MATHEMATICS

M. GIMADISLAMOV

ON EXPANSION IN EIGENFUNCTIONS OF A NON-SELF-ADJOINT SYSTEM OF DIFFERENTIAL EQUATIONS ON THE WHOLE AXIS

(Presented by Academician I. G. Petrovskii, 17 IV 1962)

In the present paper an expansion in eigenfunctions is obtained for a non-self-adjoint operator generated by a system of second-order differential equations on the whole axis.

We consider a system of differential expressions of the 2nd order, which is written briefly in the form

\[ l(y)=-y''+P(x)y, \tag{1} \]

where \(y(x)\) is a \(k\)-dimensional column vector; \(P(x)\) is a complex-valued matrix of order \(k\).

If \((1+x^2)^{1/2}|P(x)|\in L'_k(-\infty,\infty)\), then the differential expression (1) generates a closed operator \(L\) in \(L_k^2(-\infty,\infty)\) with domain \(D\), consisting of vector-functions \(y(x)\in L_k^2(-\infty,\infty)\) which have an absolutely continuous derivative and \(l(y)\in L_k^2(-\infty,\infty)\). In what follows we shall assume that

\[ e^{a|x|}|P(x)|\in L'_k(-\infty,\infty). \tag{2} \]

Denote by \(s=\sqrt{\lambda}\), \(s=\sigma+it\), \(0\leq \arg s\leq \pi\), and by \(Y_1(x,s)\) and \(Y_2(x,s)\) linearly independent solutions of the matrix equation

\[ -Y''+P(x)Y=s^2Y. \tag{3} \]

Under condition (2), these solutions may be chosen so that, for any fixed \(x\), they are analytic in \(s\) in the domain \(\operatorname{Im}s>-a/2\) and have the following asymptotics in \(x\):

\[ Y_1(x,s)=e^{isx}[1+o(1)] \qquad \text{as } x\to\infty, \tag{4} \]

\[ Y_1(x,s)=e^{isx}\left[1-\int_{-\infty}^{\infty}\frac{e^{-is\xi}}{2is}P(\xi)Y_1(\xi,s)\,d\xi+o(1)\right] \qquad \text{as } x\to-\infty; \]

\[ Y_2(x,s)=e^{-isx}[1+o(1)] \qquad \text{as } x\to-\infty, \]

\[ Y_2(x,s)=e^{-isx}\left[1-\int_{-\infty}^{\infty}\frac{e^{is\xi}}{2is}P(\xi)Y_2(\xi,s)\,d\xi\right] \qquad \text{as } x\to+\infty \tag{5} \]

uniformly in \(s\) for \(\operatorname{Im}s>-a/2\).

Denote

\[ W(s)=\det \begin{pmatrix} Y_1(x,s) & Y_2(x,s)\\ Y'_1(x,s) & Y'_2(x,s) \end{pmatrix}. \]

Theorem 1. If condition (2) is satisfied, the spectrum of the operator consists of a finite number of eigenvalues—the zeros of the function \(W(\lambda^{1/2})\)—and of a continuous spectrum on the nonnegative half-axis \(\lambda \geqslant 0\).

Let us note that the point \(\lambda = 0\) is not an eigenvalue. In what follows, for simplicity, we shall assume that: 1) \(W(\lambda^{1/2})\) has no zeros on the axis \(\sigma\); 2) the zeros of \(W(\lambda^{1/2})\) are simple.

Let \(\lambda_1, \lambda_2, \ldots, \lambda_r\) be the eigenvalues, and let \(y_1(x), y_2(x), \ldots, y_r(x)\) be the corresponding eigenvector-functions of the operator \(L\). Then the following holds:

Theorem 2. If conditions 1) and 2) are satisfied and \(G(x,\xi,\lambda)\) is the resolvent kernel of the operator \(L\), then for any point \(\lambda\) not belonging to the spectrum of the operator \(L\),

\[ G(x,\xi,\lambda) = \sum_{j=1}^{r} \frac{y_j(x) z_j^{*}(\xi)}{\lambda_j-\lambda} + \frac{1}{\pi} \int_{0}^{\infty} \sum_{i=1}^{2} \frac{\Theta_i(x,\sigma) A_i(\sigma)\Psi_i(\xi,\sigma)} {\sigma^2-\lambda}\,d\sigma , \tag{6} \]

where the integral on the right converges absolutely and uniformly with respect to \(x,\xi\) in the interval \(-\infty < x,\xi < \infty\). Here \(\Theta_i(x,\sigma)\) are linearly independent solutions of the matrix equation (3), and \(\Psi_i(x,\sigma)\) are linearly independent solutions of the matrix equation

\[ -Z'' + ZP(x)=\sigma^2 Z; \tag{7} \]

\(A_i(\sigma)\) are matrices of order \(k\).

