UDC 517.94

MATHEMATICS

L. B. ZELENKO

THE DEFECT INDEX AND THE SPECTRUM OF A SELF-ADJOINT SYSTEM OF FIRST-ORDER DIFFERENTIAL EQUATIONS

(Presented by Academician L. S. Pontryagin, November 3, 1967)

1°. We shall consider self-adjoint operators generated in the Hilbert space of \(n\)-component vector-functions \(L_n^2(a,b)\) by the differential expression:

\[ l\bar y=\frac{i}{2}(\Lambda \bar y)' + \frac{i}{2}\Lambda \bar y' + Q(x)\bar y \qquad (a<x<b), \]

where \(\Lambda(x)\), \(Q(x)\) are Hermitian matrix-functions in the unitary space \(E_n\), with \(\Lambda(x)\) nondegenerate and absolutely continuous for \(a<x<b\); \(Q(x)\) is summable on every segment \([\alpha,\beta]\subset(a,b)\).

Let us agree on some notation: \(\mu A\) and \(\nu A\) are, respectively, the smallest and the largest eigenvalues of a Hermitian matrix \(A\);
\(\operatorname{Re} A=(A+A^*)/2\), \(\operatorname{Im} A=(A-A^*)/2i\).

Let \(L_0\) be the minimal closed symmetric operator generated by the operation \(l\), with domain of definition \(D_0\). In this section we shall be interested in the question of the defect index of the operator \(L_0\).

Theorem 1. Suppose that, for the operator \(L_0\), the endpoint \(a\) is regular, and in a neighborhood of the endpoint \(b\) the condition

\[ \int_{x_0}^{\infty}\frac{dt}{\|\Lambda(t)\|}=\infty \qquad (a<x_0<b) \tag{1} \]

is satisfied.

Then the defect index of the operator \(L_0\) has the form \((\nu_+,\nu_-)\), where \(\nu_+\) and \(\nu_-\) are, respectively, the number of positive and the number of negative eigenvalues of the matrix \(\Lambda(x)\) (counting their multiplicities).

Corollary 1.1. Suppose the endpoint \(b\) is regular, and in a neighborhood of \(x=a\) the condition

\[ \int_a^{x_0}\frac{dt}{\|\Lambda(t)\|}=\infty \qquad (a<x_0<b) \]

is satisfied.

Then the defect index of the operator \(L_0\) has the form \((\nu_-,\nu_+)\).

Corollary 2.1. Suppose that in neighborhoods of the endpoints \(a\) and \(b\) the conditions

\[ \int_{x_0}^{b}\frac{dt}{\|\Lambda(t)\|}=\infty, \qquad \int_a^{x_0}\frac{dt}{\|\Lambda(t)\|}=\infty \]

are satisfied.

Then the operator \(L_0\) is self-adjoint.

Corollary 3.1. Suppose the endpoint \(a\) is regular and condition (1) is satisfied. Then, if \(\nu_-\ne\nu_+\), the core of the spectrum of the operator \(L_0\) fills the entire real axis. In particular, the latter will always occur if \(n\) is odd.

2°. In the usual standard way one constructs the resolvents of self-adjoint extensions \(L\) of the operator \(L_0\) (under the assumption that such extensions exist). In this case the resolvents turn out to be integral operato-

with kernels of Carleman type:

\[ R_{\lambda}\bar f=\int_a^b K(x,\xi,\lambda)\bar f(\xi)d\xi \qquad (\operatorname{Im}\lambda\ne 0); \]

\(K(x,\xi,\lambda)\) is a matrix kernel of the form:

\[ K(x,\xi,\lambda)=Y(x,\lambda)\left[M(\lambda)+\frac{1}{2}\operatorname{sgn}(x-\xi)\Lambda_0^{-1}\right]Y^*(\xi,\bar\lambda), \tag{2} \]

where \(Y(x,\lambda)\) is the matrix of the fundamental system of solutions of the homogeneous system \(l\bar y=\lambda\bar y\), satisfying the condition \(Y(a,\lambda)=E\); \(M(\lambda)\) is the characteristic matrix-function of the operator \(L\); \(\Lambda_0=\Lambda(a)\). The relations

\[ \int_a^b \|K(x,\xi,\lambda)\|^2\,d\xi<\infty \qquad (a<x<b), \]

\[ \int_a^b \|K(x,\xi,\lambda)\|^2\,dx<\infty \qquad (a<\xi<b). \]

hold.

The case when the defect index of the operator \(L_0\) is \((n,n)\) will be called quasiregular. As is seen from the representation (2), in this case the kernel \(K(x,\xi,\lambda)\) for \(\operatorname{Im}\lambda\ne 0\) is a Hilbert–Schmidt kernel; consequently, the spectrum of any self-adjoint extension \(L\) of the operator \(L_0\) is discrete. Here we shall formulate some conditions for quasiregularity of the operator \(L_0\).

