MATHEMATICS

I. V. STANKEVICH

ON THE THEORY OF PERTURBATION OF THE CONTINUOUS SPECTRUM

(Presented by Academician I. G. Petrovskii, 16 XII 1961)

The stability of the absolutely continuous part of the spectrum of a self-adjoint operator \(H_0\) in a Hilbert space \(\mathscr H\) with respect to perturbation by a symmetric operator \(V\) was investigated in the works \((^{1-10})\). Kuroda in \((^{10})\) showed that this problem reduces to the study of families of operators of the form \(U_s(H,H_0)=e^{isH}e^{-isH_0}P_0\), \(U_s(H_0,H)=e^{isH_0}e^{-isH}P\), where \(H=H_0+V\)* and \(P_0\) \((P)\) is the projector onto the absolutely continuous subspace \(M_0\in\mathscr H\) \((M\in\mathscr H)\) with respect to the operator \(H_0\) \((H)\). He proved that from the existence of the operators
\[ U_{\pm}(H,H_0)=\lim_{s\to\pm\infty}U_s(H,H_0),\qquad U_{\pm}(H_0,H)=\lim_{s\to\pm\infty}U_s(H_0,H) \]
(the limits being understood in the sense of strong convergence of sequences of operators in \(\mathscr H\)) there follows the unitary equivalence of the absolutely continuous parts of the operators \(H_0\) and \(H\):
\[ HP=U_+(H,H_0)H_0P_0U_+(H_0,H) \]
\[ \bigl(U_+(H,H_0)U_+(H_0,H)=P\bigr). \]
The operators \(U_{\pm}(H,H_0)\) define the generalized scattering operator \(S\),
\[ S=U_+^*(H,H_0)U_-(H,H_0), \]
which plays an important role in physics.

As Kuroda showed in \((^7)\), the operators \(U_{\pm}(H,H_0)\) and \(U_{\pm}(H_0,H)\) exist if the symmetric operator \(V\) satisfies the conditions: 1) \(\mathscr D(V)\supset \mathscr D(H_0)\), where \(\mathscr D(A)\) denotes the domain of definition of the operator \(A\); 2) for any element \(f\), \(f\in\mathscr D(H_0)\),
\[ \|Vf\|\le a\|H_0f\|+b\|f\|, \]
where \(0\le a<1\), \(b>0\); 3) \(|V|^{1/2}(H_0+i)^{-1}\in\mathfrak S_2\)**.

However, Kuroda’s result does not cover a number of important cases. Thus, for example, for the Schrödinger operator in the space \(L_2(E_k)\) (\(E_k\) is \(k\)-dimensional real Euclidean space), even in the case of an arbitrarily smooth potential function \(V(x)\), this result is applicable only for \(k\le 3\).

In the present work we generalize Kuroda’s result in the following way.

Theorem 1. Let
\[ H_0=\int \lambda\, dE_\lambda^0 \]
be a self-adjoint operator with domain of definition \(\mathscr D(H_0)\), and let \(V\) be such a symmetric operator that: 1) \(\mathscr D(V)\supset\mathscr D(H_0)\); 2) there exists an integer positive number \(n\) such that, for every integer \(l\), \(0\le l\le n-1\), and \(f,\ f\in\mathscr D(H_0^{l+1})\):
\[ \text{a) }Vf\in\mathscr D(H_0^l);\qquad \text{b) }\|H_0^lVf\|\le a\|H_0^{l+1}f\|+\sum_{j=0}^{l} b_{l,j}\|H_0^j f\| \]
and
\[ 0\le a<\frac{1}{1+2^{2(n-1)}},\qquad b_{l,j}\ge 0; \]
3) \(|V|^{1/2}(H_0+i)^{-n}\in\mathfrak S_2\).

Then the operators \(U_{\pm}(H,H_0)\) and \(U_{\pm}(H_0,H)\) exist***, where \(H=H_0+V\).

In particular, for \(n=1\) Kuroda’s theorem follows from our result****. In addition, our result is applicable to the multidimensional Schrödinger operator, for example, with a potential function \(V(x)\) satisfying the conditions: 1) \(V(x)\in L_1(E_k)\); 2) \(V(x)\) has bounded derivatives up to order \(2m\), where \(m\) satisfies the condition \(4m-k>0\). In this case

* The operator \(H=H_0+V\) is assumed to be self-adjoint.

