single-valued in \(D_2\), and \(\xi(D_2)\) is the strip \(a<\operatorname{Re}\xi<b\).

Condition 3) is approximately equivalent to the following: \(q(z)\) has no asymptotic value \(0\), i.e. the point \(z=\infty\) is not a turning point. Condition 4) is immaterial; our methods fully permit the study of the case in which \(q(z)\) has multiple zeros. We note that examples of entire functions \(q(z)\) for which condition 1) or 2) is not fulfilled are unknown.

A domain \(D\subset R_z\) is called canonical if \(\xi(z)\) is single-valued in \(D\) and \(\xi(D)\) is the whole \(\xi\)-plane with a finite or infinite number of vertical cuts. All cuts are infinite. It is shown in [2] that if \(q(z)\in Q_0\), then every S. L. can be enclosed in some canonical domain.

2. Elementary fundamental systems of solutions (e.f.s.s.). The e.f.s.s. \(u(z), v(z)\) of equation (1.3) are determined by specifying \((l,z_0,D)\), where \(l\) is an S. L., \(z_0\in l\) is a turning point, and \(D\supset l\) is a canonical domain (see [2]). \(z_0\) fixes the orientation of \(l\). In this case \(u, v\) must satisfy the following conditions:

\[ u(z)\sim c\,[q(z)]^{-1/4}\exp(\lambda\xi(z_0,z)), \tag{2.2.1} \]

\[ v(z)\sim c\,[q(z)]^{-1/4}\exp(-\lambda\xi(z_0,z)), \]

where

\[ |c|=1,\quad \lim_{\substack{z\to z_0\\ z\in l}}\arg c\,[q(z)]^{-1/4}=0, \tag{2.2.2} \]

\[ \operatorname{Im}\xi(z_0,z)>0\quad \text{for } z\in l. \tag{2.2.3} \]

Let \(\widetilde D_\varepsilon\) denote the domain \(D\) from which the preimages of \(\varepsilon\)-neighborhoods of the cuts lying in \(\xi(D)\) have been removed. Formula (2.2.1) must hold in \(\widetilde D_\varepsilon\) uniformly with respect to \(z\in \widetilde D_\varepsilon\) as \(\lambda\to+\infty\) for both \(u\) and \(v\), and, in addition, for \(u(z)\) uniformly with respect to \(\lambda\) for \(\lambda>\lambda_0(\varepsilon,D,q)>0\) as \(z\to\infty\), \(\operatorname{Re}\xi(z)\to+\infty\), and for \(v(z)\) as \(\operatorname{Re}\xi(z)\to-\infty\).

3. Classes of potentials \(q(z)\)

Denote by \(D_\varepsilon\) the domains \(D\) from which the \(\varepsilon\)-neighborhoods of the preimages of the turning points have been removed. By \(l_{z_0}^{+}\) we denote a curve issuing from the point \(z_0\) and such that \(\xi(l_{z_0}^{+})\) is a broken line, and \(\operatorname{Re}\xi(z_0,z)\) does not decrease along \(l_{z_0}^{+}\), and

\[ \sup_{z\in l_{z_0}^{+}}\operatorname{Re}\xi(z_0,z)=+\infty . \tag{2.3.1} \]

The curve \(l_{z_0}^{-}\) is defined analogously. Put

\[ \delta(z)=\frac{[q'(z)]^2}{[q(z)]^{5/2}}+\frac{q''(z)}{[q(z)]^{3/2}}, \tag{2.3.2} \]

\[ \rho_\varepsilon(z,D)=\inf_{l_z^{\pm}}\int_{l_z^{\pm}}|\delta(t)|\,|dt|, \tag{2.3.3} \]

where \(z,l_z^{\pm}\subset D_\varepsilon\).

Class \(Q_1\). \(q(z)\in Q_1\) if \(q(z)\in Q_0\) and, for any canonical domain \(D\),

\[ \sup_{z\in D_\varepsilon}\rho_\varepsilon(z,D)=\delta_\varepsilon<\infty,\quad \lim_{\substack{z\to\infty\\ z\in D_\varepsilon}}\rho_\varepsilon(z,D)=0. \tag{2.3.4} \]

If \(q(z)\in Q_1\), then in any canonical domain there exists a fundamental system of solutions.

Introduce the functions (see [2], § 4):

\[ \beta_0^{+}(z)=\beta_1^{+}(z)=0,\quad \gamma_0^{+}(z)=1, \]

\[ \beta_{k+1}^{+}(z)=-\int_{l_z^{+}}\delta(t)\bigl(\beta_k^{+}(t)+\gamma_k^{+}(t)\bigr)\,dt, \]

\[ \gamma_{k+1}^{+}(z)=\sum_{n=0}^{k-1}[q(z)]^{-1/2}L^{k-n-1}\bigl(\delta(z)(\beta_n^{+}(z)+\gamma_n^{+}(z))\bigr), \]

where

\[ L\varphi(z)=\frac{d}{dz}\bigl([q(z)]^{-1/2}\varphi(z)\bigr), \]

and \(\beta_k^{-}(z),\gamma_k^{-}(z)\) are defined analogously.

Put:

\[ \rho_0(z)=\rho(z,D),\quad \rho_{k+1}(z)= \]

\[ =\inf_{l_z^{\pm}}\left\{ \int_{l_z^{\pm}}|\delta(t)|\,|\rho_k(t)|\,|dt|+ \right. \]

\[ \left. +\sum_{n=0}^{k}|q(z)|^{-1/2}\left| L^{n-k}\delta(z)\bigl(\beta_n^{\pm}(z)+ \right. \right. \]

\[ \left. \left. +\gamma_n^{\pm}(z)\bigr)\right| +\sum_{n=0}^{k}\int_{l_z^{\pm}}\left| L^{k-n+1}\delta(t)\times \right. \right. \]

\[ \left. \left. \times\bigl(\beta_n^{\pm}(t)+\gamma_n^{\pm}(t)\bigr) \right|\,|dt| \right\}. \]

Class \(Q_\infty\). An entire function \(q(z)\in Q_\infty\) if \(q(z)\in Q_0\) and, for any canonical domain \(D\), and for all \(k\geqslant 0\) (except \(\gamma_0^{\pm}(z)\))

\[ \sup_{z\in D_\varepsilon}\left(\beta_k^{\pm}(z),\ \gamma_k^{\pm}(z),\ \rho_k(z)\right)<\infty, \]

\[ \lim_{\substack{z\to\infty\\ z\in D_\varepsilon}} \left(\beta_k^{\pm}(z),\ \gamma_k^{\pm}(z),\ \rho_k(z)\right)=0. \tag{2.3.5} \]

All polynomials \(q(z)\in Q_{\infty}\).

