Full Text
M. Sh. Birman and M. G. Krein
On the Theory of Wave Operators and Scattering Operators
(Presented by Academician V. I. Smirnov on 5 I 1962)
In the present communication the concept of wave operators is extended to the case of a pair of unitary operators. The existence of these wave operators is established for unitary operators that differ by a trace-class operator. The use of the Cayley transform then makes it possible to obtain the existence of wave operators for a pair of self-adjoint operators under the single condition that the difference of the resolvents be trace class*.
By means of the wave operators, in the usual way, one constructs the scattering operator, which in turn gives rise to the \(S\)-matrix (the scattering matrix). In the second part of the communication, some spectral characteristics of the scattering matrix are established.
- Let \(U_1, U_2\) be unitary operators in a Hilbert space \(\mathfrak H\), and let \(P_k\) \((k=1,2)\) be the projector onto the absolutely continuous subspace \(\mathfrak H_k \subset \mathfrak H\) of the operator \(U_k\).
Theorem 1. If the operator \(V = U_2 - U_1\) is trace class, then:
1) there exist the strong limits (wave operators)
\[ W_{\pm}(U_2,U_1)=\lim_{n\to\pm\infty} U_2^n U_1^{-n}P_1; \tag{1} \]
\[ W_{\pm}(U_1,U_2)=\lim_{n\to\pm\infty} U_1^n U_2^{-n}P_2. \tag{2} \]
2) the operators \(W_{\pm}(U_2,U_1)\) isometrically map \(\mathfrak H_1\) onto \(\mathfrak H_2\); the operators \(W_{\pm}(U_1,U_2)=W_{\pm}^{*}(U_2,U_1)\) isometrically map \(\mathfrak H_2\) onto \(\mathfrak H_1\).
3) The absolutely continuous parts of the operators \(U_1\) and \(U_2\) are unitarily equivalent, and for every \(f\in\mathfrak H_1\)
\[ W_{\pm}(U_2,U_1)U_1 f = U_2 W_{\pm}(U_2,U_1)f. \]
The proof of this theorem is carried out essentially by the method of T. Kato \((^{2,3})\). Namely, in the case of a one-dimensional perturbation the proof is based on the study of the resolvents of the operators \(U_1, U_2\). The passage to the case of finite-dimensional \(V\) is made on the basis of the “multiplication theorem” for wave operators \((^{2,5})\). Finally, in the general case a limiting passage from finite-dimensional perturbations is used. In this connection we note that the approximation of a perturbation by simpler operators \(V_n\) must be carried out without destroying the unitarity of the operator \(U_1+V_n\).
* Wave operators for abstract self-adjoint operators were studied in \((^{1-6})\). A brief comparison of the results is given in the note \((^{6})\); see there also the definition of a wave operator, of the absolutely continuous part of an operator, etc. In contrast to note \((^{6})\), we denote the wave operators by \(W_{\pm}\), and not by \(U_{\pm}\). Concerning the definition of the class of trace-class operators, see, for example, \((^{7})\). In the note \((^{9})\) of one of the authors this class was denoted by \(\mathfrak S\). The condition that the difference of the resolvents \(R_z(H_2), R_z(H_1)\) be trace class means that for some common regular point \(z\) (and then for every such point) \(R_z(H_2)-R_z(H_1)\in\mathfrak S\).
The possibility of such an approximation was earlier indicated by one of the authors \(^{(8)}\) in extending the “trace formulas” to unitary operators.
- Let now \(H_1, H_2\) be self-adjoint operators in \(\mathfrak H\). We shall say that the operators \(H_1, H_2\) satisfy condition (A) if the difference of their resolvents is nuclear. Let \(n(>1)\) be an integer; we shall say that the operators \(H_1, H_2\) satisfy condition \((B_n)\) if they are positive definite, the operator \(H_2^{-1}-H_1^{-1}\) is completely continuous, and the operator \(H_2^{-n}-H_1^{-n}\) is nuclear. For the projection onto the absolutely continuous subspace \(\mathfrak H_k\) of the operator \(H_k\) we retain the notation \(P_k\) \((k=1,2)\).