Theorem 2. Let the quadratic form \(\Lambda[\bar\xi]=(\Lambda\bar\xi,\bar\xi)\) be sign-definite and let the condition

\[ \int_a^b \frac{dt}{|\lambda(t)|}<\infty, \]

be satisfied, where

\[ \lambda(x)= \begin{cases} \Delta\Lambda(x), & \text{if } \Lambda[\bar\xi] \text{ is positive definite},\\ \nabla\Lambda(x), & \text{if } \Lambda[\bar\xi] \text{ is negative definite}. \end{cases} \]

Then the operator \(L_0\) is quasiregular.

In what follows we shall need the following estimates for solutions of a linear system:

Lemma 1. For any solution \(\bar y(x)\) of the linear system \(d\bar Y/dx=A(x)\bar y\), the estimates

\[ \|\bar y(x)\|^2\le c_0\exp\left[2\int_{x_0}^x \nu\,\operatorname{Re} A(\tau)d\tau\right], \]

\[ \|\bar y(x)\|^2\ge \varepsilon_0\exp\left[2\int_{x_0}^x \mu\,\operatorname{Re} A(\tau)d\tau\right], \]

hold, where \(c_0=\|Y(x_0)\|^2\).

On the basis of Lemma 1 the following assertion is proved:

Theorem 3. Let the endpoint \(a\) be regular and let the following conditions be satisfied for the matrices \(\Lambda(x)\) and \(Q(x)\):

1) There exists an absolutely continuous function \(\delta(x)>0\) \((a<x<b)\) such that: a)

\[ \int_a^b \frac{dt}{\delta(t)}<\infty; \]

b) the elements of the matrix

\[ \Lambda_0(x)=\frac{1}{\delta(x)}\Lambda(x) \]

are functions whose moduli and arguments are monotone and bounded in a neighborhood of the endpoint \(b\), and moreover

\[ \Lambda_0(b)=\lim_x \Lambda_0(x) \]

is a nondegenerate matrix.

2)
\[ \int_a^b \left\| \operatorname{Im}\bigl(\Lambda^{-1}(t)Q(t)\bigr)\right\|\,dt<\infty. \]

Then the operator \(L_0\) is quasiregular.

\(3^\circ\). Suppose that the endpoint \(a\) is regular and that condition (1) is satisfied for \(\Lambda(x)\). We shall assume, moreover, that \(n\) is even \((n=2m)\).

By applying Rapoport’s asymptotics for the solutions of the linear system \(l\bar y=0\) (see (1), p. 228), the following criterion for discreteness of the spectrum of the operator \(L\) is established:

Theorem 4. Let the following conditions be satisfied for the matrices \(\Lambda(x)\) and \(Q(x)\):

1) There exists an absolutely continuous function \(\delta(x)>0\) \((a\le x<b)\) such that
\[ \int_{x_0}^{b}\frac{dt}{\delta(t)}=\infty \qquad (a<x_0<b), \]
and, for
\[ \Lambda_0(x)=\frac{1}{\delta(x)}\Lambda(x), \]
condition \(1b\) of Theorem 3 is satisfied.

2) The matrix \(Q(x)\) is absolutely continuous, the eigenvalues \(\lambda_1,\lambda_2,\ldots,\lambda_n\) of the matrix \(i\Lambda_0^{-1}Q\) are simple, and \(\operatorname{Re}\lambda_i\), \(\operatorname{Re}(\lambda_i-\lambda_k)\) \((1\le i\le k\le n)\) do not change sign in a neighborhood of the endpoint \(b\).

3) The matrix \(B(x)\), whose columns are eigenvectors of the matrix \(\Lambda_0^{-1}Q\), satisfies the condition: the moduli and arguments of the elements of the matrix \(B(x)\) are monotone and bounded in a neighborhood of the endpoint \(b\), and
\[ B(b)=\lim_{x\to b}B(x) \]
is nonsingular.

Then, for the spectrum of any self-adjoint extension \(L\) of the operator \(L_0\) to be discrete, it is necessary and sufficient that the following conditions be satisfied: for every \(\omega>0\),
\[ \int_{\alpha}^{\beta}\frac{|\operatorname{Re}\lambda_i(t)|}{\delta(t)}\,d\tau \to \infty \qquad (1\le i\le n), \]
when \(\alpha\to b\) and
\[ \int_{\alpha}^{\beta}\frac{d\tau}{\delta(\tau)}=\omega. \]

\(4^\circ\). The following assertion holds, which is an analogue of a theorem of I. E. Shnol’ (²), established by him for the Sturm–Liouville operator:

Theorem 5. Let the endpoint \(a\) be regular and
\[ \int_{x_0}^{b}\frac{dt}{\|\Lambda(t)\|}=\infty \qquad (a<x_0<b). \]
Suppose that the system \(l\bar y=\lambda\bar y\) has a solution \(\bar y(x,\lambda)\) satisfying, for some sequences \(\xi_p\to b\) and \(\varepsilon_p\to0\) as \(p\to\infty\), one of the following estimates:

1)
\[ \int_a^{\xi_p}\|\bar y(t,\lambda)\|^2\,dt \le c\exp\left[\varepsilon_p\int_a^{\xi_p}\frac{dt}{\|\Lambda(t)\|}\right], \]
if \(\bar y(x,\lambda)\) does not belong to \(L_n^2[a,b)\).

