Mathematics

A. G. SIGALOV

A NEW ALGORITHM IN THE PERTURBATION THEORY OF THE CONTINUOUS SPECTRUM

(Presented by Academician V. I. Smirnov on 16 III 1964)

No. 1. Let \(H\) be a self-adjoint operator with simple spectrum, acting in a Hilbert space \(\mathfrak H\); \(E^H_\Delta = E_\Delta\) is its spectral function; \(\psi\) is a generating element; \(\sigma(\Delta) = (E_\Delta \psi,\psi)\). Put \(A \in \mathfrak M(H)\), if \(D_A \supset D_H\), \(A\) is symmetric on \(D_H\), and the following hold: 1) \(|A\varphi|^2 \le C_1 |H\varphi|^2 + C_2 |\varphi|^2\) for all \(\varphi \in D_H\); 2) if
\[ G_\eta^N=\{(\alpha,\beta),\ |\alpha-\beta|\ge \eta,\ |\alpha|\le N,\ |\beta|\le N\}, \]
then for any \(\eta>0,\ N>0\) there is a \(C=C(N,\eta)>0\) such that from \(\Delta\alpha \times \Delta\beta \subset G_\eta^N\) it follows that
\[ |(AE_{\Delta\alpha}\psi,E_{\Delta\beta}\psi)| \le C |E_{\Delta\alpha}\psi|^2 |E_{\Delta\beta}\psi|^2; \]
\(\Delta\alpha,\Delta\beta\) are always intervals open on the right. From 1) it follows: 3) \(|(AE_{\Delta\alpha}\psi,E_{\Delta\alpha}\psi)| \le C_3 |E_{\Delta\alpha}\psi|^2\), if \(\Delta\alpha \in [-N,N]\), \(C_3=C_3(N)\).

Let \(m(E)\) \((\sigma_2(E))\) be the measure, defined on the Borel field of sets \(\mathfrak B_m\) \((\mathfrak B_{\sigma_2})\), which is the minimal extension of the interval functions
\[ m(\Delta\alpha,\Delta\beta)=(AE_{\Delta\alpha}\psi,E_{\Delta\beta}\psi) \]
(respectively, of the function \(\sigma(\Delta\alpha)\cdot\sigma(\Delta\beta)\)), \(\Delta\alpha\times\Delta\beta\in G_\eta^N\) \((^1)\). We have \(\mathfrak B_m \supset \mathfrak B_{\sigma_2}\); \(m(E)\) is absolutely continuous with respect to \(\sigma_2(E)\) \((^1)\). By the Radon–Nikodym theorem, on \(G_\eta^N\) a function
\[ A_{\alpha\beta}^H=\frac{dm}{d\sigma_2} \]
is defined almost everywhere with respect to \(\sigma_2\). The function \(A_{\alpha\beta}\) does not depend on \(N\) or \(\eta\).

No. 2. Let \(A>0\), \(A\in\mathfrak M(H)\), and let \(\Delta'=[\alpha,\alpha')\) be a finite interval,
\[ \Pi=\{\alpha_0<\alpha_1<\cdots<\alpha_N=\alpha'\},\quad \Delta_i=[\alpha_{i-1},\alpha_i), \]
\[ V(\Pi)=\sum (AE_{\Delta_i}\psi,E_{\Delta_i}\psi), \]
\[ d(\Pi)=\max_i(\alpha_i-\alpha_{i-1}). \]
Put
\[ m_1(\Delta')=\lim_{d(\Pi)\to 0} V(\Pi). \]
The interval function \(m_1(\Delta)\) is additive and absolutely continuous with respect to \(\sigma(\Delta)\). Extending them to the corresponding measures \(m_1(E)\), \(E\in\mathfrak B_{m_1}\), \(\sigma(E)\), \(E\in\mathfrak B_\sigma\), and using the Radon–Nikodym theorem, define
\[ A_\alpha^H=\frac{dm_1}{d\sigma}. \]
If: 1) \(\sigma(\alpha)\) is absolutely continuous and 2) \(C(N,\eta)\) does not depend on \(\eta\), then \(A_\alpha^H\) can be defined as
\[ \lim_{\Delta\to\alpha}\frac{1}{\sigma(\Delta)}(AE_\Delta\psi,E_\Delta\psi) \]
independently of the assumption \(A>0\).

