MATHEMATICS

M. Sh. BIRMAN, S. B. ENTINA

ON THE STATIONARY APPROACH IN ABSTRACT SCATTERING THEORY

(Presented by Academician V. I. Smirnov on 6 XII 1963)

1. In the present note, within the framework of the abstract theory of operators, a justification is given for the so-called stationary version of scattering theory. Under the assumption that the perturbation is nuclear, convenient representations are given (see formulas (3)—(5)) for the wave operators* and scattering operators in terms of boundary values of resolvents; also justified is one practically convenient method for computing the \(S\)-matrix. At the same time, a new, purely “stationary,” proof is obtained of the Rosenblum—Kato theorem \((^{4-6})\) on the existence of wave operators. A methodological advantage of the method proposed here consists, among other things, in the absence of a limiting passage from finite-dimensional perturbations. This makes it possible to transfer the results obtained to the case of more general perturbations, which, in particular, makes them applicable in the three-dimensional problem of quantum scattering. Let us also note here that relation (3) is the “correct” form of the well-known Lippmann—Schwinger equation \((^7)\).

2. Let us first record several auxiliary propositions. Let \(A\) be a self-adjoint operator in a Hilbert space \(\mathfrak H\), \(F_\lambda\) its resolution of the identity, \(\Gamma_z\) its resolvent, and \(D\) some operator of class** \(\mathfrak S_2\). Then:

\(1^\circ.\) For any element \(f \in \mathfrak H\), for almost all (a.e.) \(\lambda\) there exists the strong derivative \(d(DF_\lambda f)/d\lambda\).

\(2^\circ.\) For a.e. \(\lambda\) there exist the strong limits of the element \(D\Gamma_z f\), when \(z \to \lambda \pm i0\), i.e., when \(z \to \lambda\) along a path not tangent to the real axis in the upper (lower) half-plane.

\(3^\circ.\) For a.e. \(\lambda\) there exists the strong derivative

\[ d(D^*F_\lambda D)/d\lambda \equiv K_\lambda \in \mathfrak S_1. \]

\(4^\circ.\) For a.e. \(\lambda\), as \(z \to \lambda \pm i0\), there exist the strong limits \(M_\lambda^{(\pm)}\) of the operators \(D^*\Gamma_zD\), with \(M_\lambda^{(\pm)} \in \mathfrak S_2\) and
\[ M_\lambda^{(+)} - M_\lambda^{(-)} = -2\pi iK_\lambda \in \mathfrak S_1. \]

Propositions \(1^\circ\) and \(3^\circ\) are essentially known. They follow easily from I. M. Gelfand’s theorem \((^8)\) on the differentiability of functionals with strongly bounded variation. Propositions \(2^\circ\) and \(4^\circ\) are derived from \(1^\circ\) and \(3^\circ\) with the aid of the theorem on boundary values of Cauchy—Stieltjes integrals. Let us note that the exceptional null sets of values \(\lambda\) in \(1^\circ\) and \(2^\circ\) depend, generally speaking, on the element \(f\).

3. Everywhere in what follows, \(H_0, H\) are self-adjoint operators in \(\mathfrak H\) with common domain of definition \(\mathfrak D\); \(E_\lambda^0, E_\lambda\) and \(R_z^0, R_z\) are their corresponding resolutions of the identity and resolvents; \(V = H - H_0\); \(P_0, P\) are the projectors onto the absolutely continuous subspaces \(\mathfrak G_0, \mathfrak G\) of the operators \(H_0, H\)

* For the definition of wave operators, the scattering operator, and other notions of abstract scattering theory, see, for example, \((^1)\) or \((^2)\). An abstract definition of the \(S\)-matrix is given in \((^3)\).

** By \(\mathfrak S_2\) we denote the class of Hilbert—Schmidt operators, and by \(\mathfrak S_1\) the class of nuclear operators.

