MATHEMATICS

M. S. BRODSKII

AN INVERSE PROBLEM FOR SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS CONTAINING A PARAMETER

(Presented by Academician I. G. Petrovskii, 19 IX 1956)

Consider the system of differential equations

\[ \frac{dy_i}{dx}=\frac{i}{\lambda}\sum_{j=1}^{n} b_{ij}(x)y_j \quad (i=1,2,\ldots,n;\; 0\leq x\leq l), \tag{1} \]

where \(\lambda\) is a complex parameter, and denote by \(W(x,\lambda)\) the fundamental matrix of solutions of this system:

\[ \frac{dW(x,\lambda)}{dx}=\frac{i}{\lambda}b(x)W(x,\lambda),\quad W(0,\lambda)=I \quad (b(x)=\|b_{ij}(x)\|). \]

In the present article we establish certain sufficient conditions under which the coefficients \(b_{ij}(x)\) of system (1) are uniquely recovered from the matrix \(W(\lambda)=W(l,\lambda)\).

I. We shall say that a square matrix-function \(W(\lambda)\) belongs to the class \(M^+\) if it satisfies the following conditions: 1) the elements of the matrix \(W(1/z)\) are entire functions of \(z=1/\lambda\), 2) \(\lim_{\lambda\to\infty} W(\lambda)=I\), 3) \(W(\lambda)W^*(\lambda)=I\) for \(\operatorname{Im}\lambda=0\), 4) \(W(\lambda)W^*(\lambda)\geq I\) for \(\operatorname{Im}\lambda>0\). From conditions 1)—4) it follows that the expansion of the matrix-function \(W(\lambda)\) in a series in negative powers of \(\lambda\) has the form \(W(\lambda)=I+\frac{i}{\lambda}H+\cdots\), where \(H\) is a Hermitian nonnegative matrix. The trace of the matrix \(H\) we agree to call the weight of the matrix-function \(W(\lambda)\). If \(W(\lambda)=W_2(\lambda)W_1(\lambda)\) \((W_1(\lambda)\in M^+,\; W_2(\lambda)\in M^+)\), then the matrix \(W_1(\lambda)\) will be called a divisor of the matrix-function \(W(\lambda)\).

Consider a bounded linear operator \(A\) acting in a Hilbert space \(H\). We shall assign the operator \(A\) to the class \(K^+\) if: 1) the spectrum of the operator \(A\) consists of only the single point \(0\), 2) the space

\[ E=\frac{A-A^*}{i}\,H \]

is finite-dimensional, and all eigenvalues \(\omega_\alpha\) \((\alpha=1,2,\ldots,r)\) of the operator \(\frac{A-A^*}{i}\) in \(E\) are positive, 3) the operator \(A\) is simple, i.e. \(H\) coincides with the closure of the linear span of vectors of the form \(A^n e_\alpha\) \((n=0,1,2,\ldots;\ \alpha=1,2,\ldots,r)\), where \(e_\alpha\) is an orthonormal basis of eigenvectors of the operator \(\frac{A-A^*}{i}\) in \(E\) \(\left(\frac{A-A^*}{i}e_\alpha=\omega_\alpha e_\alpha\right)\). The number

\[ l=\sum_{\alpha=0}^{r}\omega_\alpha \]

will be called the non-Hermitian trace of the operator \(A\). Every matrix-function

\[ W(\lambda)=I-i\Pi\|((A-\lambda E)^{-1}e_\alpha,e_\beta)\|\Pi^*, \]

where \(\Pi\) is an arbitrary square or rectangular matrix, satis-

satisfying the condition \(\Pi^*\Pi=\Omega\),

\[ \Omega=\left\|\begin{matrix} \omega_1 & & \\ & \omega_2 & \\ & & \ddots \omega_r \end{matrix}\right\|, \]

is called characteristic \({}^{(1,2)}\) for the operator \(A\). Let \(H_0\) be some subspace in \(H\), \(P_0\) the projection operator onto \(H_0\), and \(A_0\) the operator acting in \(H_0\) for which \(A_0 f=P_0Af\), \((f\in H_0)\). If \(e_\alpha^{(0)}\) \((\alpha=1,2,\ldots,r_0)\) is an orthonormal basis of eigenvectors of the operator \(A_0\) in the subspace

\[ E_0=\frac{A_0-A_0^*}{i}H_0, \]

then the matrix-function

\[ W_0(\lambda)=I-i\Pi_0\left\|\bigl((A_0-\lambda E)^{-1}e_\alpha^{(0)},\,e_\beta^{(0)}\bigr)\right\|\Pi_0^* \quad (\Pi_0=\Pi U_0,\; U_0=\|(e_\alpha,e_\beta^{(0)})\|) \]

is characteristic for the operator \(A_0\) and is called a projection \({}^{(2)}\) of the matrix-function \(W(\lambda)\) onto \(H_0\).