We outline the proof of Theorem 2. First we construct the Green’s function \(G(x,\xi,\lambda)\) for the operator \(L\), i.e., we solve the equation \(l(y)-\lambda y=f\), where \(f(x)\in L_k^2(-\infty,\infty)\):

\[ G(x,\xi,\lambda)=K(x,\xi,\sqrt{\lambda}) = \frac{1}{W(s)} \begin{cases} -Y_1(x,s)S_{12}(\xi,s), & \text{for } \xi < x,\\ \phantom{-}Y_2(x,s)S_{22}(\xi,s), & \text{for } \xi > x, \end{cases} \tag{8} \]

where

\[ \frac{1}{W(s)} \begin{pmatrix} Y_1 & Y_2\\ Y_1' & Y_2' \end{pmatrix} \begin{pmatrix} S_{11} & S_{12}\\ S_{21} & S_{22} \end{pmatrix} =1. \tag{9} \]

Denote by \(C_{R,\delta}\) the contour in the \(s\)-plane consisting of the straight line \(\tau=\delta>0\) from \(\sigma=-\sqrt{R^2-\delta^2}\) to \(\sigma=\sqrt{R^2-\delta^2}\) and the arc of the circle \(s=Re^{i\theta}\) from \(\theta=\eta=\arcsin \frac{\delta}{R}\) to \(\theta=\pi-\eta\). Choose \(\delta\) and \(R_0\) so that, if \(\lambda_0\) is an eigenvalue, then \(\operatorname{Im}\lambda_0^{1/2}\ne\delta\) and \(R_0^2>|\lambda_0|\).

Consider the integral

\[ I_{R,\delta} = \int_{C_{R,\delta}} \frac{K(x,\xi,s)}{s^2-\lambda}\,ds \tag{10} \]

for \(R\geqslant R_0\); the point \(\sqrt{\lambda}\) lies inside the contour \(C_{R,\delta}\). Applying the residue theorem and letting \(R\to\infty\), we obtain:

\[ G(x,\xi,\lambda) = \sum_{j=1}^{r} \frac{G^{(j)}(x,\xi)}{\lambda_j-\lambda} + \frac{1}{\pi i} \int_{-\infty}^{\infty} \frac{(\sigma+i\delta)K(x,\xi,\sigma+i\delta)} {(\sigma+i\delta)^2-\lambda}\,d\sigma , \tag{11} \]

where \(G^{(j)}(x,\xi)=y_j(x)z_j^{*}(\xi)\), and \(n(\delta)\) denotes the number of eigenvalues lying inside \(C_{R,\delta}\).

By virtue of conditions 1) and 2), as \(\delta\to 0\) equality (11) takes the form

\[ G(x,\xi,\lambda) = \sum_{j=1}^{r} \frac{y_j(x)z_j^{*}(\xi)}{\lambda_j-\lambda} + \frac{1}{\pi i} \int_{0}^{\infty} \frac{\sigma K(x,\xi,\sigma)-\sigma K(x,\xi,-\sigma)} {\sigma^2-\lambda}\,d\sigma . \tag{12} \]

Taking into account relation (9) and the asymptotic formulas (4) and (5), one obtains that:
\[ \begin{aligned} S_{12}(x,s)&=(-1)^k(2is)^{k-1}e^{-isx}[W_1(s)\cdot 1+o(1)] &&\text{as } x\to\infty,\\ S_{12}(x,s)&=(-1)^k(2is)^{k-1}e^{-isx}[1-A(s)+o(1)] &&\text{as } x\to-\infty,\\ S_{22}(x,s)&=(-1)^{k+1}(2is)^{k-1}e^{isx}[1-B(s)+o(1)] &&\text{as } x\to\infty,\\ S_{22}(x,s)&=(-1)^{k+1}(2is)^{k-1}e^{isx}[W_1(s)\cdot 1+o(1)] &&\text{as } x\to-\infty, \end{aligned} \]
where
\[ W_1(s)=\det\left(1-\int_{-\infty}^{\infty}\frac{e^{-is\xi}}{2is}PY_1\,d\xi\right) =\det\left(1-\int_{-\infty}^{\infty}\frac{e^{is\xi}}{2is}PY_2\,d\xi\right), \]
\[ [W_1(s)]^{k-1}=\det(1-A(s))=\det(1-B(s)). \]

