Abstract
Full Text
UDC 517.948:513.88
MATHEMATICS
V. E. LYANTSE
ON PERTURBATION OF THE CONTINUOUS SPECTRUM
(Presented by Academician I. M. Vinogradov on 21 II 1969)
This paper gives a generalization of the results obtained in \((^1)\) to the case of infinite-dimensional perturbations.
1. As the unperturbed operator we still take the operator \(S\) of multiplication by the independent variable in the Hilbert space \(H = L_2(-\infty,\infty)\), with domain \(D(S)\) consisting of those \(f(x) \in H\) for which also \(x f(x) \in H\), and \(S f(x) = x f(x)\). To describe the perturbed operator we shall need several definitions.
Let \(\varepsilon\) be some positive number. By \(\Phi = \Phi(\varepsilon)\) we denote the set of those \(\varphi \in H\) for which there exists a continuation \(\varphi(z)\) of the function \(\varphi(x)\), holomorphic for \(|\operatorname{Im} z| < \varepsilon\) and satisfying the condition
\[ \sup_{|y|\leq \varepsilon_1} \int_{-\infty}^{\infty} |\varphi(x+iy)|^2\,dx < \infty,\qquad \varepsilon_1 < \varepsilon. \tag{1} \]
Functions \(\varphi \in \Phi\) will be called completely regular elements of \(H\).
Let \(G\) be some (auxiliary abstract) Hilbert space and \(A\) a linear continuous operator: \(H = D(A) \to G\). If the range \(R(A^*)\) of the adjoint operator \(A^*: G = D(A^*) \to H\) consists of completely regular elements, i.e.
\[ R(A^*) \subset \Phi, \tag{2} \]
then, as is not hard to prove, there exists a vector-valued function \(\alpha(z)\) with values in \(G\), holomorphic for \(|\operatorname{Im} z| < \varepsilon\), such that
\[ Af = \int_{-\infty}^{\infty} f(x)\alpha(x)\,dx,\qquad f \in H, \tag{3} \]
where the integral exists in the sense of weak convergence in \(G\). If one assumes that
\[ \int_{-\infty}^{\infty} \|\alpha(x)\|_G^2\,dx < \infty, \tag{4} \]
then \(A\) will be a completely continuous operator \(H \to G\). Such an operator (i.e. an operator satisfying conditions (2) and (4)) we call completely regular.
Definition 1. An operator \(V: H = D(V) \to H\) is called a completely regular perturbation if it admits a representation of the form \(V = A^*B\), where \(A\) and \(B\) are completely regular operators \(H \to G\).
In this paper the operator \(T \overset{\mathrm{df}}{=} S+V,\ D(T) \overset{\mathrm{df}}{=} D(S)\), is studied, where \(V\) is a completely regular perturbation*. Since the norm of \(V\) is not assumed to be small, the operator \(T\), generally speaking, is not similar to \(S\). In addition to eigenvalues of the usual type, the operator \(T\) may have spectral singularities (see Sec. 2), which, in particular, may be eigenvalues with nonzero index. The eigenfunctions of the continuous spectrum of the operator \(T\) turn out to be \(\delta\)-functions perturbed by boundary values of certain Cauchy-type integrals (see Sec. 3). The Parseval equality corresponding to the operator \(T\) is written by means of regularization of a certain integral that diverges near the spectral singularities.
- It is convenient to describe the spectral properties of the operator \(T\) in terms of the operator function \(\mathrm K(\zeta): G \to G\)
\[ \mathrm K(\zeta) \overset{\mathrm{df}}{=} 1+B S_\zeta A^*, \qquad \operatorname{Im}\zeta \ne 0, \tag{5} \]
where \(S_\zeta \overset{\mathrm{df}}{=} (S-\zeta)^{-1}\). For example, it is not hard to see that a nonreal \(\lambda\) is an eigenvalue of the operator \(T\) if and only if the operator \(\mathrm K(\lambda)\) is not invertible in \(G\). If \(\operatorname{Im}\zeta \ne 0\) and \(\mathrm K(\zeta)^{-1}\) exists, then \(\zeta\) belongs to the resolvent set of the operator \(T\), and
\[ T_\zeta \overset{\mathrm{df}}{=} (T-\zeta)^{-1} = S_\zeta - S_\zeta A^* \mathrm K(\zeta)^{-1} B S_\zeta . \tag{6} \]
The set of eigenvalues of the operator \(T\) is finite, and to each eigenvalue there corresponds a finite-dimensional root subspace.
Formula (6) makes it possible to conclude that, for any \(\varphi,\psi \in \Phi\), the function \((T_\zeta \varphi,\psi)\) (of the variable \(\zeta\)) has continuations \((T_\zeta\varphi,\psi)_+\) and \((T_\zeta\varphi,\psi)_-\) from the half-planes respectively \(\operatorname{Im}\zeta>0\) and \(\operatorname{Im}\zeta<0\), analytic in the half-planes respectively \(\operatorname{Im}\zeta>-\varepsilon\) and \(\operatorname{Im}\zeta<+\varepsilon\), and having no singularities other than poles.
