Full Text
M. V. Fedoruk
ASYMPTOTICS OF A ONE-DIMENSIONAL SCATTERING PROBLEM
(Presented by Academician L. S. Pontryagin, December 2, 1964)
1. Statement of the problem
Consider the equation
\[ y''(x)-\lambda^2 q(x)y(x)=0, \tag{1} \]
where the function \(q(x)\) is real, has a finite number of zeros \(x_j\), \(1\le j\le k\) \((x_j<x_{j+1})\), and
\[ -\infty<q(\pm\infty)=q_{\pm}<0. \tag{2} \]
Let
\[ \int^{\pm\infty}\left|\sqrt{q(x)}-\sqrt{q_{\pm}}\right|\,dx<\infty,\qquad \int_{-\infty}^{\infty}|\delta(x)|\,dx<\infty, \tag{3} \]
\[ \delta(x)=|q'^2(x)|+|q''(x)|. \]
It is known that if conditions (2), (3) are satisfied and \(\lambda>0\), then equation (1) has solutions \(y_j^{+}(x)\) such that, as \(x\to+\infty\),
\[ y_{1,2}^{+}(x)\sim |q_+|^{-1/4}\exp\left(\pm i\lambda\sqrt{|q_+|}\,x\right), \tag{4} \]
and the same formulas hold for \(y_j^{-}(x)\) as \(x\to-\infty\). Then
\[ \binom{y_1^{+}}{y_2^{+}}=S(\lambda)\binom{y_1^{-}}{y_2^{-}}; \tag{5} \]
\(S(\lambda)\) is a matrix of order two, with \(s_{22}=\bar{s}_{11}\), \(s_{21}=\bar{s}_{12}\). The quantities
\[ D_+=\sqrt{\frac{q_+}{q_-}}\,|s_{11}|^{-2},\qquad R_+=|s_{21}s_{11}^{-1}|^2 \tag{6} \]
are called the transmission and reflection coefficients, respectively, and \(D_++R_+=1\).
Our aim is to find the asymptotics of \(S(\lambda)\) and \(D_+(\lambda)\) as \(\lambda\to+\infty\). We shall consider this problem under the assumption that \(q(z)\) is an entire function of \(z\), having only simple zeros. On \(q(z)\) we impose such conditions that make it possible to construct solutions of equation (1) admitting asymptotic expansions in series in powers of \(\lambda^{-1}\). Because of the unwieldiness of these conditions, we shall not write them down here. Conditions on \(q(z)\) that make it possible to obtain the first terms of the asymptotic expansion are given in [1].
2. The problem of transmission through a barrier
In this case \(q(x)\) has no zeros. Introduce the notation
\[ \xi(z_1,z_2)=\int_{z_1}^{z_2}\sqrt{q(z)}\,dz; \tag{7} \]
\[ \alpha_j=\left|\xi(x_{2j},x_{2j+1})\right|,\qquad c_j=\xi(x_{2j-1},x_{2j})>0; \tag{8} \]
\[ C_- = x_1\sqrt{|q_-|}+\int_{-\infty}^{x_1}\left(\sqrt{|q(x)|}-\sqrt{|q_-|}\right)\,dx,\qquad A_- = e^{i\lambda C_-}; \tag{9} \]
\[ C_+ = -x_k\sqrt{|q_+|}+\int_{x_k}^{+\infty}\left(\sqrt{|q(x)|}-\sqrt{|q_+|}\right)\,dx,\qquad A_+ = e^{i\lambda C_+}. \tag{10} \]
Theorem 1. Let \(q(x)\) have \(2m\) zeros and satisfy the conditions formulated above. Then, as \(\lambda\to +\infty\), \(m>1\),
\[ s_{11}=-i(A_+A_-)^{-1}s_{11}^{0},\qquad s_{21}=A_-A_+^{-1}s_{21}^{0}; \tag{11} \]
\[ s_{11}^{0}=2^{m-i}\exp\left(\lambda\sum_{j=1}^{m}c_j\right) \left(\prod_{j=1}^{m-1}\cos\lambda\alpha_j+O(\lambda^{-1})\right); \tag{12} \]
\(s_{21}^{0}\) has the same form as \(s_{11}^{0}\). For \(m=1\), the last factor in (12) should be replaced by \(1+O(\lambda^{-1})\).