Theorem 1. In order that the matrix-function \(W(\lambda)\) belong to the class \(M^+\), it is necessary and sufficient that it be characteristic for some operator belonging to the class \(K^+\).

Theorem 2. Let \(W(\lambda)\) be the characteristic matrix-function of an operator \(A\in K^+\) acting in the space \(H\). In order that the matrix \(W_1(\lambda)\) be a divisor of \(W(\lambda)\), it is necessary and sufficient that it be a projection of \(W(\lambda)\) onto some subspace \(H_1\subseteq H\), invariant with respect to \(A\).

Every matrix-function \(W(\lambda)\in M^+\) having weight \(l\) admits the multiplicative representation \({}^{(3)}\)

\[ W(\lambda)=\overset{l}{\int_0} e^{\frac{i}{\lambda}\,dE(t)} = \lim_{\Delta t_i\to 0} \left( e^{\frac{i}{\lambda}\Delta E_p}\cdots e^{\frac{i}{\lambda}\Delta E_2}e^{\frac{i}{\lambda}\Delta E_1} \right), \tag{2} \]

where

\[ E(t)=\int_0^t b(x)\,dx, \]

\(b(x)\) is a certain Hermitian nonnegative summable matrix on \([0,l]\), for which \(\operatorname{Sp} b(x)\equiv 1\), \(0=t_0<t_1<\cdots<t_p=l\), \(\Delta E_k=E(t_k)-E(t_{k-1})\), \(\Delta t_k=t_k-t_{k-1}\). From representation (2) it follows that \(\|W(\lambda)\|<e^{l/|\lambda|}\). From the same representation it is seen that the matrix-function \(W(\lambda)\) has the divisor

\[ \overset{l_1}{\int_0} e^{\frac{i}{\lambda}\,dE(t)} \]

of any weight \(l_1<l\).

Theorem 3. Let the matrix-function \(W(\lambda)\in M^+\) have weight \(l\). If for every \(\varepsilon>0\) there exists a sequence \(\lambda_k\to 0\) for which \(\|W(\lambda_k)\|>e^{(l-\varepsilon)/|\lambda_k|}\) \((k=1,2,3,\ldots)\), then \(W(\lambda)\) has one and only one divisor of the given weight \(l_1<l\).

Proof. By virtue of Theorem 1, the matrix-function \(W(\lambda)\) is characteristic for some operator \(A\in K^+\). Let \(W_1(\lambda)\) and \(W'_1(\lambda)\) be divisors of the matrix-function \(W(\lambda)\) having the common weight \(l_1\). By virtue of Theorem 2, \(W_1(\lambda)\) and \(W'_1(\lambda)\) serve as projections of \(W(\lambda)\) onto certain invariant subspaces \(H_1\) and \(H'_1\) of the operator \(A\). In \({}^{(4)}\) it is shown that, under the conditions of the theorem being proved, the operator \(A\) is unicellular. Consequently, one of the subspaces \(H_1,H'_1\) is part of the other. Since the operators \(A_1\) and \(A'_1\), generated by the operator \(A\) respectively in \(H_1\) and \(H'_1\), have a common non-Hermitian trace \(l_1\), it follows that \(H=H'_1\), and therefore \(W_1(\lambda)=W'_1(\lambda)\).

II. The system of differential equations

\[ \frac{dy_i}{dx}=\frac{i}{\lambda}\sum_{j=1}^{n} b_{ij}(x)y_j \quad (i=1,2,\ldots,n;\;0\leq x\leq l) \tag{3} \]

we shall call it normalized if the trace \(\sum_{i=1}^n b_{ii}(x) \equiv 1\). If system (3) is not normalized, but the trace \(\sum_{i=1}^n b_{ii}(x)\) differs from zero almost everywhere, then it can be normalized by making the change of independent variable

\[ t=\int_0^x \sum_{i=1}^n b_{ii}(x)\,dx . \]

Theorem 4. Suppose that on the segment \([0,l]\) there are given two normalized systems of differential equations

\[ \frac{dy_i}{dx}=\frac{i}{\lambda}\sum_{j=1}^n b_{ij}^{(1)}(x)y_j,\qquad \frac{dy_i}{dx}=\frac{i}{\lambda}\sum_{j=1}^n b_{ij}^{(2)}(x)y_j, \tag{4} \]

whose coefficient matrices \(b^{(1)}(x)=\|b_{ij}^{(1)}(x)\|\) and \(b^{(2)}(x)=\|b_{ij}^{(2)}(x)\|\) are Hermitian nonnegative, with the functions \(b_{ij}^{(1)}(x)\) possessing absolutely continuous first derivatives and the rank of the matrix \(b^{(1)}(x)\) equal to one for every \(x\in[0,l]\). Denote by \(W^{(1)}(x,\lambda)\) and \(W^{(2)}(x,\lambda)\) the fundamental matrices of solutions of these systems. If \(W^{(1)}(l,\lambda)\equiv W^{(2)}(l,\lambda)\), then \(b_{ij}^{(1)}(x)\equiv b_{ij}^{(2)}(x)\).