The pair of functions \(A_{\alpha\beta}^H,\ A_\alpha^H\) will be called the diagonally singular matrix (DS-matrix) of the operator \(A\) relative to \(H\).

No. 3. For
\[ \varphi=\int c(\alpha)\,dE_\alpha\psi\in D_H,\qquad \chi=\int e(\alpha)\,dE_\alpha\psi\in\mathfrak H \]
with continuous \(c(\alpha)\), \(e(\beta)\), the equality holds
\[ (A\varphi,\chi) = \iint c(\alpha)\overline{e(\beta)}\,A_{\alpha\beta}\,d\sigma(\alpha)\,d\sigma(\beta) + \int c(\alpha)\overline{e(\alpha)}\,A_\alpha\,d\sigma(\alpha). \tag{B} \]
Under assumptions 1) and 2) (\(A>0\) is not assumed), the integrals on the right can be understood in the Lebesgue–Stieltjes sense. Without these assumptions, equality (B) requires a generalization of the concept of integral. It can be given starting from such \(\varphi,\chi\) for which \(c(\alpha)\overline{e(\beta)}=0\) outside a rectangle \(\Delta\alpha\times\Delta\beta\) not intersecting the straight line \(\alpha=\beta\), and using property 3) of No. 1.

No. 4. If \(A\in\mathfrak M(H)\) satisfies one of the conditions ensuring equality (B), then there exists one and only one decomposition \(A=B+C\) possessing the properties: a) \(B,C\in\mathfrak M(H)\); b) \(B_\alpha=0\); c) \(CH=HC\). The bilinear form of the operator \(B\) is defined by the double integral of the right-hand side of equality (B), while the bilinear form of the operator \(C\) is defined by the single integral from (B).

No. 5. Let \(H^\varepsilon = H^0+\varepsilon W\) be bounded self-adjoint operators with simple spectrum and common generating element \(\psi\), \(W\in \mathfrak M(H^0)\);
\(W_{\alpha\beta}^{\varepsilon}=W_{\alpha\beta}^{H^\varepsilon}\), \(W_{\alpha}^{\varepsilon}=W_{\alpha}^{H^\varepsilon}\),
\(\sigma'(\varepsilon,\alpha)=\dfrac{\partial}{\partial\varepsilon}(E_{\alpha}^{H^\varepsilon}\psi,\psi)\) are continuous, \(|\varepsilon|\leqslant \varepsilon_0\).

Then, for any continuous \(c(\varepsilon,\alpha)\), \(e(\varepsilon,\beta)\),
\[ \varphi=\int c(\varepsilon,\alpha)\,dE_{\alpha}^{\varepsilon}\psi, \]
\[ \chi=\int e(\varepsilon,\beta)\,dE_{\beta}^{\varepsilon}\psi \]
the following equality holds:
\[ \frac{\partial}{\partial\varepsilon}(E_{\nu}^{\varepsilon}\varphi,\chi) = \iint g_{\nu}(\alpha,\beta)\, W_{\alpha\beta}^{\varepsilon}c(\varepsilon,\alpha)\overline{e(\varepsilon,\beta)}\, d\sigma(\varepsilon,\alpha)\,d\sigma(\varepsilon,\beta) + \sigma'(\varepsilon,\nu)c(\varepsilon,\nu)\overline{e(\varepsilon,\nu)}W_{\nu}^{\varepsilon}, \]
where
\[ g_{\nu}(\alpha,\beta)= \begin{cases} 0, & \text{if }(\alpha-\nu)(\beta-\nu)>0,\\ -|\alpha-\beta|^{-1}, & \text{if }(\alpha-\nu)(\beta-\nu)<0. \end{cases} \]

The proof is based on the following assertions, formulated, for simplicity of notation, for \(H>0\).