Accordingly. Put also \(Q_z^0=E+VR_z^0,\ Q_z=E-VR_z\), and note that \(Q_z^0=Q_z^{-1}\), and

\[ R_z^0-R_z=Q_z^{0*}(R_z-R_{\bar z})Q_z^0 . \tag{1} \]

If \(V\in\mathfrak S_1\), then, by virtue of \(2^\circ\), a.e. there exist strong limits \(Q_{\lambda\pm i0}^0 f\) of the elements \(Q_z^0 f\). Similarly for \(Q_z f\). In addition, by virtue of \(1^\circ\)—\(3^\circ\), for any \(f,g\in\mathfrak H\), a.e. in \(\lambda\) and a.e. in \(\mu\) there exist the derivatives
\(d(E_\lambda Q_{\mu\pm i0}^0 f,g)/d\lambda\) and \(d(E_\lambda Q_{\mu\pm i0}^0 f,Q_{\mu\pm i0}^0 g)/d\lambda\). Finally, passing to the limit in (1), with the aid of \(1^\circ\)—\(4^\circ\) one can find that a.e.

\[ \bigl[d(E_\lambda Q_{\mu\pm i0}^0 f,Q_{\mu\pm i0}^0 g)/d\lambda\bigr]_{\mu=\lambda} = d(E_\lambda^0 f,g)/d\lambda . \tag{2} \]

From (2) it follows immediately that

\[ \int_{-\infty}^{+\infty} \left| \bigl[d(E_\lambda Q_{\mu\pm i0}^0 f,g)/d\lambda\bigr]_{\mu=\lambda} \right|\,d\lambda \leqslant \|P_0 f\|\,\|Pg\|. \]

We can now introduce bounded operators \(W_\pm\) by means of the relation

\[ (W_\pm f,g)= \int_{-\infty}^{+\infty} \bigl[d(E_\lambda Q_{\mu\pm i0}^0 f,g)/d\lambda\bigr]_{\mu=\lambda}\,d\lambda . \tag{3} \]

Theorem 1. Under the condition \(V\in\mathfrak S_1\), the wave operators \(W_\pm(H,H_0)\) coincide with the operators \(W_\pm\) defined by the relations (3). Along with (3), for \(W_\pm\) the representation

\[ (W_\pm f,g)= \int_{-\infty}^{+\infty} \bigl[d(E_\lambda^0 f,Q_{\mu\pm i0}g)/d\lambda\bigr]_{\mu=\lambda}\,d\lambda \tag{4} \]

is valid.

Let us note that the wave operators exist by the Rosenblum—Kato theorem. They map \(\mathfrak G_0\) isometrically onto \(\mathfrak G\) and implement a unitary equivalence of the absolutely continuous parts of \(H_0\) and \(H\). These properties of the operators \(W_\pm\) can also be derived directly from relations (2)—(4). Taking into account the connection between the definition (3) and the usual nonstationary definition of the wave operators, we arrive at a new (“stationary”) proof of the existence of wave operators for nuclear perturbations. Let us point out in this connection that, in the case of a one-dimensional perturbation, a stationary proof was given earlier by T. Kato \((^5)\). Formula (3) was indicated in a somewhat different form by one of the authors (see formula (7) in \((^9)\)).

The scattering operator \(S=W_+^*W_-\) admits representations of the same type as (3), (4). Namely, for any \(f,g\in\mathfrak H\),

\[ (Sf,g)= \int_{-\infty}^{+\infty} \bigl[d(E_\lambda Q_{\mu-i0}^0 f,Q_{\lambda+i0}^0 g)/d\lambda\bigr]_{\mu=\lambda}\,d\lambda, \]

\[ (Sf,g)= \int_{-\infty}^{+\infty} \bigl[d(E_\lambda^0 Q_{\mu+i0}Q_{\mu-i0}^0 f,g)/d\lambda\bigr]_{\mu=\lambda}\,d\lambda . \tag{5} \]

4. Let us now recall the abstract definition of the \(S\)-matrix given in \((^3)\). Let \(\mathfrak G_0\) be decomposed into a continuous direct sum \((^{10})\) of Hilbert spaces

\[ \mathfrak G_0=\int_\Lambda \oplus\,\mathfrak H_\lambda\,d\lambda \tag{6} \]

so that the part of the operator \(H_0\) in \(\mathfrak G_0\) becomes the operator of multiplication by \(\lambda\). Here \(\Lambda\) is the spectrum of \(H_0\) in \(\mathfrak G_0\). The operator \(S\) is unitary in \(\mathfrak G_0\) and commutes with \(H_0\), and therefore in the decomposition (6) there corresponds to it a measurable family of unitary operators \(S_\lambda\) in \(\mathfrak H_\lambda\). The operator \(S_\lambda\), defined for a.e. \(\lambda\in\Lambda\),

is called the \(S\)-matrix (the scattering suboperator). The bilinear form \(S_\lambda\) coincides with the integrand in formula (5). We describe another method for computing the \(S\)-matrix.