Theorem 2. Let \(H_1,H_2\) be self-adjoint operators in \(\mathfrak H\) satisfying condition (A). Then there exist the strong limits
\[ W_{\pm}(H_2,H_1)=\lim_{t\to\pm\infty} e^{iH_2t}e^{-iH_1t}P_1 . \tag{3} \]
If \(U_k=(H_k-iI)(H_k+iI)^{-1}\), \((k=1,2)\), are the Cayley transforms of the operators \(H_k\), then the difference \(U_2-U_1\) is nuclear and
\[ W_{\pm}(H_2,H_1)=W_{\pm}(U_1,U_2). \tag{4} \]
Obviously, assertions analogous to assertions 2) and 3) of Theorem 1 are also valid. We note that Kato’s theorem \(^{(3)}\) on perturbations of a self-adjoint operator by a nuclear one follows already from Theorem 1 (it suffices to apply Theorem 1 to the operators \(\exp(iH_k)\)). Both of Kuroda’s criteria \(^{(4)}\) for the existence of wave operators are contained in Theorem 2. In proving Theorem 2 it is enough to establish the existence of the weak limits (3), as well as relation (4). Then, by Theorem 1, the operators \(W_{\pm}(H_2,H_1)\) are isometric and the limits (3) are strong. By this method the existence of the strong limits (3) was earlier proved by one of the authors \(^{(6)}\) under the additional condition of bounded invertibility of the operators \(H_1,H_2\). Instead of formula (4), in that case the relation
\(W_{\pm}(H_2,H_1)=W_{\mp}(H_2^{-1},H_1^{-1})\) was established. In the same work the existence of strong limits of the form (3) was proved for the operators \(H_1,H_2\) and the operators \(H_1^{-1},H_2^{-1}\) under condition \((B_n)\). It was shown there that
\[ W_{\pm}(H_2,H_1)=W_{\mp}(H_2^{-1},H_1^{-1})=W_{\mp}(H_2^{-n},H_1^{-n}). \tag{5} \]
- With the aid of the wave operators one constructs the scattering operator
\[ S\equiv S_{21}\equiv S(H_2,H_1)=W_+^*(H_2,H_1)\,W_-(H_2,H_1), \]
widely used in physics. The operator \(S\) is unitary in the space \(^*\mathfrak H_1\) and commutes there with \(H_1\). From the multiplication theorem for wave operators there easily follows the multiplication rule for scattering operators, which we give for the case of three factors:
\[ S(H_4,H_1)\equiv S_{41}=S'_{43}S'_{32}S_{21}, \tag{6} \]
where
\[ S'_{k+1,k}=W_+^*(H_k,H_1)\,S(H_{k+1},H_k)\,W_+(H_k,H_1)\quad (k=2,3) \]
are unitary in \(\mathfrak H_1\) operators commuting with \(H_1\). Suppose \(\mathfrak H_1\) is decomposed \(^{(9)}\) into a continuous direct sum of Hilbert spaces \(\mathfrak h_\lambda\) over the ring of bounded functions of the absolutely continuous part of the operator \(H_1\). Since all spectral measures of the operator \(H_1\) in \(\mathfrak H_1\) are absolutely continuous, we may assume that the measure associated with the decomposition of \(\mathfrak H_1\) into the sum of the spaces \(\mathfrak h_\lambda\) is Lebesgue measure on the spectrum \(\Lambda\) of the operator
\[ \text{* Without restricting generality, we shall regard } \mathfrak H_1 \text{ as separable, since for perturbations of the types considered by us the orthogonal complement to the set of non-fixed elements of the operator } S \text{ is separable.} \]