If
\[ K_t(r,\nu)=\frac{r(r-\nu\cos t)}{r^2-2r\nu\cos t+\nu^2}, \qquad \widetilde E_{r}^{\varepsilon}=\frac12\{E_{r+0}^{\varepsilon}+E_{r}^{\varepsilon}\}, \]
\[ \widetilde E_{r}^{\varepsilon,\delta} = \frac1\pi\int_{\delta}^{\pi} K_t(r,H)\,dt, \]
then:

a) \(\widetilde E_{r}^{\varepsilon,\delta}\to \widetilde E_{r}^{\varepsilon}\) strongly;

b)
\[ \frac{\partial \widetilde E_{r}^{\varepsilon,\delta}}{\partial\varepsilon} = \frac{r}{\pi}\int_{\delta}^{\pi} A^{-1}SA^{-1}\,dt, \]
where
\[ A=r^2E-2r\cos t\,H^\varepsilon+(H^\varepsilon)^2, \]
\[ S=(r^2W+H^\varepsilon W H^\varepsilon)\cos t -r(H^\varepsilon W+WH^\varepsilon); \]

c)
\[ \frac{\partial}{\partial\varepsilon}(\widetilde E_{r}^{\varepsilon,\delta}\varphi,\chi) \to \frac{\partial}{\partial\varepsilon}(E_{r}^{\varepsilon}\varphi,\chi) \quad(\delta\to0). \]

a) and b) do not require simplicity or continuity of the spectrum.

No. 6. Let \(c(\varepsilon,\alpha)\), \(e(\varepsilon,\alpha)\),
\(\rho(\varepsilon,\alpha)=d\sigma(\varepsilon,\alpha)/d\alpha\),
\(W_{\alpha\beta}^{\varepsilon}\) satisfy a Hölder condition in the variables \(\alpha,(\alpha,\beta)\),
\(|\alpha|\leqslant M\), \(|\beta|\leqslant M\), \(|\varepsilon|\leqslant\varepsilon_0\). Put
\(F(\varepsilon,\nu)=(E_{\nu}^{\varepsilon}\varphi,\chi)\), where \(E_{\nu}^{\varepsilon}=E_{\nu}^{H^\varepsilon}\). Then
\[ \lim_{\Delta\nu\to0} \frac{1}{2\Delta\nu} \left\{ \frac{\partial F(\nu+\Delta\nu,\varepsilon)}{\partial\varepsilon} - \frac{\partial F(\nu-\Delta\nu,\varepsilon)}{\partial\varepsilon} \right\} = \]
\[ = -\int \frac{W_{\alpha\nu}^{\varepsilon}}{\alpha-\nu} c(\varepsilon,\alpha)\overline{e(\varepsilon,\nu)} \,d\sigma(\varepsilon,\alpha)\rho(\varepsilon,\nu) - \int \frac{W_{\nu\beta}^{\varepsilon}}{\beta-\nu} c(\varepsilon,\nu)\overline{e(\varepsilon,\beta)} \,d\sigma(\varepsilon,\beta)\rho(\varepsilon,\nu). \]

The integrals on the right-hand side are understood in the sense of the principal value. The limit exists uniformly with respect to \(\nu\) in every closed interval in which \(\rho(\varepsilon,\nu)>0\). If the exponent and the constant multiplier in the Hölder condition do not depend on \(\varepsilon\), \(|\varepsilon|\leqslant\varepsilon_0\), then the limit exists uniformly with respect to \(\varepsilon\). It follows from this that \(\rho(\varepsilon,\alpha)\) and \(c(\varepsilon,\alpha)\) satisfy the equations
\[ \frac{\partial \rho(\varepsilon,\alpha)}{\partial\varepsilon} = -\int \frac{2\operatorname{Re} W_{\alpha\nu}^{\varepsilon}}{\alpha-\nu} \rho(\varepsilon,\alpha)\,d\alpha\,\rho(\varepsilon,\nu), \tag{1} \]
\[ \frac{\partial c(\varepsilon,\nu)}{\partial\varepsilon} = -\int \frac{W_{\alpha\nu}^{\varepsilon}}{\alpha-\nu} [c(\varepsilon,\alpha)-c(\varepsilon,\nu)]\rho(\varepsilon,\alpha)\,d\alpha. \tag{2} \]