** \(\mathfrak S_2\) denotes the class of Hilbert—Schmidt operators.

*** From conditions 1) and 2b) of Theorem 1 there follows, as was shown, for example, by Kato in \((^9)\), the self-adjointness of the operator \(H\) on \(\mathscr D(H_0)\).

**** From our result Kuroda’s theorem follows for \(0\le a<1/2\). But, as Kuroda showed in \((^7)\), this is sufficient for the validity of the theorem for any \(a\) in \([0,1)\).

the operator \(H_0\) is the operator \(-\Delta\), where \(\Delta\) is the Laplace operator in \(L_2(E_k)\), and the operator \(V\) is the operator of multiplication by the real function \(V(x)\).

Let us verify that the conditions of Theorem 1 are satisfied for \(n\) satisfying the inequalities \([k/4] < n \leqslant m\). Indeed, if \(f \in \mathcal D(H_0^{l+1})\), \(0 \leqslant l \leqslant n-1\), \(l\) an integer, then the function \(g = Vf\) has all derivatives of order \(2l\) from \(L_2(E_k)\), and therefore
\[ g=\int_{E_k}\frac{e^{i(p,x)}\varphi(p)}{(p^2+i)^l}\,d^k p\quad(\varphi(p)\in L_2(E_k)), \]
and, consequently, \(g \in D(H_0^l)\). Moreover, the kernel of the operator \(|V|^{1/2}(H_0+i)^{-n}\) is equal to
\[ K(x,y)=|V(x)|^{1/2}\int_{E_k}\frac{e^{i(p,x-y)}}{(p^2+i)^n}\,d^k p, \]
therefore
\[ \int |K(x,y)|^2\,d^k x\,d^k y<\infty \]
and
\[ |V|^{1/2}(H_0+i)^{-n}\in \mathfrak S_2. \]

The fulfillment of conditions 1) and 2b) of Theorem 1 is obvious. For the proof of Theorem 1 we shall need several auxiliary theorems.

Theorem 2*. Let \(H_0\) and \(V\) be the same as in Theorem 1. Then: 1) the operator \(H=H_0+V\) on the set \(\mathcal D(H_0)\) is self-adjoint; 2) \(\mathcal D(H_0^n)=\mathcal D(H^n)\).

Proof. It is enough to prove that \(\mathcal D(H_0^n)=\mathcal D(H^n)\), since the first assertion of the theorem was proved by Kato in \((^9)\).

Let us show that \(\mathcal D(H_0^n)\supset \mathcal D(H^n)\). Let \(f\in\mathcal D(H^n)\). Then \(Hf\in\mathcal D(H_0^{n-1})\). Suppose that we have proved for some integer \(l\), \(0\leqslant l<n\), that \(H^l f\in\mathcal D(H_0^{n-l})\). Let us show that \(H^{l+1}f\in\mathcal D(H_0^{n-l-1})\). Indeed,
\[ H^{l+1}f=(H_0+V)H^l f=H_0H^l f+VH^l f, \]
and, by condition 2a) of Theorem 1, \(H^{l+1}f\in\mathcal D(H_0^{n-l-1})\). It follows that \(f\in\mathcal D(H^n)\) and \(\mathcal D(H_0^n)\supset\mathcal D(H^n)\).

Let us prove that the operator \(H^n\) on the set \(\mathcal D(H_0^n)\) is self-adjoint. For this, we shall show that, for sufficiently large \(k\), \(k>0\), the operators \(H^n\pm ik\) map \(\mathcal D(H_0^n)\) onto all of \(\mathcal H\). From condition 2b) of Theorem 1 it follows that, for sufficiently large \(k\),
\[ \left\|T(H_0^n\pm ik)^{-1}\right\|<1, \]
where \(Tf=(H^n-H_0^n)f\) for \(f\in\mathcal D(H_0^n)\). Therefore the operator
\[ E+T(H_0^n\pm ik)^{-1}, \]
where \(E\) is the identity operator, maps \(\mathcal H\) onto all of \(\mathcal H\). From the equality
\[ H^n\pm ik=H_0^n+T\pm ik=(1+T(H_0^n\pm ik)^{-1})(H_0^n\pm ik) \]
it follows that, for sufficiently large \(k\), the range of the operator \(H^n\pm ik\) coincides with the whole space. Therefore \(H^n\) on the set \(\mathcal D(H_0^n)\) is self-adjoint and \(\mathcal D(H_0^n)=\mathcal D(H^n)\). Theorem 2 is proved.