We shall say that \(q(z)\in Q_2\) if \(q(z)\in Q_0\) and condition (2.3.5) is satisfied for \(k=0,1\).

Remark. Let the canonical domain \(D\) be such that all cuts in \(\xi(D)\) are directed to one side, say downward. Let the domain \(D_0\subset D_\varepsilon\) and
\[ \sup_{z\in D_0}\operatorname{Im} z=+\infty . \]
Then formulas (2.2.1) hold as \(z\to\infty\) in \(D_0\), provided that \(\operatorname{Im}\xi(z)\to+\infty\), uniformly in \(\lambda\) for \(\lambda\geq\lambda_0>0\) (where \(\lambda_0\) is arbitrary but fixed). This fact was proved in [2], and we take it into account in constructing canonical domains (see § 2, item 5).

4. Transition matrices. If we have two F.s.f.s. \(u_1,v_1\) and \(u_2,v_2\), then any solution \(w_0(z)\) can be written in the form
\[ w_0=\alpha_1u_1+\beta_1v_1=\alpha_2u_2+\beta_2v_2; \]
\(\alpha_j,\ \beta_j\) depend only on \(\lambda\). Then
\[ \begin{pmatrix} \alpha_2\\ \beta_2 \end{pmatrix} =\Omega \begin{pmatrix} \alpha_1\\ \beta_1 \end{pmatrix}. \]

The matrix \(\Omega\) is called the transition matrix from \(u_1,\ v_1\) to \(u_2,\ v_2\) and is denoted by \((1,2)\). This matrix does not depend on \(w_0(z)\). We note that in [2], \((\Omega')^{-1}\) is used instead of \(\Omega\). As shown in [2], every transition matrix is a product of certain elementary transition matrices, which we shall now write out. Everywhere in what follows it is assumed that \(q(z)\in Q_1\).

If \(u_1,\ v_1\) and \(u_2,\ v_2\) are determined by \((l,z_1,D)\) and \((l,z_2,D)\), then
\[ (1,2)=e^{i\varphi_0} \begin{pmatrix} 0 & e^{-i\lambda\alpha}\\ e^{i\lambda\alpha} & 0 \end{pmatrix}, \quad \alpha=|\xi(z_1,z_2)|, \tag{2.4.1} \]
\[ e^{i\varphi_0}=\frac{c_2}{c_1}. \]

If \(u_1,\ v_1\) and \(u_2,\ v_2\) are determined by \((l_1,z_1,D)\) and \((l_2,z_2,D)\), and \(l_2\) lies to the left of \(l_1\), then
\[ (1,2)=e^{i\varphi_0} \begin{pmatrix} e^{-\lambda c} & 0\\ 0 & e^{\lambda c} \end{pmatrix}, \quad c=\xi(z_1,z_2),\quad \operatorname{Re}c>0. \tag{2.4.2} \]

If \(z_1,\ z_2\) and \(q(x)\) are real, then
\[ \varphi_0=0 \quad \text{in (2.4.1) and} \quad \varphi_0=-\frac{\pi}{6} \quad \text{in (2.4.2).} \]

If \(u_j,\ v_j\) are determined by \((l_j,z_0,D_j)\), \(j=1,2,3\) (in the subsequent formulas all indices are taken mod 4), \(z_0\) is a simple zero of \(q(z)\), \(l_{j+1}\) lies to the left of \(l_j\), and the part of \(D_j\) lying to the left of \(l_j\) coincides with the part of \(D_{j+1}\) lying to the right of \(l_{j+1}\), then
\[ (j,j+1)=e^{-i\pi/6} \begin{pmatrix} 0 & \alpha_{j,j+1}^{-1}\\ 1 & i\alpha_{j+1,j+2} \end{pmatrix}, \tag{2.4.3} \]

\[ a_{12}a_{23}a_{31}=1. \tag{2.4.4} \]

We shall call such a triple of regions consistent. Formally, we have

\[ a_{j,j+1}\sim \exp\left(\sum_{k=1}^{\infty}(-\lambda)^{-k}a_{jk}\right), \tag{2.4.5} \]

\[ a_{jk}=\int_{l_{j,j+1}}\alpha_k(z)\,dz, \tag{2.4.6} \]

\[ \alpha_0(z)=-\frac{q'(z)}{4q(z)},\qquad \alpha_{k+1}(z)=-\frac{1}{2\sqrt{q(z)}}\times \tag{2.4.7} \]

\[ \times\left(\sum_{m=0}^{k}\alpha_m(z)\alpha_{k-m}(z)+\alpha'_k(z)\right). \]

Here the path \(l_{j,j+1}\subset D_j\cup D_{j+1}\) begins in \(D_{j+1}\) where \(\operatorname{Re}\xi(z_0,z)\to+\infty\), and ends in \(D_j\) where \(\operatorname{Re}\xi(z_0,z)\to-\infty\). The branch of \(\sqrt{q(z)}\) is chosen in the same way as for \(u_j,v_j\).

If \(q(z)\in Q_1\,(Q_2)\), then formula (2.4.5) holds with accuracy up to \(O(\lambda^{-1})\) (respectively with accuracy up to \(O(\lambda^{-2})\)), i.e. \(a_{ij}=1+O(\lambda^{-1})\) for \(q(z)\in Q_1\). If \(q(z)\in Q_\infty\), then formula (2.4.5) holds.

5. Topological lemmas. In what follows, in order to solve the scattering problem we shall choose the F.S.R. in a special way. In this subsection we indicate the choice of the corresponding canonical regions and clarify some properties of the Stokes lines.

Everywhere below it is assumed that \(q(x)\) has a finite number of zeros \(x_1<x_2<\cdots<x_k\).