Definition 2. The real poles of the functions \((T_\zeta\varphi,\psi)_+\) and \((T_\zeta\varphi,\psi)_-\) are called spectral singularities of the operator \(T\).
The set of spectral singularities of the operator \(T\) is also finite and coincides with the set of spectral singularities of the operator \(T^*\). In order that a real \(\sigma\) be an eigenvalue of the operator \(T\) or \(T^*\), it is necessary (but not sufficient) that \(\sigma\) be a spectral singularity. In this case the dimension of the root subspace corresponding to \(T^*\), generally speaking, is not equal to the dimension of the root subspace corresponding to \(T\), so that, generally speaking, \(\sigma\) has nonzero index. The spectral singularities coincide with the real poles of the operator-valued functions \(\mathrm K_+(\zeta)^{-1}\) and \(\mathrm K_-(\zeta)^{-1}\), where \(\mathrm K_+(\zeta)\) and \(\mathrm K_-(\zeta)\) are the analytic continuations of \(\mathrm K(\zeta)\) from the upper and lower half-planes, respectively.
To each nonreal eigenvalue \(\lambda\) of the operator \(T\) we assign the eigenprojection
\[ P_\lambda \overset{\mathrm{df}}{=} -\frac{1}{2\pi i}\oint T_\zeta\,d\zeta, \tag{7} \]
where inside the contour of integration there are no singularities of the resolvent \(T_\zeta\) other than \(\lambda\). To each spectral singularity \(\sigma\) we assign two “eigenprojections” \(P_{\sigma+}\) and \(P_{\sigma-}\),
\[ (P_{\sigma\pm}\varphi,\psi) = -\frac{1}{2\pi i}\oint (T_\zeta\varphi,\psi)_\pm\,d\zeta, \qquad \varphi,\psi \in \Phi, \tag{8} \]
* Some of the results formulated below are also valid under more general assumptions on the perturbation \(V\).
with an analogous condition on the contour of integration. We note that the “projectors” \(P_{\sigma\pm}\) act from \(\Phi\) into the set of semilinear functionals defined on \(\Phi\). All the operators \(P_\lambda\), \(P_{\sigma+}\), and \(P_{\sigma-}\) are finite-dimensional. It is useful to apply formula (6) in order to express \(P_\lambda\), \(P_{\sigma+}\), and \(P_{\sigma-}\) in terms of the residues of \(K(\zeta)\), \(K_+(\zeta)\), and \(K_-(\zeta)\) at \(\zeta=\lambda\) and \(\zeta=\sigma\), respectively.
- If the real \(\xi\) is not a spectral singularity, then \(\xi\) belongs to the continuous spectrum of the operator \(T\). In order to describe the jump of the resolvent \(T_\xi\) on the continuous spectrum, we introduce the operators:
\[ \mathfrak A\varphi(\xi)\overset{\mathrm{df}}{=}\varphi(\xi)-B^*K_+(\bar\xi)^{*-1}AS_\xi\varphi_-(\xi),\qquad \varphi\in\Phi,\ |\operatorname{Im}\zeta|<\varepsilon, \tag{9} \]
\[ \mathfrak B\varphi(\xi)\overset{\mathrm{df}}{=}\varphi(\xi)-A^*K_-(\xi)^{-1}BS_\xi\varphi_-(\xi); \]
here \(AS_\xi\varphi_-\) and \(BS_\xi\varphi_-\) are continuations from the lower half-plane of the functions \(AS_\xi\varphi\) and \(BS_\xi\varphi\), which exist and are holomorphic (in the metric of \(G\)) if \(\varphi\in\Phi\). It turns out that, for arbitrary \(\varphi,\psi\in\Phi\) and for all \(\xi\) from some neighborhood of the real axis, with the exception of spectral singularities,
\[ (T_\xi\varphi,\psi)_+-(T_\xi\varphi,\psi)_-=2\pi i\,\mathfrak B\varphi(\xi)\overline{\mathfrak A\psi(\xi)}. \tag{10} \]
From the “jump” formula (10) it is easy to derive that
\[ \mathfrak A T^*\varphi(\xi)=\xi\mathfrak A\varphi(\xi),\qquad \mathfrak B T\varphi(\xi)=\xi\mathfrak B\varphi(\xi) \tag{11} \]
for all real \(\xi\) distinct from the spectral singularities, which gives grounds for regarding the operators \(\mathfrak A\) and \(\mathfrak B\) as, respectively, the \(T^*\)- and \(T\)-Fourier transforms on the continuous spectrum. To describe the continuity properties of these operators, put*