Introduce into consideration the operator \(T_z=P_0Q_zVP_0\). In the expansion (6), the operator \(T_z\) corresponds to the “kernel” \(T_z(\lambda,\mu)\) in the following sense. \(T_z(\lambda,\mu)\) is a measurable family of nuclear operators, defined for a.e. \(\lambda\in\Lambda\) and a.e. \(\mu\in\Lambda\), and acting from \(\mathfrak h_\mu\) into \(\mathfrak h_\lambda\). Moreover, if \(f(\lambda)\in\mathfrak h_\lambda\) is the representation of the element \(f\in\mathfrak G_0\) in the expansion (6), then

\[ (T_z f)(\lambda)=\int_\Lambda T_z(\lambda,\mu)f(\mu)\,d\mu . \tag{7} \]

Denote by \(I_\lambda\) the identity operator in \(\mathfrak h_\lambda\). The following holds.

Theorem 2. For a.e. real \(\nu\), a.e. \(\lambda\in\Lambda\), and a.e. \(\mu\in\Lambda\), the operators \(T_z(\lambda,\mu)\) converge in the nuclear norm as \(z\to\nu+i0\). The limiting operator \(T_{\nu+i0}(\lambda,\mu)\) is related to the scattering suboperator \(S_\lambda\) by the relation

\[ S_\lambda=I_\lambda-2\pi iT_{\lambda+i0}(\lambda,\lambda) \quad(\text{for a.e. }\lambda\in\Lambda). \]

The results of §§ 3, 4 are easily carried over to the case of a pair of unitary operators, first considered in \((^3)\).

1. We now give one generalization of the preceding results, which makes it possible to use them directly in applications. By the same method as Theorem 1, one proves the following.

Theorem 3. Let \(|V|^{1/2}(H_0-iE)^{-n}\in\mathfrak S_2\), \(|V|^{1/2}(H-iE)^{-n}\in\mathfrak S_2\) \((n=1,2,\ldots)\). Then there exist the wave operators \(W_\pm(H,H_0)\), \(W_\pm(H_0,H)\).

We note that under the hypotheses of Theorem 3 the representation (3) remains valid for arbitrary \(g\in\mathfrak H\) and for a dense set of \(f\in\mathfrak H\). For \(n=1\), Theorem 3 contains the criterion for the existence of wave operators due to S. T. Kuroda \((^{11})\), and, in turn, is contained in the criterion obtained by M. Sh. Birman and M. G. Kreĭn in \((^3)\). For \(n>1\), Theorem 3 gives a new criterion for the existence of wave operators, containing the results of I. V. Stankevich \((^{12})\). It is interesting that for \(n>1\) the result obtained does not follow from the general criterion for the existence of wave operators \((^{13},\,^2)\), formulated in terms of functions of operators.

Under the hypotheses of Theorem 3 the operator \(T_z\) is in any case defined on elements \(f\in\mathfrak D\). It is easy to show that now also the operator \(T_z\) admits the integral representation (7) with kernel \(T_z(\lambda,\mu)\in\mathfrak S_1\).

Theorem 4. Under the hypotheses of Theorem 3, the assertions of Theorem 2 are valid.

In conclusion we note that the rule for computing \(S_\lambda\) given by Theorems 2 and 4 is, from a formal point of view, known to physicists. For the three-dimensional problem of quantum scattering this rule was justified by L. D. Faddeev \((^{14})\). However, even in application to this problem, Theorem 4 gives new information, since in \((^{14})\) the limiting passage is understood in a different sense and is justified in a special (momentum) representation.

Leningrad State University
named after A. A. Zhdanov

Received
29 XI 1963

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