We shall state two lemmas proved in [11].

Lemma 2.1. Let \(q(z)\in Q_0\) and satisfy the conditions of problem \(1^\circ\). Then there exists a region \(D\supset(x_k,+\infty)\) such that \(\xi(z)\) is single-valued in \(D\), \(D=D^*\) (\(D^*\) is the region symmetric to \(D\) with respect to \(Ox\)), and \(\xi(D)\) is the half-plane \(\operatorname{Re}\xi>a\) (or \(\operatorname{Re}\xi<a\)) with a finite or infinite number of vertical cuts.

Let \(q(x_1)=q(x_2)=0,\ q(x)>0\) for \(x_1<x<x_2\). Denote by \(l_j,\ j=1,2,\) the S.L. issuing from \(x_j\), \(\operatorname{Im}z>0\) for \(z\in l_j\).

Lemma 2.2. Let \(q(z)\in Q_0\), and let it satisfy the conditions formulated above and the conditions of one of the problems \(1^\circ\)–\(3^\circ\). Then there exists a region \(D\) such that \(\partial D\supset l_j,\ j=1,2,\ D=D^*\); \(\xi(z)\) is single-valued in \(D\), and \(\xi(D)\) is the strip \(a<\operatorname{Re}\xi<b\) with a finite number of vertical cuts.

Lemma 2.3. Let \(q(z)\in Q_0\) and satisfy the conditions of problem \(2^\circ\). Then there exists a canonical region \(D\supset l_0\)—the ray \((x_k,+\infty)\), \(D=D^*\), and all cuts in \(\xi(D)\) are directed to the side opposite to \(\xi(l_0)\).

Proof. Since \(q(x)<0\) for \(x>x_k\), \(l_0\) is an S.L. By virtue of the reality of \(q(x)\), all S.L. are symmetric with respect to \(Ox\), and we shall construct only the upper half \(D_0\) of the region \(D\). Let \(l_1\) be the S.L. issuing from \(x_1\) and lying in the upper half-plane. Denote by \(D_1\) a region such that \(D_1\subset R_z\setminus\Phi\), \(\partial D_1\subset\Phi\), and \(\partial D_1\supset l_0\cup l_1\). If \(D_1\) is a region of half-plane type, then put \(D=D_1\cup l_0\cup D_1^*\). Then \(\xi(z)\) is single-valued in \(D\), and \(\xi(D)\) is the whole \(\xi\)-plane with a cut along the ray \(\xi(l_1\cup l_1^*)\), which is directed to the side opposite to \(\xi(l_0)\), and the lemma is proved. If \(D_1\) is a region of strip type, then \(\partial D_1\) has a connected compo-

the boundary \(\Gamma_1\), not containing \(l_0\). Put \(\sqrt{q(x)}=i|\sqrt{q(x)}|\) for \(x>x_k\). Then \(\xi(0,x)\) maps \(D_1\) onto the strip \(a<\operatorname{Re}\xi<0\), \(l_0\) onto the ray \(\operatorname{Re}\xi=0\), \(\operatorname{Im}\xi\gg 0\), and \(\Gamma_1\) onto the straight line \(\operatorname{Re}\xi=a\). Further, \(\Gamma_1\) contains zeros of \(q(z)\), and their number is, of course, finite; let \(\xi_0\) be the image of a zero of \(q(z)\) such that on the ray \(\operatorname{Re}\xi=a\), \(\operatorname{Im}\xi<\operatorname{Im}\xi_0\), there are no images of zeros of \(q(z)\). Let \(\Gamma'_1\) be the preimage of this ray, and let \(D_2\subset R\setminus\Phi\) be a domain such that \(\partial D_2\supset \Gamma'_1\). Consider the domain \(D'_2=D_1\cup \Gamma'_1\cup D_2\). If \(D_2\) is a domain of half-plane type, then \(D'_2\) may be taken as \(D_0\), and the lemma is proved. If, however, \(D_2\) is a domain of strip type, then we continue this process and adjoin \(\Gamma'_2\cup D_3\) to \(D'_2\). Put
\[ D_0=\left(\bigcup_{i=1}^{n} D_i\right)\cup\left(\bigcup_{i=1}^{n-1}\Gamma'_i\right), \]
where \(D_n\) is the first domain of half-plane type among the domains \(D_i\) (in particular, such a domain may fail to exist, and then \(n=\infty\)). By construction, \(D_0\) satisfies all the conditions of the lemma.

Lemma 2.4. Let \(q(z)\in Q_0\) and satisfy the conditions of problem \(3^\circ\). Then:

1) There exists a domain \(D_1=D_1^*\supset Ox\) of strip type.

2) There exists a canonical domain \(D=D^*\supset D_1\), and all cuts in \(\xi(D)\) are directed to the side opposite to \(\xi(l_0)\), where \(l_0\) is the ray \((0,+\infty)\).

Proof. Since \(q(x)<0\) on the real axis \(Ox\), \(\operatorname{Re}\xi(0,x)=0\), i.e., \(Ox\) is a level line of the function \(\operatorname{Re}\xi(0,z)\), and therefore \(Ox\subset R_z\setminus\Phi\). The maximal connected component \(D_1\subset R\setminus\Phi\) containing \(Ox\) is symmetric with respect to \(Ox\) (since \(q(x)\) is real). Therefore \(\partial D_1\) consists of no fewer than two connected components, and hence \(D_1\) is a domain of strip type. The domain \(D\supset D_1\) is constructed in the same way as in Lemma 2.3.

Lemma 2.5. Let \(q(z)\in Q_0\) and satisfy the conditions of one of the problems \(1^\circ\)—\(2^\circ\). Let \(q(x_1)=q(x_2)=0\), \(q(x)<0\) for \(x\in I\), where \(I\) is the interval \((x_1,x_2)\). Then there exists a canonical domain \(D=D^*\supset I\).

The proof of this lemma is analogous to the proof of Lemma 2.3.

6. Connection of the f.s.r. and \(\psi_j^{\pm}(x)\). We shall now show that the f.s.r. corresponding to the canonical domains constructed in § 2, item 5, are very simply related to \(\psi_j^{\pm}(x)\).

