Abstract
Full Text
MATHEMATICS
Z. A. AGRANOVICH and V. A. MARCHENKO
RECOVERY OF TENSOR FORCES FROM SCATTERING DATA
(Presented by Academician S. N. Bernstein, 9 X 1957)
- In our note ((^{1})) an analysis and a method of solution were given for the inverse problem of scattering theory for a system of differential equations whose coefficients have a first absolute moment on the half-axis (0 < x < \infty). Starting from these results, one can carry out an analogous investigation also in the case most interesting for physics, when the coefficients of the differential equations contain singularities of order (x^{-2}) (both at zero and at infinity).
We shall confine ourselves here to considering a system of two equations, equivalent to the matrix equation
[
Y''-\left[V(x)+6x^{-2}P\right]Y+\lambda^2Y=0
\qquad (0<x<\infty),
\tag{A}
]
where (V(x)=|v_{jk}(x)|_1^2) is a Hermitian matrix,* satisfying, for some (\varepsilon>0), the condition
[
\int_0^\infty t^{1+\theta}|V(t)|\,dt<\infty
\qquad (-\varepsilon<\theta<\varepsilon),
\tag{1}
]
and
[
P=\begin{pmatrix}0&0\0&1\end{pmatrix}.
]
As is known ((^{2,3})), the Schrödinger equation for the deuteron reduces to such a system if the tensor forces of interaction are taken into account.
- Let us consider the boundary-value problem generated by the system (A) and the boundary condition
[
Y(0)=0.
\tag{B}
]
Theorem 1. The boundary-value problem (A)—(B) has a continuous spectrum on the entire positive half-axis ((\lambda^2>0)) and, possibly, a finite number of nonpositive eigenvalues (0\ge \lambda_1^2>\lambda_2^2>\cdots>\lambda_p^2). For (\lambda^2) from the spectrum there exist solutions (U(x,\lambda)) of equation (A), vanishing at zero for (x=0) and generating Parseval’s equality, equivalent to the following expansion of the (\delta)-function:
[
\delta(x-y)\cdot I=
\sum_{k=1}^{p} U(x,\lambda_k)U^(y,\lambda_k)
+\frac{1}{2\pi}\int_0^\infty U(x,\lambda)U^(y,\lambda)\,d\lambda,
\tag{2}
]
where (I) is the identity matrix, and (U^) is the matrix Hermitian conjugate to (U).*
The matrices (U(x,\lambda)) entering formula (2) can be normalized so that, as (x\to\infty), the asymptotic equalities hold
[
U(x,\lambda)\sim e^{i\lambda x}\cdot I-e^{i\lambda x}\cdot S(-\lambda)
\qquad (\lambda^2>0),
\tag{3′}
]
[
U(x,\lambda_k)\sim e^{-|\lambda_k|x}\cdot M_k
\qquad (\lambda_k^2<0),
\tag{3″}
]
* We note that all matrices occurring in the paper—(Y(x), U(x,\lambda), S(\lambda), M_k), etc.—are square matrices of order two.
If (A=|a_{jk}|_2^1), then, by definition,
[
|A|=\max{|a_{11}|+|a_{12}|,\ |a_{21}|+|a_{22}|}.
]
and, if (\lambda_1=0), then
[
U(x,0)\sim x^{-2}\cdot M_1,\qquad M_1=mP\quad (m>0),
\tag{3'''}
]
where (S(\lambda)) is a unitary matrix, called the scattering matrix; (M_k) is a Hermitian matrix whose rank is equal to the multiplicity of the eigenvalue (\lambda_k^2).
The collection of the scattering matrix (S(\lambda)), the eigenvalues (\lambda_k^2\leqslant 0), and the corresponding matrices (M_k) will be called, for brevity, the scattering data of the boundary-value problem (A)—(B). Below a method is presented for reconstructing the potential (V(x)) from the scattering data of problem (A)—(B), and the necessary and sufficient conditions that these data must satisfy are found.
- The scattering matrix (S(\lambda)) of problem (A)—(B) has the following properties:
(1^\circ). The matrix (I-S(\lambda)) is the Fourier transform of a Hermitian matrix (F_1(t)), so that
[
F_1(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}[I-S(\lambda)]e^{i\lambda t}\,d\lambda.
\tag{4}
]
The elements of the matrix (F_1(t)), (-\infty<t<\infty), can be represented as the sum of two functions, one of which is summable, while the other is square-summable and bounded; for all (t>0) there exists (F_1'(t)) and
[
\int_0^\infty t^{1+\theta}\,\lvert F_1'(t)\rvert\,dt<\infty
\quad (-\varepsilon<\theta<\varepsilon).