From this asymptotics and from the properties of the Green function it follows that \(S_{12}(x,s)\) and \(S_{22}(x,s)\) are linearly independent solutions of the matrix equation (7). We choose linearly independent solutions of equation (7) so that
\[ S_{12}(x,s)=(-1)^k(2is)^{k-1}[W_1(s)]^{\frac{k-1}{k}}Z_2(x,s), \]
\[ S_{22}(x,s)=(-1)^k(2is)^{k-1}[W_1(s)]^{\frac{k-1}{k}}Z_1(x,s), \]
\[ \det \begin{pmatrix} Z_2 & Z_1\\ Z_2' & Z_1' \end{pmatrix} =W(s). \]

Then relation (9) takes the form:
\[ \frac{(-1)^k(2is)^{k-1}[W_1(s)]^{\frac{k-1}{k}}}{W(s)} \begin{pmatrix} Y_1 & Y_2\\ Y_1' & Y_2' \end{pmatrix} \begin{pmatrix} Z_2' & Z_2\\ Z_1' & Z_1 \end{pmatrix} =1. \tag{13} \]

Let us compute \(\sigma K(x,\xi,\sigma)-\sigma K(x,\xi,-\sigma)\) for \(\xi\le x\):
\[ \begin{aligned} \sigma K(x,\xi,\sigma)-\sigma K(x,\xi,-\sigma) &=\frac{(-1)^{k+1}\sigma(2i\sigma)^{k-1}[W_1(\sigma)]^{\frac{k-1}{k}}}{W(\sigma)} Y_1(x,\sigma)Z_2(\xi,\sigma)\\ &\quad+\frac{(-1)^k\sigma(-2i\sigma)^{k-1}[W_1(-\sigma)]^{\frac{k-1}{k}}}{W(-\sigma)} Y_1(x,-\sigma)Z_2(\xi,-\sigma)\\ &=\frac{(-1)^{k+1}\sigma(2i\sigma)^{k-1}[W_1(\sigma)]^{\frac{k-1}{k}}}{W(\sigma)} Y_1(x,\sigma)\sum_{i=1}^k e_i v_2^{(i)*}(\xi,\sigma)\\ &\quad+\frac{(-1)^k\sigma(-2i\sigma)^{k-1}[W_1(-\sigma)]^{\frac{k-1}{k}}}{W(-\sigma)} \sum_{i=1}^k y_1^{(i)}(x,-\sigma)e_i^* Z_2(\xi,-\sigma), \end{aligned} \tag{14} \]
where \(e_i\) is the unit column vector; \(e_i^*\) is the unit row vector; \(v_2^{(i)*}(\xi,\sigma)\) is the \(i\)-th row of the matrix \(Z_2(\xi,\sigma)\); \(y_1^{(i)}(x,-\sigma)\) is the \(i\)-th column of the matrix \(Y_1(x,-\sigma)\).

Since \(Z_1(\xi,-\sigma)\) and \(Z_2(\xi,-\sigma)\) are fundamental matrices of (7), it follows that
\[ v_2^{(i)*}(\xi,\sigma)=a_1^{(i)*}Z_1(\xi,-\sigma)+a_2^{(i)*}Z_2(\xi,-\sigma), \tag{15} \]
where \(a_1^{(i)*}, a_2^{(i)*}\) are constant row vectors.

We determine the row vectors \(a_1^{(i)*}\) and \(a_2^{(i)*}\) as follows: we differentiate equality (15); the equality obtained, together with (15), gives the following system of equations:
\[ v_2^{(i)*}(\xi,\sigma)=a_1^{(i)*}Z_1(\xi,-\sigma)+a_2^{(i)*}Z_2(\xi,-\sigma), \]
\[ v_2^{(i)*'}(\xi,\sigma)=a_1^{(i)*}Z_1'(\xi,-\sigma)+a_2^{(i)*}Z_2'(\xi,-\sigma). \]

Taking (13) into account, from this system we obtain

\[ a_1^{(i)*}= \frac{(-1)^k(-2i\sigma)^{k-1}[W_1(-\sigma)]^{\frac{k-1}{k}}}{W(-\sigma)} \,[v_2^{(i)*}(\xi,\sigma)Y_2'(\xi,-\sigma)+v_2^{(i)*'}(\xi,\sigma)Y_2(\xi,-\sigma)], \]

\[ a_2^{(i)*}= \frac{(-1)^k(-2i\sigma)^{k-1}[W_1(-\sigma)]^{\frac{k-1}{k}}}{W(-\sigma)} \,[v_2^{(i)*}(\xi,\sigma)Y_1'(\xi,-\sigma)+v_2^{(i)*'}(\xi,\sigma)Y_1(\xi,-\sigma)]. \]