Lemma 2.6. Let the conditions of Lemma 2.1 be fulfilled, \(q(z)\in Q_1\). Denote by \(l_k\) the Stokes line issuing from \(x_k\), \(\operatorname{Im}z>0\) for \(z\in l_k\), and let \(u_k(z),v_k(z)\) be the f.s.r. determined by \((l_k,x_k,D_k)\). Here \(D_k\) is a canonical domain, \(D_k\supset D\) (\(D\) is the domain constructed in Lemma 2.1). If \(\psi_0(x)\) is a solution of equation (1.3) such that, as \(x\to+\infty\), \(\psi_0(x)\to 0\) and
\[ \psi_0(x)\sim [q(x)]^{-\frac14}\exp(-\lambda \xi(x_k,x)), \tag{2.6.1} \]
then
\[ \psi_0(x)=e^{-\frac{i\pi}{12}}v_k(x). \tag{2.6.2} \]

This lemma is proved in [11].

Lemma 2.7. Let the conditions of Lemma 2.3 and condition (1.9) be fulfilled, \(q(z)\in Q_1\). Introduce the f.s.r. \(u_k(z),v_k(z)\), determined by \((l_k,x_k,D_k)\). Here \(D_k=D\) (see Lemma 2.3), \(l_k\) is the ray \(x>x_k\). Then
\[ u_k(x)=A_+\psi_1^+(x),\qquad v_k(x)=A_+^{-1}\psi_2^+(x), \tag{2.6.3} \]
where \(A_+\) is determined by formula (1.13).

Proof. Since all cuts in \(\xi(D_k)\) are directed in the direction opposite to \(\xi(l_k)\), it follows (see item 2, § 2) that \(u_k(x)\), \(v_k(x)\) have, as \(x\to+\infty\), the asymptotics (2.2.1). Moreover, for \(x>x_k\),

\[ c_k[|q(x)|]^{-1/4}=|q(x)|^{-1/4},\qquad \operatorname{Im}\xi(x_k,x)>0, \]

\[ \sqrt{q(x)}=i|\sqrt{q(x)}|\qquad \text{for } x>x_k . \]

Next,

\[ \int_{x_k}^{x}\sqrt{|q(t)|}\,dt = x\sqrt{|q_+|}+C_+ + \int_x^\infty\left(\sqrt{|q(t)|}-\sqrt{|q_+|}\right)\,dt \]

and the last term is \(o(1)\) as \(x\to+\infty\). Consequently,

\[ \lim_{x\to+\infty} u_k(x)[\psi_1^+(x)]^{-1}=A_+, \]

whence (2.6.3) follows, and similarly for \(v_k,\ \psi_2^+\).

Corollary. Suppose the conditions of Lemma 2.3 and condition (1.5) are satisfied, \(q(z)\in Q_1\). Let \(l_0\) denote the ray \(x<x_1\), and introduce the s.f.s. \(u_0(x), v_0(x)\), determined by \((l_0,x_1,D_0)\). Here \(D_0\) is such a canonical domain that all cuts in \(\xi(D_0)\) are directed in the direction opposite to \(\xi(l_0)\) (\(D_0\) can be constructed in the same way as the domain \(D\) in Lemma 2.3). Then

\[ u_0(x)=A_-\psi_2^-(x),\qquad v_0(x)=A_-^{-1}\psi_1^-(x), \tag{2.6.4} \]

where \(A_-\) is defined by formula (1.13).

We now consider problem \(3^\circ\). Let \(D_1\) be the domain constructed in Lemma 2.4, \(\Gamma^+=\partial D_1\cap\{\operatorname{Im}z>0\}\). On \(\Gamma^+\) there lies a finite number of zeros of \(q(z)\); let \(z_1\) be the leftmost and \(z_k\) the rightmost zeros of \(q(z)\) on \(\Gamma^+\). Denote by \(l_0\) the part of \(\Gamma^+\) lying to the left of \(z_1\), and by \(l_k\) the part of \(\Gamma^+\) lying to the right of \(z_k\),

\[ C_\pm^0=\int_0^{\pm\infty}\left(\sqrt{|q(x)|}-\sqrt{|q_\pm|}\right)\,dx, \tag{2.6.5} \]

\[ c_j=\xi(0,z_j),\quad j=1,k,\qquad \operatorname{Re}c_j>0, \tag{2.6.6} \]

the path of integration for \(c_j\) lies in \(D_1\) (see Lemma 2.4),

\[ A_\pm^0=e^{i\lambda C_\pm^0}, \tag{2.6.7} \]

\[ \varphi_j=\Delta\arg [q(z)]^{1/4},\quad j=1,k, \tag{2.6.8} \]

under continuation of \(\sqrt[4]{q(x)}\) from \(Ox\) to \(l_0,l_k\), respectively, inside \(D_1\). Construct the canonical domains \(D_k,D_0\); \(D_k\) coincides with the domain \(D\) constructed in Lemma 2.4, \(D_0\supset D_1\), \(D_0^*=D_0\), and all cuts in \(\xi(D_0)\) are directed in the same direction as \(\xi(l)\). Introduce s.f.s. with numbers \(0,k\), determined by \((l_0,z_1,D_0)\) and \((l_k,z_k,D_k)\), respectively.

Lemma 2.8. Suppose the conditions of Lemma 2.4 and conditions (1.6), (1.9) are satisfied, \(q(z)\in Q_1\). Then

\[ u_k(x)=e^{i\varphi_k}e^{\lambda c_k}A_+^0\psi_1^+(x),\qquad v_k(x)=e^{i\varphi_k}e^{-\lambda c_k}(A_+^0)^{-1}\psi_2^+(x), \tag{2.6.9} \]

\[ u_0(x)=e^{i\varphi_0}e^{-\lambda c_0}A_-^0\psi_2^-(x),\qquad v_0(x)=e^{i\varphi_0}e^{\lambda c_0}(A_-^0)^{-1}\psi_1^-(x). \tag{2.6.10} \]

Proof. By virtue of the choice of \(u_k, v_k\), we have \(\operatorname{Im}\xi(z_k,z)>0\) on \(l_k\) and, consequently, on \(Ox\) for sufficiently large \(x>0\). Further,

\[ \xi(z_k,x)=c_k+C_+^0+o(1) \tag{2.6.11} \]

as \(x\to+\infty\). By virtue of the choice of \(D_k\), the asymptotics (1.11) for \(u_k, v_k\) is suitable for fixed \(\lambda>0\) and \(x\to+\infty\). Therefore (2.6.9) follows from (2.6.11) and the normalization of \(u_k, v_k\), and is proved in the same way as (2.6.3). Similarly we obtain (2.6.10).