no. 7. Put, for \(p(\alpha)=p(\alpha_1,\ldots,\alpha_n)\),
\[ \Delta_i p=|p(\beta)-p(\alpha)|:|\beta-\alpha|^\gamma, \]
where \(\beta=(\alpha_1,\ldots,\alpha_{i-1},\alpha_i+\Delta\alpha_i,\alpha_{i+1},\ldots,\alpha_n)\),
\[ \|p\|=\sup |p(\alpha)|+\sum_i \sup |\Delta_i p(\alpha)|+\sum_{j\ne i}\sup |\Delta_i\Delta_j p(\alpha)|. \tag{2} \]

Let \(A^\varepsilon\in \mathfrak{M}(H)\), \(A^\varepsilon_{\alpha}=0\), \(\|A^\varepsilon_{\alpha\beta}\|<\infty\); \(dA^\varepsilon/d\varepsilon=A^\varepsilon_1\in \mathfrak{M}(H)\) exist in the sense of strong convergence and \(\|(A^\varepsilon_1)_{\alpha\beta}\|<\infty\),
\[ P_2=\{(\nu,\mu);\ \rho(\varepsilon,\nu),\rho(\varepsilon,\mu)>0\},\qquad P_1=\{\nu,\rho(\varepsilon,\nu)>0\}. \]
Then for \((\nu,\mu)\in P_2\)
\[ \frac{dA^\varepsilon_{\nu\mu}}{d\varepsilon} = \left(\frac{dA^\varepsilon}{d\varepsilon}\right)_{\nu\mu} + A^\varepsilon_{\nu\mu}\left(\Phi^\varepsilon_\nu+\Phi^\varepsilon_\mu\right) + (A^\varepsilon V^\varepsilon-V^\varepsilon A^\varepsilon)^{H^\varepsilon}_{\nu\mu}. \]
Here
\[ \Phi^\varepsilon_\nu=\int \frac{W^\varepsilon_{\alpha\nu}}{\alpha-\nu}\,d\sigma(\varepsilon,\nu),\qquad \frac{W^\varepsilon_{\alpha\beta}}{\alpha-\beta}=V^\varepsilon_{\alpha\beta}. \]
The second term on the right-hand side should be understood as the convolution of DC-matrices of multiplier operators. Putting in this equality \(A=W\), we obtain the equation
\[ \frac{\partial W^\varepsilon_{\nu\mu}}{\partial\varepsilon} = W^\varepsilon_{\nu\mu}\left(\Phi^\varepsilon_\nu+\Phi^\varepsilon_\mu\right) + (WV^\varepsilon-V^\varepsilon W)^{H^\varepsilon}_{\nu\mu}, \tag{3} \]
which expresses \(\partial W^\varepsilon_{\nu\mu}/\partial\varepsilon\) in terms of \(W^\varepsilon_{\nu\mu}\) and \(\rho(\varepsilon,\nu)\).