Theorem 3. Let \(H_0\) and \(V\) be the same as in Theorem 1. Then there exist constants \(a_l'\) and \(b_{lj}\), \(0\leqslant a_l'<1\), \(b_{lj}\geqslant0\), \(l=0,\ldots,n-1\), \(j=0,\ldots,l\), such that for any \(f\), \(f\in\mathcal D(H^{l+1})\),
\[ \|H^l Vf\|\leqslant a_l'\|H^{l+1}f\|+\sum_{i=0}^{k} b_{li}\|H^i f\|. \tag{1} \]

Let us outline the proof. Inequality (1) for \(l=1\) was proved by Kuroda in \((^7)\). Suppose that (1) is true for \(l=0,1,\ldots,j-2\), where \(j<n+1\). Using condition 2b) of Theorem 1, it is not difficult to show that then (1) is also fulfilled for \(l=j-1\).

Theorem 4**. Let the operators \(H_0\) and \(V\) satisfy the conditions of Theorem 1 and, in addition, let \(V\) be self-adjoint and \(0\leqslant a<1\). Then there exists a sequence \(\{V_k\}\) of self-adjoint operators possessing the following properties:

1) \(V_k\in\mathfrak S_1\)***;

2) \(\bigl\||V_k|^{1/2}(H_0+i)^{-n}\bigr\|_2<c<\infty\) \((c\) does not depend on \(k)\);

* Theorem 2 is a generalization of Kato’s result \((^9)\) for the case \(n=1\).

** This theorem is a generalization of Kuroda’s analogous theorem for \(n=1\).

*** This means that \(|V_k|^{1/2}\in\mathfrak S_2\).

3) \(\left\|(H_0+i)^n(H_k+i)^{-n}\right\| < M < \infty,\ H_k = H_0 + V_k,\ M\) does not depend on \(k\);

4) \(\displaystyle \lim_{k\to\infty} e^{itH_k}=e^{itH}\) (in the strong sense);

5) \[ \lim_{k\to\infty}\int_s^\infty \left\|\,|V_k|^{1/2}e^{-itH_0}f\,\right\|^2\,dt = \int_s^\infty \left\|\,|V|^{1/2}e^{-itH_0}f\,\right\|^2\,dt \]
for any \(f,\ f\in \mathcal L\), where \(\mathcal L\) denotes the set of elements \(f,\ f\in \mathcal D(H_0+i)^n\cap M_0\), such that
\[ d\left\|E^0_\lambda(H_0+i)^n f\right\|^2/d\lambda \leqslant m^2,\quad (E_l^0-E_{-l}^0)f=f \]
for some positive numbers \(m^2\) and \(l\).

Proof. Put
\[ A_k=|V|^{1/2}(1+ik^{-1}H_0)^{-n} \]
and
\[ B_k=(1-ik^{-1}H_0)^{-n}|V|^{1/2}W, \]
where \(V=|V|^{1/2}W|V|^{1/2}\) and \(\|W\|=1\). Then define the operator \(V_k=B_kA_k\). Since \(B_k\) and \(A_k\) are of \(\mathfrak S_2\), it follows that \(V_k\in\mathfrak S_1\),
\[ V_k=(1-ik^{-1}H_0)^{-n}V(1+ik^{-1}H_0)^{-n}. \]
On the set \(\mathcal D(H_0)\) define the operator \(H_k=H_0+V_k\). Since \(V_k\) is a bounded operator, \(H_k\) is self-adjoint on \(\mathcal D(H_0)\). Since
\[ |V_k|=(1-ik^{-1}H_0)^{-n}|V|(1+ik^{-1}H_0)^{-n}, \]
we have
\[ \left\|\,|V_k|^{1/2}f\,\right\|^2=(|V_k|f,f)=\|A_kf\|^2, \]
and, consequently,
\[ \left\|\,|V_k|^{1/2}(H_0+i)^{-n}\right\|_2 = \left\|A_k(H_0+i)^{-n}\right\|_2 \leqslant \left\|\,|V|^{1/2}(H_0+i)^{-n}\right\|_2 =c<\infty. \]
Assertion 3) is proved with the aid of Theorem 2 and inequalities 2b) of Theorem 1.