7. Refinement of the transition matrices. The reality of \(q(x)\) makes it possible, in a number of cases, to compute the transition matrices more accurately, which in turn will allow us to obtain more accurate results in the scattering problem.

Lemma 2.9. Let \(q(z)\in Q_1\), let \(q(x)\) be real, \(q(x)>0\) for \(x>x_0\), where \(x_0\) is a simple zero of \(q(x)\), and let \(q_+>0\). Denote by \(l_0,l_1,l_{1'}\) the Stokes lines issuing from \(x_0\), where \(l_0=l_0^*\), \(\operatorname{Im} z>0\) for \(z\in l_1\). Introduce the e.f. s. r. defined by \((l_j,x_0,D_j)\), \(j=0,1,1'\). Here \(D_0=D_0^*\), \(D_{1'}^*=D_1\) and \(D_1\supset D\), where \(D\) is the domain constructed in Lemma 2.1. Then

\[ \overline{\alpha_{10}}\alpha_{01'}=1,\qquad |\alpha_{1'1}|=1. \tag{2.7.1} \]

Proof. By virtue of the choice of the canonical domains, we have

\[ u_0(\bar z)=\overline{v_0(z)},\qquad u_{1'}(z)=e^{-\,i\pi/12}\psi(z), \tag{2.7.2} \]

where \(\psi(x)\) is real. Further,

\[ \psi(x)\equiv A u_0(x)+B v_0(x), \tag{2.7.3} \]

and from the reality of \(\psi(x)\) it follows that \(\bar A=B\). By virtue of the definition of the transition matrices, we have

\[ u_{1'}=e^{i\pi/6}\bigl(\alpha_{01'}u_0-i\alpha_{10}^{-1}v_0\bigr). \tag{2.7.4} \]

From (2.7.2)—(2.7.4) follows (2.7.1).

Let \(q(x_j)=0\), \(j=1,2\), and \(q(x)>0\) for \(x_1<x<x_2\). Denote by \(l_j,l_{j'}\) the Stokes lines issuing from \(x_j\), \(\operatorname{Im} z>0\) for \(z\in l_j\), \(l_{j'}=l_j^*\), and by \(l_0,l_3\) the Stokes lines issuing from \(x_1,x_2\) and forming part of \(Ox\). Introduce e.f. s. r. \((l_j,x_j,D_j)\) with numbers \(0\le j\le 3\) and \(1',2'\); here \(x_j=x_1\) or \(x_2\), depending on which of the latter points is the end of \(l_j\), \(D_3=D_3^*\), \(D_0=D_0^*\). Further, let \(D\) be the domain constructed in Lemma 2.2, \(\partial D\supset l_1\cup l_2\). Put

\[ D_2=D_0^+\cup D\cup D_3^+\cup l_1\cup l_2, \]

\[ D_{2'}=D_0^+\cup D\cup D_3^-\cup l_1\cup l_{2'}, \]

\[ D_1=D_2,\qquad D_{1'}=D_{2'}^*, \]

where \(D^+=D\cap\{\operatorname{Im} z>0\}\), and \(D^-\) is defined analogously.

Lemma 2.10. Let \(q(z)\in Q_1\) and satisfy the conditions formulated above. Choose the e.f. s. r. as indicated above. Then

\[ \alpha_{1'1}=(1+\delta)^{-1/2}e^{i\zeta_1},\qquad \alpha_{22'}=(1+\delta)^{+1/2}e^{i\zeta_2}, \tag{2.7.5} \]

\[ \alpha_{1'1}=\alpha_{32}\,\overline{\alpha}_{2'3},\qquad \delta=e^{-2\lambda c}\bigl(1+O(\lambda^{-1})\bigr), \]

\[ c=\int_{x_1}^{x_2}\sqrt{q(x)}\,dx, \tag{2.7.6} \]

\[ \operatorname{Im}\left(a_{10}\,\overline{\alpha_{32}}\right)=0 \tag{2.7.7} \]

and \(\varphi_1,\ \varphi_2\) are real.

Proof. By virtue of the choice of the e.f.s. we have

\[ u_j(\bar z)=\overline{v_j(z)},\quad j=0,\ 3. \tag{2.7.8} \]

Further,

\[ u_0=Au_3+Bv_3,\quad v_0=Cu_3+Dv_3. \tag{2.7.9} \]

Replacing \(z\) in (2.7.9) by \(\bar z\) and then replacing all quantities by their complex conjugates, we find (taking (2.7.8) into account) that

\[ v_0=\overline{A}v_3+\overline{B}u_3,\quad u_0=\overline{C}v_3+\overline{D}u_3. \tag{2.7.10} \]

From (2.7.9), (2.7.10) we obtain

\[ A=\overline{D},\quad B=\overline{C}. \tag{2.7.11} \]

Further,

\[ \begin{pmatrix} u_0\\ v_0 \end{pmatrix} = \left(((1,0)\cdot(2,1)\cdot(3,2))'\right)^{-1} \begin{pmatrix} u_3\\ v_3 \end{pmatrix}, \]

where the matrices \((1,0)\), \((3,2)\) have the form (2.4.2), and the matrix \((2,1)\) has the form (2.4.3). Hence we find (the relation (2.4.4) is satisfied, since the canonical domains are matched)

\[ A=ie^{\lambda c}\left(-(a_{1'1}\alpha_{2'3})^{-1}+e^{2\lambda c}a_{10}\right), \]

\[ B=e^{\lambda c}a_{32}\alpha_{1'1}^{-1},\quad C=e^{\lambda c}\alpha_{2'3}^{-1}, \]

\[ D=ie^{\lambda c}\alpha_{32}. \tag{2.7.12} \]