no. 8. Put \(T^\varepsilon_{\nu\mu}=W^\varepsilon_{\nu\mu}\rho(\varepsilon,\nu)\). From (1), (3) it follows that
\[ \frac{\partial T^\varepsilon_{\nu\mu}}{\partial\varepsilon} = A^\varepsilon_{\nu\mu}T^\varepsilon_{\nu\mu} + B^\varepsilon_{\nu\mu},\qquad A^\varepsilon_{\nu\mu} = -\int \frac{T^\varepsilon_{\nu\mu}}{\alpha-\mu}\,dx + \int \frac{T^\varepsilon_{\nu\mu}}{\alpha-\nu}\,d\alpha, \]
\[ B^\varepsilon_{\nu\mu} = -\int T^\varepsilon_{\nu\alpha}T^\varepsilon_{\alpha\mu} \left\{\frac{1}{\alpha-\nu}+\frac{1}{\alpha-\mu}\right\}\,d\alpha. \tag{4} \]
Let \(\|T^0_{\nu\mu}\|\), \(\|\rho(0,\nu)\|<\infty\). Equation (4) has a solution \(T^\varepsilon_{\nu\mu}\) in the class of functions with finite norm \(\|T^\varepsilon_{\nu\mu}\|\), vanishing for \(|\alpha|,|\beta|\ge R\), \(T^\varepsilon_{\nu\mu}\big|_{\varepsilon=0}=T^0_{\nu\mu}\). From the solution of equation (4) we obtain solutions of equations (1), (3) on \(P_1\), \(P_2\), which at \(\varepsilon=0\) reduce respectively to \(\rho(0,\nu)\) and
\[ W^0_{\nu\mu}=T^0_{\nu\mu}\rho(0,\nu)^{-1}. \]

no. 9. From equations (1)—(3) it follows that they have first integrals:
\[ \Phi(\varepsilon)=\int c(\varepsilon,\nu)\overline{e(\varepsilon,\nu)}\,\rho(\varepsilon,\nu)\,d\nu\equiv c, \]
\[ \Phi_1(\varepsilon)=\iint W^\varepsilon_{\nu\mu}c(\varepsilon,\nu)\overline{e(\varepsilon,\mu)}\,\rho(\varepsilon,\nu)\rho(\varepsilon,\mu)\,d\nu d\mu\equiv c_1, \]
\[ \Phi_2(\varepsilon)=\int \nu c(\varepsilon,\nu)\overline{e(\varepsilon,\nu)}\,\rho(\varepsilon,\nu)\,d\nu -\varepsilon\Phi_1(\varepsilon)\equiv c_2. \]
Let \(\mathfrak{B}\) be the totality of all \(\varphi\in\mathfrak{H}\) for which \(c(\nu)\) in the expansion
\[ \varphi=\int c(\nu)\,dE^0_\nu\psi \]
has finite norm \(\|c(\nu)\|\), and \(c(\varepsilon,\nu)\) is the solution of equation (2) equal to \(c(\nu)\) at \(\varepsilon=0\). Then, for \(\varphi,\chi\in\mathfrak{B}\),
\[ (W\varphi,\chi)=\Phi_1(\varepsilon). \]
\[ (\varphi,\chi)=\Phi(\varepsilon),\qquad (H^\varepsilon\varphi,\chi)-\varepsilon(W\varphi,\chi)=\Phi_2(\varepsilon), \]
for the left- and right-hand sides do not depend on \(\varepsilon\) and coincide when \(\varepsilon=0\). Hence it follows that
\[ (E^\varepsilon_\nu\varphi,\chi) = \int_{\alpha\le \nu} c(\varepsilon,\alpha)\overline{e(\varepsilon,\alpha)}\rho(\varepsilon,\alpha)\,d\alpha. \]
Thus, the solutions of equations (1)—(3) make it possible to reproduce the spectral expansion of the operator \(H^\varepsilon\).

no. 10. For
\[ \varphi=\int c(0,\nu)\,dE^0_\nu\psi \]
put
\[ U^\varepsilon\varphi=\int c(0,\nu)\rho(0,\nu)^{1/2}\times \rho(\varepsilon,\nu)^{-1/2}\,dE^\varepsilon_\nu\varphi. \]
Then \(U^\varepsilon\) is a unitary operator and
\[ U^\varepsilon E^0_\Delta=E^\varepsilon_\Delta U \]
for any \(\Delta\). The notion of a DC-matrix is carried over to unitary operators. From equations (1)—(3) it is not difficult to obtain an equation for \(U^\varepsilon_{\alpha\beta}\) and an expression for \(\partial U^\varepsilon_{\alpha\beta}/\partial\varepsilon\) in terms of \(V^\varepsilon_{\alpha\beta}\) and \(\Phi^\varepsilon_\nu\).