]
(2^\circ). (S(0)P=P).
From property (1^\circ) of the matrix (S(\lambda)), according to Theorem 4 of note ((^1)), it follows that if we construct the matrix
[
F(t)=\sum' M_k^2 e^{-|\lambda_k|t}+F_1(t),
\tag{5}
]
where the summation on the right-hand side is extended over all nonzero eigenvalues of problem (A)—(B), then the equation
[
F(x+y)+K(x,y)+\int_x^\infty K(x,t)F(t+y)\,dt=0
\tag{6}
]
has, for all (x>0), a unique solution (K(x,y)), with (K(x,x)) differentiable for every (x>0), and the set of matrix solutions (Z(x,\lambda)) of the equation
[
Z''-W(x)Z+\lambda^2Z=0,
\tag{7}
]
[
W(x)=-2\frac{d}{dx}K(x,x),
\tag{8}
]
which are asymptotically equal as (x\to\infty) to the right-hand sides of formulas ((3')) and ((3''')), gives an expansion of the delta function analogous to (2).
For the further characterization of the scattering data of problem (A)—(B), let us consider (as in ((^1))) the equations
[
x(t)+\int_0^\infty x(\xi)F(t+\xi)\,d\xi=0,\qquad 0\leqslant t<\infty,
\tag{I}
]
[
-y(t)+\int_\infty^0 y(\xi)F_1(t+\xi)\,d\xi=0,\qquad -\infty<t\leqslant 0,
\tag{II}
]
[
z(t)+\int_0^\infty z(\xi)F_1(t+\xi)\,d\xi=0,\qquad 0\leqslant t<\infty.
\tag{III}
]
and denote the number of their linearly independent vector-solutions, respectively, by (n_1), (n_2), and (n_3), and the Fourier transform of any nonzero solution of equation (I) by (\tilde{x}(\lambda)).
Let (r) denote the sum of the ranks of all matrices (M_k) (if (\lambda_1=0), then the rank of the matrix (M_1=mP) is included in (r)).
Then the following four cases are possible:
a) all (\lambda_k\ne 0), (n_1=0), (n_2=0), (n_3=r), the matrix (K(x,y)) (see (6)) satisfies the equality
[
P\left{I+\int_0^\infty K(0,t)\,dt\right}=\gamma P,
]
where (\gamma) is a nonzero number;
b) (\lambda_1=0), (M_1=mP), (n_1=1), (n_2=0), (n_3=r), (\lambda^2\tilde{x}(\lambda)(I-P)\to 0) as (\lambda\to\infty);
c) all (\lambda_k\ne 0), (n_1=1), (n_2=0), (n_3=r), (\lambda^2\tilde{x}(\lambda)(I-P)\to 0) as (\lambda\to\infty), and, if (S(0)=I), then (\tilde{x}(0)(I-P)\ne 0);
d) all (\lambda_k\ne 0), (n_1=1), (n_2=1), (n_3=r+1), (\lambda^2\tilde{x}(\lambda)(I-P)\to 0) as (\lambda\to\infty), and, if (S(0)=I), then (\tilde{x}(0)(I-P)\ne 0).
The listed properties are characteristic for the scattering data of problem (A)—(B), i.e., the following theorem is valid.
Theorem 2. In order that the given unitary matrix (S(\lambda)), the numbers (\lambda_k^2\le 0), and the Hermitian matrices (M_k) ((k=1,2,\ldots,p)) be the scattering data of problem (A)—(B) with a Hermitian potential (V(x)) satisfying condition (1), it is necessary and sufficient that conditions (1^\circ), (2^\circ), and one of the conditions a), b), c), or d) be fulfilled.
Moreover, if (S(0)=I), then the potential (V(x)) is determined uniquely by the scattering data. If, however, (S(0)\ne I) (i.e., (\lambda=0) is a virtual level), then the potential (V(x)) is determined uniquely by the scattering data only in case a), while in cases b), c), and d) there exists a one-parameter family of potentials (V(x)) satisfying condition (1) and such that the corresponding boundary-value problems (A)—(B) have identical scattering data.
- To reconstruct the potential (V(x)) of problem (A)—(B) (in the case of non-uniqueness—the whole family of potentials) from the scattering data, we use the relation between (V(x)) and the matrix (W(x)) constructed by formula (8) from the solution of equation (6).