Substituting (2.7.12) into (2.7.11) and putting

\[ \alpha_{1'1}=x,\quad \alpha_{32}\overline{\alpha_{2'3}}=y,\quad \delta=e^{-2\lambda c}a_{10}\overline{\alpha_{32}}^{-1}, \]

we obtain the system of equations

\[ x=y,\quad x=\overline{y}^{-1}(1+\delta)^{-1}. \]

From this system we find that \(\delta\) is real and that

\[ |x|=(1+\delta)^{-1/2},\quad \delta=e^{-2\lambda c}(1+O(\lambda^{-1})). \tag{2.7.13} \]

Since

\[ \alpha_{22'}\alpha_{2'3}\alpha_{32}=1, \]

we have

\[ \alpha_{22'}=\frac{\overline{\alpha_{2'3}}}{\alpha_{2'3}}\,y^{-1},\quad |\alpha_{22'}|=|x|^{-1}. \]

The lemma is proved.

This lemma is also of independent interest. Indeed, formulas (2.4.3), (2.4.5) determine \(\alpha_{ij}\) only with algebraic accuracy \(O(\lambda^{-n})\), whereas formula (2.7.4) makes it possible to determine \(|\alpha_{1'1}|\), \(|\alpha_{22'}|\) with exponential accuracy in \(\lambda\).

§ 3. REFLECTION FROM A BARRIER AND PASSAGE THROUGH A BARRIER

In this section we shall consider problems \(1^\circ, 2^\circ\) (see § 1) and obtain the simplest formulas. More delicate questions will be considered in § 4.

1. Problem \(1^\circ\). We shall retain the notation of § 1, item 1 and Lemmas 2.1, 2.6.

Theorem 3.1. Let \(q(z)\in Q_1\) and satisfy the conditions of problem \(1^\circ\). Denote by \(x_j\), \(1\le j\le 2m+1\), \(x_j<x_{j+1}\), the real zeros of \(q(x)\). Then, as \(\lambda\to+\infty\), \(m>1\),

\[ s_{11}=Ce^{-\frac{i\pi}{4}}A_-s_{11}^0, \tag{3.1.1} \]

\[ s_{11}^0=2^m\exp\left(\lambda\sum_{j=1}^m c_j\right) \prod_{j=1}^m\left(\cos\lambda\alpha_j+O(\lambda^{-1})\right), \tag{3.1.2} \]

where \(C\) is an arbitrary constant, and \(A_-\), \(c_j\), \(\alpha_j\) are defined by formulas (1.12), (1.14). For \(m=1\) the last factor in (3.2) should be replaced by \(1+O(\lambda^{-1})\). For \(s_{12}\) analogous formulas hold.

Proof. The method of proof is the same for all the theorems of the present paper, and this time we shall carry out the proof in detail. Our plan is as follows. The function \(\Psi(x)\) must satisfy the condition \(\Psi(+\infty)=0\) (see (1.7)). We shall express \(\Psi(x)\) in terms of some e.f.s. \(r\) (see Lemma 2.6), for one of whose solutions the asymptotic formula (2.2.1) holds for \(\lambda>0\), \(x\to+\infty\). Then we shall express \(\Psi(x)\) in terms of another e.f.s. \(r\), for which the asymptotic formula (2.2.1) is suitable for \(\lambda>0\), \(x\to-\infty\). The transition from one e.f.s. \(r\) to another will be carried out with the help of transition matrices; in doing so we shall have to introduce some auxiliary e.f.s. \(r\). Expressing then the latter e.f.s. \(r\) in terms of \(\Psi_j^-(x)\), \(j=1,2\) (see § 2, item 6), we shall find \(s_{11}\), \(s_{12}\). Let us also note that if we were interested only in the leading term of the asymptotics of \(s_{lj}(\lambda)\), then we could refrain from taking care over the choice of canonical domains, and simply pass from one Stokes line to another (see [2], § 5).

1. Choice of canonical domains and e.f.s. Since all zeros of \(q(x)\) are simple, three Stokes lines issue from \(x_j\). Denote by \(l_j\) the Stokes line such that \(x_j\in l_j\) and \(\operatorname{Im} z>0\) on \(l_j\). Put \(l_{j'}=l_j^*\), \(l_0=(-\infty,x_1)\), \(l_{0j}=(x_{2j},x_{2j+1})\), \(1\le j\le m\). Construct canonical domains \(D_j\supset l_j\), \(D_{0j}\supset l_{0j}\). We shall choose the domain \(D_0\) so that \(D_0=D_0^*\) and all cuts in \(\xi(D_0)\) are directed to the side opposite to \(\xi(l_0)\). This can be done by virtue of Lemma 2.3. We shall choose the domain \(D_{01}\) so that \(D_{01}^*=D_{01}\), which can be done by virtue of Lemma 2.5. Further, by Lemma 2.2 there exists a domain \(G_0\) such that \(\partial G_0\supset l_1\cup l_2\), \(G_0=G_0^*\), and \(\xi(G_0)\) is a strip \(a<\operatorname{Re}\xi<b\) with a finite number of cuts. Put

\[ D_1=D_0^+\cup G_0\cup D_{01}^+\cup l_1\cup l_2, \]

\[ D_{1'}=D_0^-\cup G_0\cup D_{01}^-\cup l_{1'}\cup l_{2'}, \]

\[ D_2=D_1,\qquad D_{2'}=D_{1'}^*. \]

These domains are coordinated with one another (see § 2, item 4), i.e., if \(a,b\) are two neighboring Stokes lines issuing from \(x_1\) or \(x_2\), then \(\xi(D_a\cap D_b)\) contains a half-plane with a finite or infinite number of vertical cuts. This gives us the possibility of using formulas (2.4.3), (2.4.4). Let us also note that the canonical domains have been chosen by us in the same manner as in Lemma 2.10.

Now we can introduce e.f.s.s. with numbers \(0, 1, 2, 1', 2', 01\) and \((01)'\). The canonical domains \(D_j, D_{j'}, D_{0j}\) and the e.f.s.s. with numbers \(j, j'\), \(3\leq j\leq 2m+1\), and \((0,j), (0,j)'\), \(2\leq j\leq m\), are introduced analogously. In this case \(D_{2m+1}\) is chosen in the same way as the domain \(D\) in Lemma 2.1.