Proof. Studying the asymptotics of the solutions of the first of systems (4), we find that the matrix-function \(W^{(1)}(l,\lambda)\) satisfies the conditions of Theorem 3. Since

\[ W^{(i)}(x,\lambda)=\int_0^x e^{\frac{i}{\lambda}}\,dE^{(i)}(x),\qquad E^{(i)}(x)=\int_0^x b^{(i)}(t)\,dt\quad (i=1,2), \]

it follows that \(W^{(1)}(x,\lambda)\) and \(W^{(2)}(x,\lambda)\) are, for the matrix-function \(W^{(1)}(l,\lambda)\), divisors of one and the same weight \(x\). By virtue of Theorem 3, \(W^{(1)}(x,\lambda)\equiv W^{(2)}(x,\lambda)\) for every fixed \(x\in[0,l]\), and, consequently, \(b^{(1)}(x)\equiv b^{(2)}(x)\).

Theorem 5. Suppose that on the segment \([0,l]\) there are given normalized systems of differential equations (4), where \(b^{(2)}(x)\) is a Hermitian nonnegative matrix with summable elements, and the matrix \(b^{(1)}(x)=\xi^*(x)\xi(x)\), \(\xi(x)=\|\xi_1(x)\ldots \xi_n(x)\|\), where the vector \(\xi(x)\) assumes on each of the intervals \([x_{i-1},x_i]\) of some partition \(0=x_0<x_1<\cdots<x_p=l\) a constant value \(\xi^{(i)}\). If among the vectors \(\xi^{(1)},\xi^{(2)},\ldots,\xi^{(p)}\) there are no two neighboring ones that are mutually orthogonal, and \(W^{(1)}(l,\lambda)\equiv W^{(2)}(l)\), then \(b^{(1)}(x)\equiv b^{(2)}(x)\).

Proof. Since \(\xi^{(i)}\xi^{(i)*}=1\) \((i=1,2,\ldots,p)\), the matrix-function

\[ W^{(1)}(l,\lambda)=\exp\left[\frac{i}{\lambda}\xi^{(p)*}\xi^{(p)}\Delta x_p\right]\cdots \exp\left[\frac{i}{\lambda}\xi^{(2)*}\xi^{(2)}\Delta x_2\right] \exp\left[\frac{i}{\lambda}\xi^{(1)*}\xi^{(1)}\Delta x_1\right]= \]

\[ =\left[I+\xi^{(p)*}\xi^{(p)}\left(e^{\frac{i}{\lambda}\Delta x_p}-1\right)\right]\cdots \]

\[ \cdots \left[I+\xi^{(2)*}\xi^{(2)}\left(e^{\frac{i}{\lambda}\Delta x_2}-1\right)\right] \left[I+\xi^{(1)*}\xi^{(1)}\left(e^{\frac{i}{\lambda}\Delta x_1}-1\right)\right] \]

again satisfies the conditions of Theorem 3, and the arguments given in Theorem 4 may be applied to it.

The following assertion is a consequence of the last theorem.

Let \(A_1, A_2, \ldots, A_n, B_1, B_2, \ldots, B_m\) be Hermitian nonnegative matrices of the first rank, satisfying the conditions

\[ s_p A_i = 1 \ (i=1,2,\ldots,n), \qquad A_i \ne A_{i+1}, \quad A_i A_{i+1} \ne 0 \quad (i=1,2,\ldots,n-1); \]

\[ s_p B_j = 1 \ (j=1,2,\ldots,m), \qquad B_j \ne B_{j+1} \quad (j=1,2,\ldots,m-1), \]

\(\alpha_1,\ \alpha_2,\ldots,\alpha_n,\ \beta_1,\beta_2,\ldots,\beta_m\) be certain positive numbers. If, for every \(z\), the equality

\[ e^{z\alpha_1 A_1} e^{z\alpha_2 A_2} \ldots e^{z\alpha_n A_n} = e^{z\beta_1 B_1} e^{z\beta_2 B_2} \ldots e^{z\beta_m B_m}, \]

holds, then \(n=m\), \(\alpha_i=\beta_i\), and \(A_i=B_i\) \((i=1,2,\ldots,n)\).

Odessa State Pedagogical Institute
named after K. D. Ushinsky

Received
20 IV 1956

REFERENCES

¹ M. S. Livshits, Matem. sborn., 34 (76), 1, 145 (1954).
² M. S. Brodskii, DAN, 77, No. 5 (1950).
³ V. P. Potapov, DAN, 72, No. 5 (1950).
⁴ M. S. Brodskii, DAN, 111, No. 5 (1956).