\[ \varkappa_\pm(\zeta)\overset{\mathrm{df}}{=}\prod_\sigma \left(\frac{\zeta-\sigma}{\zeta-i\varepsilon}\right)^{n_{\sigma\pm}}, \tag{12} \]
where the product sign extends over all spectral singularities \(\sigma\), and \(n_{\sigma\pm}\) denotes the order of \(\sigma\) as a pole of \(K_\pm(\zeta)^{-1}\). Then the operators
\[ \mathfrak A_0\varphi(\xi)\overset{\mathrm{df}}{=}\overline{\varkappa_+(\xi)}\,\mathfrak A\varphi(\xi),\qquad \mathfrak B_0\varphi(\xi)\overset{\mathrm{df}}{=}\varkappa_-(\xi)\,\mathfrak B\varphi(\xi) \tag{13} \]
admit extension from \(\Phi\) to continuous operators \(H\to H\). Moreover, if \(\mathcal D\) denotes the differentiation operator in \(H\), then from \(f\in D(\mathcal D^j)\) it follows that \(\mathfrak A_0 f,\mathfrak B_0 f\in D(\mathcal D^j)\) and
\[ \|\mathcal D^j\mathfrak A_0 f\|,\ \|\mathcal D^j\mathfrak B_0 f\| \leq C_j\left(\sum_{j'=1}^{j}\|\mathcal D^{j'}f\|^2\right)^{1/2}, \qquad j=0,1,2,\ldots, \tag{14} \]
where \(C_j\) are certain constants.
The estimates (14) (for \(j=1\)) allow one to speak of the values of the functions \(\mathfrak A_0 f(\xi)\), \(\mathfrak B_0 f(\xi)\) for \(f\in D(\mathcal D)\) for all real \(\xi\). Denoting them respectively by \((f,a_\xi)\) and \((f,b_\xi)\), and taking (11) into account, we may regard the functionals \(a_\xi\) and \(b_\xi\) as (generalized) eigenfunctions of the continuous spectrum of the operators \(T\) and \(T^*\), respectively.
- In conclusion we describe one variant of Parseval’s equality for the operator \(T\). Put
\[ \int_+ \frac{f(x)\,dx}{(x-\sigma)^\nu} \overset{\mathrm{df}}{=} \lim_{\tau\to -0}\int_{-\infty}^{\infty} \frac{f(x)\,dx}{[x-(\sigma+i\tau)]^\nu} \tag{15} \]
* Since the operator \(K_\pm(\zeta)-1\) is nuclear, there exists a determinant of the operator \(K_\pm(\zeta)\). Everything that follows remains valid if relation (12) is replaced by the relation
\[ \varkappa_\pm(\zeta)\overset{\mathrm{df}}{=}\det K_\pm(\zeta). \]
provided that the limit on the right exists. Let \(n=\max(n_{\sigma+}+n_{\sigma-})\), where the maximum is taken over all spectral singularities \(\sigma\), and the numbers \(n_{\sigma\pm}\) are the same as in (12). Then, for all \(f,g\in D(\mathscr D^n)\),
\[ \int_{-\infty}^{\infty} f(x)\overline{g(x)}\,dx = \int_{+} (f,b_\xi)\overline{(g,a_\xi)}\, \frac{d\xi}{\varkappa_+(\xi)\varkappa_-(\xi)} + \sum_{\sigma}(P_\sigma f,g) + \sum_{\lambda}(P_\lambda f,g). \tag{16} \]
Here \(a_\xi\) and \(b_\xi\) are eigenfunctions of the continuous spectrum of \(T\) and \(T^*\) (see item 3); \(P_\lambda\) and \(P_\sigma\) are the eigenprojections of eigenvalues and spectral singularities (see (7) and (8)); let us clarify that \(P_\sigma\) is extended by continuity, in the norm \((\|f\|^2+\cdots+\|f^{(n)}\|^2)^{1/2}\) from \(\Phi\) to \(D(\mathscr D^n)\), while \(\sum_\lambda\) ranges over all nonreal eigenvalues \(\lambda\), and \(\sum_\sigma\) over all spectral singularities \(\sigma\). As is seen from (12), each spectral singularity \(\sigma\) is a pole of order \(n_{\sigma+}+n_{\sigma-}\) of the “spectral” density function \(1/\varkappa_+(\sigma)\varkappa_-(\sigma)\). In this connection, a regularized integral \(\int_+\) arises on the right-hand side of (16). We understand this integral, roughly speaking, in the sense of (15)*. The existence of such a regularization is ensured by the estimates (14).
Lviv State University
named after Iv. Franko
Received
6 II 1969
References
- V. E. Lyantse, DAN, 182, No. 5 (1968).
* The corresponding partition of unity is used, making it possible to find the limit (15) for each spectral singularity separately.