2. Connection of e.f.s.s. with solutions. Calculation of transition matrices. The solution \(\Psi(x)\) of interest to us in problem \(1^\circ\) satisfies the condition \(\Psi(+\infty)=0\). Therefore \(\Psi(x)\), up to a constant factor \(C\), coincides with \(\Psi_0(x)\) (see Lemma 2.6). Put \(C=1\). Then, by virtue of (2.6.2),

\[ \Psi(x)=e^{-\frac{i\pi}{12}}v_{2m+1}(x). \tag{3.1.3} \]

We shall now express \(\Psi(x)\) in terms of \(u_0, v_0\). For this it is necessary to compute the transition matrix

\[ (2m+1,0)=\prod_{j=1}^{2m+1}(j,j-1)\equiv \|a_{ij}\|. \tag{3.1.4} \]

First let us compute the matrix \((2j+1,\,2j-1)\). For \(j>0\) we have

\[ (2j+1,\,2j-1)=(2j,\,2j-1)(2j+1,\,2j), \]

where the matrix \((2j,\,2j-1)\) has the form (2.4.2), \(c=c_j\), \(\varphi_0=0\), and

\[ (2j+1,\,2j)=((0j)',\,2j)(0j,\,(0j)')(2j+1,\,0j)= \]

\[ = e^{-\frac{i\pi}{3}}\|b^j_{ik}\|. \tag{3.1.5} \]

In this product the outer matrices have the form (2.4.3), and the middle one has the form (2.4.1), \(\alpha=\alpha_j,\ \varphi_0=0\). Therefore

\[ b^j_{11}=0,\qquad b^j_{12}=e^{i\lambda\alpha_j}+O(\lambda^{-1}), \]

\[ b^j_{21}=e^{-i\lambda\alpha_j}+O(\lambda^{-1}),\qquad b^j_{22}=2i\cos\lambda\alpha_j+O(\lambda^{-1}). \tag{3.1.6} \]

Next,

\[ (2j+1,\,2j-1)=-i e^{\lambda c_j}\|c^j_{ik}\|, \tag{3.1.7} \]

\[ c^j_{12}=e^{-2\lambda c_j}b^j_{12}, \tag{3.1.8} \]

and the remaining \(c^j_{ik}\) coincide with \(b^j_{ik}\). Thus,

\[ (2m+1,\,0)=(1,0)\cdot\prod_{j=1}^{m}(2j+1,\,2j-1), \]

where the matrix \((1,0)\) has the form (2.4.3), and we obtain

\[ a_{11}=e^{-\frac{2}{3}\pi i}e^{-i\lambda\alpha_m}2^{m-1} \prod_{j=1}^{m-1}\bigl(\cos\lambda\alpha_j+O(\lambda^{-1})\bigr), \]

\[ a_{21}=i\bigl(1+O(\lambda^{-1})\bigr)a_{11}, \]

\[ a_{22}=e^{\frac{i\pi}{6}}s^0_{11}\bigl(1+O(\lambda^{-1})\bigr),\qquad a_{12}=-ia_{22}\bigl(1+O(\lambda^{-1})\bigr). \tag{3.1.9} \]

Since

\[ \Psi(x)=\bigl(a_{11}u_0(x)+a_{12}v_0(x)\bigr)e^{-\frac{i\pi}{12}}, \tag{3.1.10} \]

it follows from (3.1.9), (3.1.10), and Lemma 2.7 that (3.1.1), (3.1.2) hold.

Remark. For \(m>1\) the last factor in (3.1.2) is equal to

\[ b_{22}^{1}\cdots b_{22}^{m}(2i)^{-m}+O(e^{-\lambda c_0}), \tag{3.1.11} \]

\[ c_0=\min_j c_j . \]

To derive this formula it is enough simply to multiply the matrices in (3.1.4) more carefully.

2. Problem \(2^\circ\). We shall keep the notation of § 1, item 1 and of Lemmas 2.3, 2.7.

Theorem 3.2. Let \(q(z)\in Q_1\) and let it satisfy the conditions of problem \(2^\circ\). Denote by \(x_j\), \(1\le j\le 2m\), \(x_j<x_{j+1}\), the real zeros of \(q(x)\). Then as \(\lambda\to+\infty\), \(m>1\),

\[ s_{11}=-i(A_+A_-)^{-1}s_{11}^{0},\qquad s_{21}=A_-A_+^{-1}s_{21}^{0}, \tag{3.2.1} \]

\[ s_{11}^{0}=2^{m-1}\exp\left(\lambda\sum_{j=1}^{m-1}c_j\right) \prod_{j=1}^{m-1}\left(\cos\lambda\alpha_j+O(\lambda^{-1})\right), \]

\[ s_{21}^{0}=s_{11}^{0}\left(1+O(\lambda^{-1})\right), \tag{3.2.2} \]

where \(A_{\pm}\), \(c_j\), \(\alpha_j\) are determined by formulas (1.12)—(1.14). For \(m=1\) the last factor in (3.13) should be replaced by \(1+O(\lambda^{-1})\).

Proof. In this case the rays \(l_0:x<x_1\) and \(l_{2m+1}:x>x_{2m}\) are p.c., since \(q(x)<0\) on these rays. Choose the canonical domains \(D_0\supset l_0\) and \(D_{2m+1}\supset l_{2m+1}\) in the same way as in Lemma 2.3, and connect them by a chain of canonical domains analogously to how this was done in the proof of Theorem 3.1. Put

\[ \Psi(x)=u_{2m+1}(x) \tag{3.2.3} \]

and express \(\Psi(x)\) in terms of \(u_0(x)\), \(v_0(x)\). We have

\[ (2m+1,0)=\prod_{j=1}^{2m-1}(j,j-1)\cdot(2m,2m-1)(2m+1,2m)=\|d_{ij}\|. \tag{3.2.4} \]

The last two factors have the form (2.4.2), (2.4.3), respectively, and the product appearing in (3.2.4) has the form (3.1.9) with \(m\) replaced by \(m-1\). Therefore

\[ d_{11}=-is_{11}^{0}\left(1+O(\lambda^{-1})\right),\qquad d_{21}=s_{11}^{0}\left(1+O(\lambda^{-1})\right). \tag{3.2.5} \]

Since

\[ \Psi(x)=d_{11}u_0(x)+d_{21}v_0(x), \tag{3.2.6} \]

from (3.2.5), (3.2.6) and (2.6.3), (2.6.4) we obtain (3.2.1), (3.2.2). Analogously one can compute \(s_{21}\), \(s_{22}\).