\(H_1\) in \(\mathfrak H_1\). Since the operator \(S\) commutes with \(H_1\), it gives rise, in the family \(\mathfrak H_\lambda\), to a measurable family of unitary operators \(S(\lambda)\), defined for almost all \(\lambda \in \Lambda\). We shall call the family \(S(\lambda)\) the scattering matrix corresponding to the operator \(S\). If the multiplicity of the spectrum of \(H_1\) in \(\mathfrak H_1\) is finite, then all \(\mathfrak H_\lambda\) are finite-dimensional, and \(S(\lambda)\) can be realized in the form of a finite unitary matrix-function. In the contrary case, \(S(\lambda)\) can be realized (depending on the choice of realization of the spaces \(\mathfrak H_\lambda\)) in the form of infinite matrix-functions or kernels. The multiplication rule (6) is, evidently, carried over also to scattering matrices:
\[ S_{41}(\lambda)=S_{43}(\lambda)S_{32}(\lambda)S_{21}(\lambda). \tag{7} \]
- In proving the results presented below, an essential role is played by the apparatus introduced to justify the trace formula \((^{10,8})\). We give the information needed for what follows. If condition (A) is satisfied, then for the trace of the difference of the resolvents of the operators \(H_1, H_2\), for any non-real \(z\), the representation
\[ \operatorname{Sp}\{R_z(H_2)-R_z(H_1)\} = -\int_{-\infty}^{\infty}\frac{\xi(\lambda)\,d\lambda}{(\lambda-z)^2}, \tag{8} \]
is valid, where \((1+\lambda^2)^{-1}\xi(\lambda)\in L_1(-\infty,\infty)\). The spectral shift function \(\xi(\lambda)\) is determined by relation (8) up to a constant term. In what follows it is sufficient for us to determine \(\xi(\lambda)\) up to an integer summand by means of the relation
\[ \int_{-\infty}^{\infty}\frac{\xi(\lambda)\,d\lambda}{1+\lambda^2} = \frac{1}{2i}\operatorname{Sp}\ln(U_1^{-1}U_2), \tag{9} \]
where \(U_k\) \((k=1,2)\) are the Cayley transforms of the operator \(H_k\). If \(H_2=H_1+V\), where \(V\) is a nuclear operator, then \(\xi(\lambda)\in L_1(-\infty,\infty)\), and condition (9) can be replaced by the condition
\[ \int_{-\infty}^{\infty}\xi(\lambda)\,d\lambda=\operatorname{Sp}V. \]
If \(V\) is one-dimensional, then the signs of \(\xi(\lambda)\) and \(V\) are the same and \(|\xi(\lambda)|\leq 1\). The spectral shift function \(\xi(\lambda)\) can also be introduced under condition \((B_n)\) (see \((^8)\)) by transforming the independent variable in the spectral shift function for the operators \(H_1^{-n}, H_2^{-n}\). In this case \(\xi(\lambda)=0\) to the left of the spectra of \(H_1\) and \(H_2\), and \(\lambda^{-(n+1)}\xi(\lambda)\in L_1(0,\infty)\).
- Theorem 3. If, for the self-adjoint operators \(H_1, H_2\), condition (A) or some condition \((B_n)\) \((n=2,3,\ldots)\) is satisfied, then the scattering matrix for almost all \(\lambda\in\Lambda\) has the form
\[ S(\lambda)=I_\lambda+T(\lambda), \tag{10} \]
where \(I_\lambda\) is the identity operator and \(T(\lambda)\) is a nuclear operator in \(\mathfrak H_\lambda\). Moreover, for almost all \(\lambda\in\Lambda\)
\[ \det S(\lambda)=e^{-2\pi i\xi(\lambda)}, \tag{11} \]
where \(\xi(\lambda)\) is the spectral shift function for \(H_1\) and \(H_2\).