No. 11. Let \(H^\varepsilon=\sum_{k=0}^{n} S_k \varepsilon^k,\quad S_k\in \mathfrak M(H^0),\quad (S_k)_\alpha\equiv 0,\quad \|(S_k)_{\alpha\beta}\|<\infty.\) Put
\(W^{i,\varepsilon}=d^iH^\varepsilon/d\varepsilon^i,\ (i=1,2,\ldots,n),\ W^{0,\varepsilon}=H^\varepsilon,\ W^{n+1,\varepsilon}=0.\) Let \(W_{\alpha\beta}^{i,\varepsilon}\) be the DC-matrix of the operator \(W^{i,\varepsilon}\) relative to \(W^{i-1,\varepsilon}\) \((i=1,2,\ldots,n)\), \(E_\alpha^{i,\varepsilon}\) the spectral function of \(W^{i,\varepsilon}\), \(\psi_i\) a generating element, and
\(\rho_i(\varepsilon,\alpha)=\dfrac{\partial}{\partial\alpha}(E_\alpha^{i,\varepsilon}\psi_i,\psi_i)\). The system of equations generalizing, for \(n>1\), equations (1)—(3), has the form

\[ \frac{\partial \rho_i(\varepsilon,\alpha)}{\partial\varepsilon} = -\left(\int \frac{2\operatorname{Re} W_{\alpha\nu}^{\,i+1,\varepsilon}}{\alpha-\nu} \,\rho_i(\varepsilon,\alpha)\,d\alpha\right)\rho_i(\varepsilon,\nu) \qquad (i=0,1,2,\ldots,n-1), \]

\[ \frac{\partial c_i(\varepsilon,\alpha)}{\partial\varepsilon} = -\int \frac{W_{\alpha\nu}^{\,i-1,\varepsilon}}{\alpha-\nu} \,[c_i(\varepsilon,\alpha)-c_i(\varepsilon,\nu)]\rho_i(\varepsilon,\alpha)\,d\alpha, \]

\[ \frac{\partial W_{\alpha\beta}^{\,i,\varepsilon}}{\partial\varepsilon} = W_{\alpha\beta}^{\,i+1,\varepsilon} + W_{\alpha\beta}^{\,i,\varepsilon} (\Phi_\nu^{\,i,\varepsilon}+\Phi_\mu^{\,i,\varepsilon}) + \]

\[ + (W^{i,\varepsilon}V^{i,\varepsilon}-V^{i,\varepsilon}W^{i,\varepsilon})_{\nu\mu}^{\,i,\varepsilon} \qquad (i=1,2,\ldots,n), \]

\[ \Phi_\nu^{\,i,\varepsilon} = \int \frac{W_{\alpha\nu}^{\,i,\varepsilon}}{\alpha-\nu}\rho_{i-1}(\varepsilon,\alpha)\,d\alpha, \qquad V_{\alpha\beta}^{\,i,\varepsilon} = \frac{W_{\alpha\beta}^{\,i,\varepsilon}}{\alpha-\beta}. \]

Putting \(T_{\alpha\beta}^{\,i,\varepsilon}=W_{\alpha\nu}^{\,i,\varepsilon}\rho_{i-1}(\varepsilon,\alpha)\), we obtain the solution of this system, the reconstruction of the spectral expansion of \(W^{i,\varepsilon}\) from the expansion of \(W^{i,0}\), and the unitary equivalence of the operator \(W^{i,\varepsilon}\) to the operator \(W^{i,0}\).

Received
11 III 1964

REFERENCES

¹ P. Halmos, Measure Theory, IL, 1953.
² K. O. Friedrichs, Math. Ann., 115, No. 2, 249 (1938).