3. Asymptotics of \(\Psi(x)\) in the simplest case. The transmission coefficient \(D_+\) and the reflection coefficient \(R_+\) in problems \(2^\circ\), \(3^\circ\) for a wave moving from left to right are equal to (see [1])

\[ D_+=\sqrt{\frac{q_+}{q_-}}\,|s_{11}|^{-2},\qquad R_+=|s_{11}|^{-2}|s_{12}|^{2} \tag{3.3.1} \]

and \(D_+ + R_+ = 1\). If \(m=1\), then from (3.2.1), (3.2.2) it follows that \(D_+\sim \sqrt{\dfrac{q_+}{q_-}}\,e^{-2\lambda c_1}\), i.e. the transmission coefficient is exponentially small.

Now we shall find the asymptotics of \(\Psi(x,\lambda)\) for \(m=1\), normalizing this solution by the condition

\[ \Psi(x)=\Psi_1^{-}(x)+B\Psi_2^{-}(x)=A\Psi_1^{+}(x). \tag{3.3.2} \]

Then

\[ A=s_{11}^{-1},\quad B=s_{12}s_{11}^{-1}. \tag{3.3.3} \]

In order not to encumber the formulation of the theorem with formulas, we write out all the necessary asymptotic expansions in advance. Denote

\[ \varphi_{\pm}(l_j)=\exp\left[\sum_{k=1}^{\infty}(\pm\lambda)^{-k}\int_{l_j}a_k(t)\,dt\right], \tag{3.3.4} \]

where the \(a_k(z)\) are determined by formula (2.4.7), and the series in (3.3.4) is asymptotic in \(\lambda\). Then for \(x>x_2+\varepsilon\) and as \(\lambda\to+\infty\) (in what follows the latter condition is understood everywhere)

\[ \Psi_1^{+}(x)\sim |q(x)|^{-\frac14}\exp(\lambda\xi(x_2,x))\varphi_{-}(l_1), \tag{3.3.5} \]

where \(\sqrt{q(x)}=i|\sqrt{q(x)}|\), \(l_1\) is the ray \((x,+\infty)\). Similarly, for \(x<x_1-\varepsilon\),

\[ \Psi_{1,2}^{-}(x)\sim |q(x)|^{-\frac14}\exp(\mp\lambda\xi(x_1,x))\varphi_{\mp}(l_2), \tag{3.3.6} \]

where \(\sqrt{q(x)}=i|\sqrt{q(x)}|\), \(l_2\) is the ray \((x,-\infty)\); the plus sign is taken for \(\Psi_2^{-}(x)\), and the minus sign for \(\Psi_1^{-}(x)\). Finally, for \(x_1+\varepsilon<x<x_2-\varepsilon\),

\[ \Psi_1^{+}(x)\sim |q(x)|^{-\frac14}\exp(\lambda\xi(x_2,x))\varphi_{-}(l_3), \tag{3.3.7} \]

where \(\sqrt{q(x)}>0\), \(l_3\) is a contour joining the point \(x\) with \(+\infty\) and passing around the point \(x_2\) from above.

Further, from formulas (2.4.5), (2.4.6) we obtain

\[ \alpha_{1'1}=\varphi_{-}(l_4),\quad \alpha_{01'}=\varphi_{-}(l_5). \tag{3.3.8} \]

The contour \(l_4\) encloses the ray \(x<x_2\), passes around it counterclockwise, and \(\sqrt{q(x)}<0\) for \(x\in l_4\). The contour \(l_5\) joins \(+\infty\) with \(-\infty\) and passes around the point \(x_1\) from below; \(\sqrt{q(x)}<0\) for \(x\in l_5\), \(x_1<x<x_2\).

Theorem 3.3. Suppose the conditions of Theorem 3.2 are satisfied, \(m=1\), and \(q(z)\in Q_{\infty}\). Then, as \(\lambda\to+\infty\), the function \(\Psi(x,\lambda)\), normalized by condition (3.3.2), has the form:

1)

\[ x>x_2+\varepsilon,\quad x_1+\varepsilon<x<x_2-\varepsilon, \]

\[ \Psi(x)=e^{-\lambda c_1}\alpha_{01'}^{-1}A_{-}A_{+}\Psi_1^{+}(x); \tag{3.3.9} \]

2)

\[ x<x_1-\varepsilon, \]

\[ \Psi(x)=\Psi_1^{-}(x)-i\,\alpha_{1'1}^{-1}A_{-}^{2}\Psi_2^{-}(x). \tag{3.3.10} \]

The asymptotic expansions for all the expressions entering into (3.3.9)—(3.3.10) have been written out above.

The proof of this theorem coincides with the proof of Theorem 3.2, except that we compute the matrix \((3,0)\) more accurately. Using (2.4.1)—(2.4.4), we obtain

\[ (3,0)=e^{\lambda c_1}\|d_{ij}\|, \]

\[ d_{11}=-i\alpha_{10}^{-1},\quad d_{21}=\alpha_{1'1}. \tag{3.3.11} \]

From (3.2.6), (3.3.11), and Lemma 2.7 we find

\[ A=e^{-\lambda c_1}a_{01}^{-1}A_-A_+,\quad B=-i\alpha_{1,1}A_-^2. \tag{3.3.12} \]

Next, from (2.4.3) it follows that

\[ v_2(z)=e^{\frac{i\pi}{6}}u_3(z), \tag{3.3.13} \]

which proves the theorem.

Our methods make it possible to find the asymptotics of \(\Psi(x)\) also in the case when \(q(x)\) has a larger number of zeros.

(To be continued)

Received by the editors
December 8, 1964

Moscow Institute of Physics and Technology