For \(m=1,2\) and in some special cases with \(m>2\), these formulas were obtained earlier (see \((^2\!-\!^6)\)). We note that, for the proof of formulas (11), (12), it is enough to impose on \(q(x)\) conditions (2), (3).
As follows from Theorem 1, the coefficient \(D_+(\lambda)\) has, for \(m>1\), maxima near the points
\[ \lambda_{nj}^{0}=\pi\alpha_j^{-1}(n+1/2),\qquad 1\le j\le m-1. \]
The values of \(\lambda\) for which \(D_+(\lambda)\) has maxima are called resonance values. We shall study the following case in more detail.
3. The case \(m=2\).
Theorem 2. Let the conditions of Theorem 1 be satisfied, \(m=2\). Then, for sufficiently large \(\lambda>0\), the maxima of \(D_+(\lambda)\) are attained at the points \(\lambda_n\), \(n_0\le n<\infty\), and
\[ 1^\circ.\quad D_+(\lambda_n)\sim 4\sqrt{\frac{q_+}{q_-}}\exp[-2\lambda_n|c_1-c_2|], \tag{13} \]
if \(c_1\ne c_2\),
\[ 2^\circ.\quad D_+(\lambda_n)=1+O(n^{-2}), \tag{14} \]
if \(c_1=c_2\). If \(q(x)\) is an even function, then \(D_+(\lambda_n)=1\).
Formula (13) is unknown even in the physical literature.
For \(\lambda_n\) there is an asymptotic expansion as \(n\to\infty\)
\[ 2\lambda_n\alpha_1\sim 2n\pi+\pi-\sum_{k=1}^{\infty}(\lambda_n)^{-k}\int_C \alpha_k(z)\,dz, \tag{15} \]
where \(C\) is a closed contour enclosing the segment \([x_2,x_3]\) and not containing inside itself other zeros of \(q(z)\). For the functions \(\alpha_k(z)\), see \((^1)\). Analogous expansions hold for \(\lambda_{nj}\) \((m>2)\).
Theorem 3. Let the conditions of Theorem 1 be satisfied, \(c_1\ne c_2\). Then, for \(n>n_0\), there exist complex values \(\lambda_n^*\) such that \(D_+(\lambda_n^*)=1\) and
\[ \lambda_n^*-\lambda_n\sim \frac{i}{4}\alpha_1^{-1}\exp(-2\lambda_n c_0)\,\varepsilon, \tag{16} \]
\[ c_0=\min(c_1,c_2),\qquad \varepsilon=\operatorname{sign}(c_1-c_2). \]
4. The overbarrier reflection problem. In this case \(q(x)\) has no zeros and there exists a domain \(D\supset Ox\) such that \(\xi(D)\) is the strip \(-a<\operatorname{Re}\xi<a\) \((\xi=\xi(0,x))\), on \(\Gamma\)—the boundary of \(D\)—lie zeros \(q(z)\). The domain \(D\) is symmetric with respect to \(Ox\). Let \(\Gamma^+\) be the part of \(\Gamma\) lying in the upper half-plane. The case when \(\Gamma^+\) contains exactly 1 zero was investigated by physicists in \((^7,^8)\).
Theorem 4. Let \(q(x)\) have no zeros, satisfy the conditions formulated in item 1, and let \(\Gamma^{+}\) contain exactly two zeros \(z_1, z_2\) of the function \(q(z)\). Then, as \(\lambda \to +\infty\),
\[ s_{12}=-2i\exp\{-\lambda(c_1+c_2)\}\left(\cos\lambda\alpha+O(\lambda^{-1})\right)s_{12}^{0}, \tag{17} \]
\[ s_{12}^{0}=(A_{+}^{0}A_{-}^{0})^{-1},\qquad c_j=\xi(0,z_j),\qquad \alpha=|\xi(z_1,z_2)|. \]
Here \(\operatorname{Re} c_j>0\), and \(A_{\pm}^{0}\) are determined by formulas (9), (10), where \(x_1=x_k=0\).
In this case, near \(\lambda_n=\pi\alpha^{-1}(n+1/2)\) lie resonance values of \(\lambda\).
I express my deep gratitude to M. A. Evgrafov for a number of valuable suggestions and constant attention to the work.
Moscow Institute of Physics and Technology
Received
23 XI 1964
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