The characteristic properties of the scattering data (see Theorem 2) ensure the existence, for (\lambda=0), of a solution (Z(x)=|z_{jh}(x)|_1^2) of equation (7) such that* (Z(x)P=Z(x)) and
1) (Z(x)=-3xP+O(1)) as (x\to\infty);
2) (Z(0)=0) and ((I-P)Z'(0)=0) in case a), (Z(0)=0) in case b), (PZ(0)=0), and ((I-P)Z(0)\ne 0) in cases c) and d). Moreover, in cases c) and d)
[
\lim_{x\to 0}\left[Z^(x)Z(x,\lambda)-Z^(x)Z'(x,\lambda)\right]
=\lambda^2 h^2 Z^*(0)Z(0,\lambda),
\tag{9}
]
where (Z(x,\lambda)) is the solution of equation (7) with asymptotic (3′) as (x\to\infty), and (h^2>0) and does not depend on (\lambda). Note that in the cases of non-uniqueness the matrix (Z(x)) contains one arbitrary parameter.
In all cases the desired potential (V(x)) is found by the formula
[
V(x)=W(x)+2\frac{d}{dx}\left[Z(x)Z_1^*(x)\right]-6x^{-2}P,
\tag{10}
]
* If
[
\int_1^\infty t^2\lvert F_1'(t)\rvert\,dt<\infty,
]
then
[
Z(x)=
\begin{pmatrix}
0 & \alpha\
0 & -3x+\beta
\end{pmatrix}
\int_x^\infty
K(x,t)
\begin{pmatrix}
0 & \alpha\
0 & -3t+\beta
\end{pmatrix}
\,dt,
]
where (\alpha) and (\beta) are determined from the conditions at the point (x=0).
where
[
Z_1(x)=Z(x)\left[I-(1+c^2)P-\int_0^x Z^*(t)Z(t)\,dt\right]^{-1}
\tag{11}
]
and (c^2=0) in case a), (c^2=m^2) in case b), (c^2=h^2|z_{12}(0)|^2) in cases c) and d) (see formula (9)).
- To illustrate the results obtained, we give several examples.
I (case a)). (S(\lambda)=(\lambda+i)(\lambda-i)^{-1}(I-P)+P), the point spectrum is absent. Then (V(x)=W(x)=-2(\operatorname{ch}x)^{-2}(I-P)).
II (case b)). (S(\lambda)=(\lambda+i)(\lambda-i)^{-1}(I-P)+(\lambda+i)(\lambda+2i)[(\lambda-i)\times(\lambda-2i)]^{-1}P), (\lambda_1^2=0), (M_1=mP) ((m>0)). Then (W(x)=-2(\operatorname{ch}x)^{-2}\times(I-P)+6(\operatorname{sh}x)^{-2}P),
[
Z(x)=
\begin{pmatrix}
0 & \alpha\,\operatorname{th}x\
0 & \varphi(x)
\end{pmatrix},
]
where (\alpha) is an arbitrary number and
[
\varphi(x)=-3x+4.5-2.25\,(e^{-2x}-1+2x)(\operatorname{sh}x)^{-2};
]
the potential (V(x)) is expressed in terms of (W(x)) and (Z(x)) by formulas (10) and (11).
III (case c)). (S(\lambda)) is the same as in example II, (\lambda_1^2=-1), (M_1=I-P). Then (W(x)=-12e^{-2x}(1+1.5e^{-2x})^{-2}(I-P)+6(\operatorname{sh}x)^{-2}P),
[
Z(x)=
\begin{pmatrix}
0 & \alpha(1-1.5e^{-2x})(1+1.5e^{-2x})^{-1}\
0 & \varphi(x)
\end{pmatrix}
]
and (c^2=0.2\alpha^2).
IV (case d)). (S(\lambda)=(\lambda-i)(\lambda+i)^{-1}(I-P)+(\lambda+i)(\lambda+2i)[(\lambda-i)(\lambda-2i)]^{-1}P), the point spectrum is absent. Then (W(x)=6(\operatorname{sh}x)^{-2}P),
[
Z(x)=
\begin{pmatrix}
0 & \alpha\
0 & \varphi(x)
\end{pmatrix}
]
and (c^2=\alpha^2).
In examples III and IV, (\alpha) is arbitrary, and the function (\varphi(x)) is the same as in example II. We note that in all the examples given, (S(0)\ne I); in the first example (V(x)) is determined uniquely, while in the last three it is not unique, since the arbitrary parameter (\alpha) contained in (Z(x)) obviously also enters into expression (10) for (V(x)).
Received
9 X 1957
REFERENCES
- Z. S. Agranovich, V. A. Marchenko, DAN, 113, No. 5, 951 (1957).
- D. Blatt, V. Weisskopf, Theoretical Nuclear Physics, IL, 1954, pp. 84–88.
- R. G. Newton, Phys. Rev., 100, No. 1, 412 (1955).