2)
\[ \int_{\xi_p}^{b}\|\bar y(t,\lambda)\|^2\,dt \ge c\exp\left[-\varepsilon_p\int_a^{\xi_p}\frac{dt}{\|\Lambda(t)\|}\right], \]
if \(\bar y(x,\lambda)\) belongs to \(L_n^2[a,b)\).

Then \(\lambda\) belongs to the continuous spectrum \(C(L)\) of the operator \(L\).

On the basis of Theorem 5 and the estimates of Lemma 1, the following assertion is proved:

Theorem 6. Let the endpoint \(a\) be regular and let condition 1) of Theorem 4 be satisfied for the matrix \(\Lambda(x)\). Then, if
\[ \liminf_{x\to b} \left\{ \int \left\|\operatorname{Im}\bigl(\Lambda^{-1}Q\bigr)\right\|\,dt \Big/ \int_a^x\frac{dt}{\delta(t)} \right\} =0, \]
then the spectrum of the operator \(L\) fills the entire real axis.

5°. Consider the operator \(L\), generated in \(L_n^2(0,\infty)\) by a system of even order \((n=2m)\) of a more special form (the so-called Hamiltonian system):

\[ l\bar y=I\bar y' + Q(x)\bar y,\quad \text{where } I= \left\| \begin{array}{cccc} 0 & & & -1\\ & & \iddots & \\ & -1 & & \\ 1 & & & \\ & \ddots & & \\ 1 & & & 0 \end{array} \right\|. \]

By \(q_i\) denote the diagonal elements of the matrix \(Q(x)\): \(q_i=q_{ii}\) \((1\le i\le n)\). Put \(Q(x)=D(x)+Q_0(x)\), where \(D(x)\) is the matrix on whose main diagonal stand the \(q_i\) \((1\le i\le n)\), and all remaining elements are zeros. Divide the matrix \(IQ_0\) into 4 matrices of size \(m\times m\):

\[ IQ_0= \left\| \begin{array}{cc} P_1 & M\\ K & P_2 \end{array} \right\|. \]

Denote

\[ N= \left\| \begin{array}{cc} P_1 & 0\\ 0 & P_2 \end{array} \right\|. \]

With the aid of Rapoport’s asymptotics for solutions of the system \(l\bar y=\lambda \bar y\) (see \((^1)\), p. 232), the following assertion is proved:

Theorem 7. Suppose that the following conditions are satisfied:

1) For \(m+1\le k\le 2m\), \(q_k(x)\) are monotone and \(q_k(x)\to 0\).

2) For \(1\le k\le m\),

\[ q_k(x)\xrightarrow[x\to\infty]{}\infty, \]

\(q_k'(x)\) and \(q_k''(x)\) preserve their sign for \(x\ge x_0\), and

\[ q_k'=O\left(|q|^\alpha\right)\quad \text{for }0<\alpha<\frac32. \]

3) \(q_i(x)-q_j(x)\ge 0\) for \(1\le i\le j\le 2m\).

4) For \(q_k(x)\), \(1\le k\le m\), there exists such a monotone function \(q(x)\) \(\bigl(q(x)\xrightarrow[x\to\infty]{}\infty\bigr)\) that, for certain constants \(\alpha_k\) and \(\beta_k\),

\[ \alpha_k q(x)\le q_k(x)\le \beta_k q(x)\qquad (x\ge x_0). \]

5) The elements of the matrices

\[ \frac{1}{\sqrt q}\,K(x),\qquad \sqrt q\,M(x)\quad \text{and}\quad N(x) \]

are summable on \([x_0,\infty)\).

Then the half-axis \(\lambda>0\) contains no points of the continuous spectrum of \(L\).

Using Theorem 5, the following assertion is proved.

Theorem 8. Suppose that, in addition to the conditions of Theorem 7, the condition

\[ \liminf_{x\to\infty}\frac1x\ln\int_{x_0}^{x}\sqrt{|q(s)|}\,ds=0 \]

is satisfied.

Then the continuous spectrum of the operator \(L\) fills the half-axis \(\lambda\le 0\), i.e.

\[ C(L)=(-\infty,0]. \]

In conclusion the author expresses deep gratitude to N. P. Kuptsov for posing the problem, and also to V. V. Martynov and Prof. M. A. Naimark for discussing the results.

Saratov State University
named after N. G. Chernyshevsky

Received
31 X 1967

REFERENCES

1. M. A. Naimark, Linear Differential Operators, Moscow, 1954.
2. I. E. Shnol’, UMN, 9, no. 4 (62), 113 (1954).