Assertions 4) and 5) are proved analogously to the case \(n=1\), which was considered by Kuroda in (7).

Theorem 5. Let the operators \(H_0\) and \(V\) be the same as in Theorem 1 and, in addition, let \(V\) be self-adjoint. Then, if \(U_\pm(H,H_0)\) exists, then for any \(f,\ f\in\mathcal L\) (see Theorem 4), the inequality
\[ \left\|(U_t-U_s)f\right\|^2 \leqslant c\,[\eta(t,f)+\eta(s,f)] \tag{2} \]
holds, where \(c\) and \(\eta\) have the form
\[ c=\left[8m^2\pi\left\|\,|V|^{1/2}(H_0+i)^{-n}\right\|_2^2 \left\|(H_0+i)^n(H+i)^{-n}\right\|^2\right]^{1/4}, \]
\[ \eta(t,f)= \left[\int_t^\infty \left\|\,|V|^{1/2}e^{-itH_0}f\,\right\|^2\,dt\right]^{1/4}. \]

Conversely, if for any \(f,\ f\in\mathcal L\), inequality (2) holds with a constant \(c\) independent of \(t\) and \(s\), then the operator \(U_\pm(H,H_0)\) exists.

Proof. Suppose, for definiteness, that the operator \(U_+(H,H_0)\) exists. Then
\[ e^{-itH}U_+=U_+e^{-itH_0}, \]
and therefore
\[ \left\|(U_s-U_t)f\right\|^2 = 2\,\operatorname{Re}\, i\int_s^\infty (Ve^{-itH_0},\,U_+e^{itH_0}f)\,dt. \]
Using the representation \(V\) in the form
\[ V=|V|^{1/2}W|V|^{1/2}, \]
where \(\|W\|=1\), and then estimating the integral with the aid of Schwarz’s inequality, we obtain
\[ \left\|(U_+-U_s)f\right\|^2 \leqslant 2\left[\int_s^\infty \left\|\,|V|^{1/2}e^{-itH_0}f\,\right\|^2\,dt\right]^{1/2} \left[\int_s^\infty \left\|\,|V|^{1/2}U_+e^{-itH_0}f\,\right\|^2\,dt\right]^{1/2}. \]
Using the equality \(U_+H_0P_0=HU_+\), we transform the integrand in the second integral to the form
\[ \left\|\,|V|^{1/2}(H_0+i)^{-n}(H_0+i)^n(H+i)^{-n}U_+e^{-itH_0}(H_0+i)^n f\,\right\|^2. \]
Using results, for example, of Kuroda (7), it is not hard to show that
\[ \left\|(U_+-U_s)f\right\| \leqslant \left[8m^2\pi\left\|\,|V|^{1/2}(H_0+i)^{-n}\right\|_2 \left\|(H_0+i)^n(H+i)^{-n}\right\|\right]^{1/4} \eta(s,f). \]
From this the direct assertion of the lemma follows.

Conversely, suppose that (2) is true for any \(f\in\mathcal L\). Since the integrals \(\eta(t,f)\) and \(\eta(s,f)\) converge, as \(t,s\to\infty\) the right-hand side of (2) tends to zero. But in (7) it is proved that \(\mathcal L\) is dense in \(M_0\). Moreover, \(\|U_t\|=1\). Therefore \(U_+(H,H_0)\) exists. Theorem 5 is proved.

We pass to the proof of Theorem 1. Just as in the case \(n=1\), Theorem 1 reduces to an analogous theorem under the additional assumption that the operator \(V\) is self-adjoint. We shall prove Theorem 1 in this case.