Similarly, for the column vector \(y_1^{(i)}(x,-\sigma)\) we have

\[ y_1^{(i)}(x,-\sigma)=Y_1(x,\sigma)b_1^{(i)}+Y_2(x,\sigma)b_2^{(i)}, \tag{16} \]

where \(b_1^{(i)}, b_2^{(i)}\) are constant column vectors, which we determine analogously to the preceding ones, and we obtain:

\[ b_1^{(i)}= \frac{(-1)^k(2i\sigma)^{k-1}[W_1(\sigma)]^{\frac{k-1}{k}}}{W(\sigma)} \,[Z_2'(x,\sigma)y_1^{(i)}(x,-\sigma)+Z_2(x,\sigma)y_1^{(i)'}(x,-\sigma)], \]

\[ b_2^{(i)}= \frac{(-1)^k(2i\sigma)^{k-1}[W_1(\sigma)]^{\frac{k-1}{k}}}{W(\sigma)} \,[Z_1'(x,\sigma)y_1^{(i)}(x,-\sigma)+Z_1(x,\sigma)y_2^{(i)'}(x,-\sigma)]. \]

Substituting the expressions found for \(v_2^{(i)*}(\xi,\sigma)\) and \(y_1^{(i)}(x,-\sigma)\) into (14), we obtain that, for \(\xi \leqslant x\),

\[ \sigma K(x,\xi,\sigma)-\sigma K(x,\xi,-\sigma)= \]

\[ = \frac{\sigma(2i\sigma)^{k-1}(-2i\sigma)^{k-1}[W_1(\sigma)W_1(-\sigma)]^{\frac{k-1}{k}}}{W(\sigma)W_1(-\sigma)} \sum_{j=1}^{2}Y_j(x,\sigma)T_j(\sigma)Z_j(\xi,-\sigma), \tag{17} \]

where

\[ T_2(\sigma)=Z_2(x,\sigma)Y_2'(x,-\sigma)+Z_2'(x,\sigma)Y_2(x,-\sigma), \]

\[ T_1(\sigma)=-[Z_1(x,\sigma)Y_1'(x,-\sigma)+Z_1'(x,\sigma)Y_1(x,-\sigma)]. \]

In a similar way we arrive at the same expression (17) also for \(\xi>x\). Denoting

\[ \Theta_j(x,\sigma)= \frac{\sigma(2i\sigma)^{k-1}[W_1(\sigma)]^{\frac{k-1}{k}}}{W(\sigma)} \,Y_j(x,\sigma), \]

\[ \Psi_j(\xi,\sigma)= \frac{\sigma(-2i\sigma)^{k-1}[W_1(-\sigma)]^{\frac{k-1}{k}}}{W(-\sigma)} \,Z_j(\xi,-\sigma), \]

\[ A_i(\sigma)=\frac{1}{i\sigma}T_i(\sigma), \]

we obtain formula (6).

From the expansion (6) it is not difficult to obtain an expansion for any vector-function \(y(x)\in D_L\) and an analogue of Parseval’s equality.

It is likewise not difficult to write the expansion for \(G(x,\xi,\lambda)\) when the zeros of \(W(s)\) are multiple.

Remark 1. If \(W(s)\) has zeros on the real axis \(\tau=0\), then the expansion for \(G(x,\xi,\lambda)\) is obtained by the method of paper (1).

Remark 2. If \((1+x^2)^{1/2}|P(x)|\in L_k'(-\infty,\infty)\), then the expansion (6) can be obtained if \(W(s)\) has no zeros on the real axis.

Remark 3. For \(k=1\), from the expansion (6) one can obtain Kemp’s result (3).

In conclusion, the author expresses his gratitude to Prof. M. A. Naimark for advice and comments.

Received
12 IV 1962

References

1. M. A. Naimark, Trans. Moscow Math. Soc., 3, 181 (1954).
2. M. A. Naimark, Linear Differential Operators, 1954.
3. R. R. D. Kemp, Canad. J. Math., 10, No. 3, 447 (1958).
4. M. G. Gimadilov, DAN, 140, No. 1 (1961).