Let us make some comments. If \(H_2=H_1+V\) and \(V\) is nuclear, then the passage from \(H_1\) to \(H_2\) can be carried out in steps, introducing at each step a one-dimensional perturbation. Such a perturbation corresponds to a scattering matrix whose spectrum consists of unity and a simple eigenvalue \(\exp[-2\pi i\xi_k(\lambda)]\), if the latter is different from unity. Here \(\xi_k(\lambda)\) is the spectral shift function corresponding to the introduction of a one-
measurable perturbation at the \(k\)-th step. Note that (see \((^{10})\))
\[ \sum_{k=1}^{\infty}\int_{-\infty}^{\infty} |\xi_k(\lambda)|\,d\lambda \leq \operatorname{Sp}|V|. \tag{12} \]
Using estimate (12), one can prove that the infinite product of scattering matrices corresponding to the process described above, constructed according to rule (7), converges, after subtracting \(I_\lambda\) from it, in the trace norm for almost all \(\lambda\in\Lambda\) to some operator \(T(\lambda)\). It is also easily verified that \(I_\lambda+T(\lambda)\) is the scattering matrix corresponding to the operator \(S(H_2,H_1)\). In the case of condition (A) (condition \((B_n)\)), formula (4) (respectively formula (5)) makes it possible to reduce our problem to the analogous problem for the Cayley transforms \(U_1, U_2\) (respectively for the operators \(H_1^{-n}, H_2^{-n}\)), after which one can carry out an argument similar to that presented above.
From (10) there follows the possibility of representing \(S(\lambda)\) in the form
\[ S(\lambda)=e^{-2\pi iK(\lambda)}\quad (\text{for almost all } \lambda\in\Lambda), \tag{13} \]
where \(K(\lambda)\) is a trace-class operator in \(\mathfrak h_\lambda\). \(K(\lambda)\) can be chosen so that
\[ \int_{\Lambda} \operatorname{Sp}|K(\lambda)|\,p(\lambda)\,d\lambda \leq C\ (<+\infty). \]
Here: 1) \(p(\lambda)=1,\ C=\operatorname{Sp}|V|\), if \(H_2=H_1+V\) and \(V\) is a trace-class operator; 2) \(p(\lambda)=(1+\lambda^2)^{-1},\ C=\operatorname{Sp}|U_2-U_1|\), if condition (A) is satisfied; 3) \(p(\lambda)=n\lambda^{-(n+1)},\ C=\operatorname{Sp}|H_2^{-n}-H_1^{-n}|\), if condition \((B_n)\) is satisfied.
It can be shown that the trace-class operator \(K(\lambda)\) in representation (13) can be chosen nonnegative, in general under the same conditions under which it was possible in \((^{10})\) to establish the lower semiboundedness of the function \(\xi(\lambda)\). Namely, the following is true:
Theorem 4. The trace-class operator \(K(\lambda)\) in representation (13) can be chosen nonnegative if condition (A) is satisfied, and also the following conditions: 1) \(\mathfrak D(H_1)=\mathfrak D(H_2)\); 2) the form \(((H_2-H_1)f,f)\) \((f\in\mathfrak D(H_1))\) has a finite number of negative squares; there exist constants \(\alpha,\beta\) \((0\leq \alpha<1)\) such that \(\|(H_2-H_1)f\|\leq \alpha\|H_1f\|+\beta\|f\|\) for all \(f\in\mathfrak D(H_1)\).
It is also easy to see that in representation (13) the trace-class operator \(K(\lambda)\) can be chosen nonnegative if, for some \(n\ (\geq 1)\), condition \((B_n)\) is satisfied and the form \(((H_1^{-n}-H_2^{-n})f,f)\) has a finite number of negative squares.
- Let us note in conclusion that, for the radial Schrödinger equation, a rigorous proof of the coincidence of the function \(-\pi\xi(\lambda)\) with the limiting phase is contained in the work \((^{12})\). In the three-dimensional problem of quantum scattering, an expression of \(\xi(\lambda)\) in terms of the scattering amplitude was recently obtained by V. S. Buslaev \((^{13})\). V. S. Buslaev’s formula is interesting in that it is considerably simpler than the expression obtained from the general formula (11).
Leningrad State University
named after A. A. Zhdanov
Odessa Civil Engineering Institute
Received
27 XII 1961
CITED LITERATURE
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* The last condition arose, for other reasons, in the works \((^{11,4})\).