From Kuroda’s result (⁷) it follows that for any natural number \(k\), \(k>0\), there exists an operator \(U_{\pm}(H_k,H_0)\), where \(H_k=H_0+V_k\) and
\[ V_k=(1-ik^{-1}H_0)^{-n}V(1+ik^{-1}H_0)^{-n},\quad V_k\in\mathfrak S_1. \]
Therefore, from Theorem 5, for any \(f\in\mathfrak L\) inequality (2) follows for \(U_t^{(k)}\), \(U_s^{(k)}\), \(c_k\), and \(\eta_k\), which are defined in Theorem 5 \((U_t^{(k)}=e^{itH_k}e^{-itH_0}P_0)\). Hence, and from Theorem 4, inequality (2) follows for \(H_0\) and \(H\) with a constant \(c\) independent of \(t\) and \(s\). By virtue of the converse assertion of Theorem 5, this proves the existence of the operator.

From Theorems 2, 3, and 4 it follows that if the operator \(H\) is taken as the unperturbed operator, then the operators \(H\) and \(-V\) satisfy the conditions of Theorem 1 with constants \(0\le a<1\), \(b_{lj}\ge 0\). This is sufficient for the existence of the operator \(U_{\pm}(H_0,H)\). Theorem 1 is proved.

In conclusion let us consider an example. Let \(\mathcal H=L_2(E_k)\); \(H_0=(-1)^t\Delta^t\) (\(t\) is a positive integer), where \(\Delta\) is the Laplace operator in \(L_2(E_k)\), and
\[ \mathfrak D(H_0)= \left\{ \psi(x),\ \psi(x)=\int \frac{\varphi(p)e^{i(p,x)}}{p^{2t}+i}\,d^kp,\ \varphi(p)\in L_2(E_k) \right\}. \]
On \(\mathfrak D(H_0)\) define a symmetric differential operator of order \(\alpha\), where \(0\le\alpha<2t\):
\[ V\psi(x)=\sum_{\substack{j_1\ldots j_\beta\\ \beta\le\alpha}} a_{j_1\ldots j_\beta}(x) \frac{\partial^\beta}{\partial x_{j_1}\ldots \partial x_{j_\beta}}\psi(x) \]
and
\[ (1+|x|^4)a_{j_1\ldots j_\beta}(x)\in L_1(E_k) \]
and \(a_{j_1\ldots j_\beta}(x)\) have bounded derivatives up to order \(2mt\), where \(m\) is a positive integer satisfying the condition \(2tm-\alpha-k>0\). The operators \(H_0\) and \(V\) thus defined satisfy the conditions of Theorem 1 for any integer \(n\),
\[ \left[\frac{\alpha+k}{2t}\right]<n\le m. \]

Indeed, the operator \(V\) maps any element \(f\), \(f\in\mathfrak D(H_0^l)\), \(0\le l\le n\), into an element \(g\), \(g=Vf\), and \(g\) has all derivatives up to order \(2t(l-1)\) inclusive belonging to \(L_2(E_k)\); therefore
\[ g=\int \frac{\varphi(p)e^{i(p,x)}}{(p^2+1)^{t(l-1)}}\,d^kp, \]
where \(\varphi(p)\in L_2(E_k)\), and \(g\in\mathfrak D(H_0^{l-1})\). Moreover, the operator
\[ A=(H_0-i)^{-n}|V|(H_0+i)^{-n} =\left|(H_0-i)^{-n}V(H_0+i)^{-n}\right|, \]
and the kernel of the operator \((H_0-i)^{-n}V(H_0+i)^{-n}\) can be computed explicitly. Using results of A. O. Gelfond (¹¹), it is not hard to show that
\[ (H_0-i)^{-n}V(H_0+i)^{-n}\in\mathfrak S_1. \]
Therefore also the operator \(A\in\mathfrak S_1\) and
\[ |V|^{1/2}(H_0+i)^{-n}\in\mathfrak S_2. \]

Condition 2b) of Theorem 1 is directly easy to verify.

Thus, the absolutely continuous part of the operator \((-\Delta)^t+V\), where \(V\) is a differential operator of order less than \(2t\), with coefficients satisfying the conditions listed above, is unitarily equivalent to the operator \((-\Delta)^t\) in \(L_2(E_k)\).

The author expresses gratitude to F. A. Berezin, under whose supervision this work was carried out.

Moscow State University
named after M. V. Lomonosov

Received
12 XII 1961

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