CONDITIONS FOR DISCRETENESS AND CONTINUITY OF THE SPECTRUM IN THE CASE OF A SELF-ADJOINT SYSTEM OF DIFFERENTIAL EQUATIONS OF EVEN ORDER

V. V. MARTYNOV

In recent years a number of profound results have been obtained in the study of the nature of the spectrum of singular differential operators. Among them belong, first of all, the theorems of A. M. Molchanov, I. M. Glazman, I. S. Kats and M. G. Krein, M. Sh. Birman, and B. S. Pavlov [1]. The present article is mainly devoted to the generalization of effective criteria to systems of ordinary differential equations.

We consider self-adjoint operators generated by the system

\[ (-1)^n y^{(2n)}+Q(x)y=\lambda y,\quad 0\le x<+\infty, \tag{1} \]

where \(Q(x)\) is a Hermitian matrix. To operators of the form (1) there carry over the criteria of M. Sh. Birman [3] for stable discreteness and semiboundedness, finiteness and infiniteness of the negative spectrum, and coincidence of the continuous spectrum with the positive semi-axis. All these criteria are formulated in terms of the eigenvalues of an integrated matrix.

The well-known criterion of A. M. Molchanov for discreteness of the spectrum does not carry over to vector-functions in these terms. Moreover, as Theorem 2 shows, the “natural” form of the criterion—the condition of A. M. Molchanov on the least eigenvalue \(\mu Q(x)\) of the matrix \(Q(x)\)—is also not realized for system (1): even if \(\mu Q(x)\equiv 0\), the operator may nevertheless have a discrete spectrum due to strong rotation of the eigenvectors of the matrix \(Q(x)\) and due to growth of its greatest eigenvalue.

It is interesting that the same problem for the equation

\[ (-1)^n y^{(2n)}=\lambda Q(x)y,\quad 0\le x<+\infty, \tag{2} \]

where the Hermitian matrix \(Q(x)>0\) [5], is solved positively: equation (2) has a discrete spectrum if and only if

\[ \lim_{x\to+\infty} x^{2n-1}\cdot\left\|\int_x^\infty Q(t)\,dt\right\|=0. \tag{3} \]

As a rule, we use the notation and terminology from the monograph [1] of I. M. Glazman. Thus, \(\|A\|\), \(\mu A\), and \(\nu A\) denote respectively the norm, the least eigenvalue, and the greatest eigenvalue of a Hermitian matrix \(A\) in a finite-dimensional unitary space \(E_r\). Everywhere below \(|A|\) is the arithmetic square root of \(A^2\), and

\[ A^-=\frac12(A-|A|)\le 0 \]

denotes the negative part of the matrix \(A\).

§ 1. DISCRETENESS OF THE SPECTRUM OF A SECOND-ORDER OPERATION ON VECTOR FUNCTIONS

Let \(L\) be generated by the differential expression

\[ ly=-y''+Q(x)y \tag{4} \]

in the space \(L_2(-\infty,+\infty)\) of vector functions \(y(x)=(y_1,\ldots,y_k)\) with scalar product

\[ [a,b]=\int_{-\infty}^{\infty}(a,b)\,dx =\int_{-\infty}^{\infty}\sum_{i=1}^{k} a_i(x)b_i(x)\,dx . \]

We assume that the continuous potential \(Q(x)\geq 0\) everywhere on the axis. Then the operator \(L\) is nonnegative and, consequently, self-adjoint.

Let us first consider the special case when \(Q(x)\) is a diagonal matrix. Then \(L\) is the direct sum of \(k\) copies of scalar operators, to each of which A. M. Molchanov’s theorem is applicable. Consequently, for an operator \(L\) with diagonal potential the spectrum is discrete if and only if each diagonal element of the matrix satisfies A. M. Molchanov’s condition, i.e., for any fixed \(\omega>0\),

\[ \lim_{|x|\to\infty}\int_x^{x+\omega} q_{ii}(t)\,dt=+\infty . \tag{5} \]

Since for a diagonal matrix \(\mu Q(x)=\min_i q_{ii}(x)\), it follows from the condition

\[ \lim_{|x|\to\infty}\int_x^{x+\omega}\mu Q(t)\,dt=+\infty \tag{6} \]

that the spectrum of the operator \(L\) is discrete. Below, in Theorem 1, it will be shown that the fulfillment of condition (6) is sufficient for discreteness of the spectrum also in the case of a nondiagonal matrix. Therefore it is natural to attempt to seek a criterion for discreteness of the spectrum in the general case in the form (6), externally very close to the form of A. M. Molchanov’s scalar criterion. However, even in the class of diagonal matrices one can construct an example\(^*\) of an operator \(L\) with discrete spectrum whose potential is degenerate everywhere, i.e., \(\mu Q(x)\equiv 0\). Take an increasing sequence of positive numbers \(a_l\to+\infty\) for which \((a_{l+1}-a_l)\to 0\). Divide each interval \(\Delta_l=(a_l,a_{l+1})\) into two equal parts \(\Delta_{l,1}\) and \(\Delta_{l,2}\). Define on \(\Delta_l\) two nonnegative continuous functions \(q_{11}(x)\geq 0\) and \(q_{22}(x)\geq 0\) as follows: \(q_{11}(x)=0\) on \(\Delta_{l,2}\), while on \(\Delta_{l,1}\) the function \(q_{11}(x)\) forms a peak of area 1. The function \(q_{22}(x)\), conversely, is zero on \(\Delta_{l,1}\), while on \(\Delta_{l,2}\) its integral is equal to one. An analogous construction is made on the half-axis \((-\infty,0)\). Then for the matrix

\[ Q(x)= \begin{pmatrix} q_{11}, & 0\\ 0, & q_{22} \end{pmatrix} \]

we have \(\mu Q(x)\equiv 0\), although each of the \(q_{ii}(x)\) satisfies A. M. Molchanov’s condition.

\(^*\) This example was communicated to us by B. V. Fedosov.

The following fact deserves attention: condition (5), in the case of a diagonal matrix, admits the equivalent form

\[ \lim_{|x|\to\infty}\mu\int_x^{x+\omega} Q(t)\,dt=+\infty \qquad (\omega\ \text{fixed}). \tag{7} \]

The following theorem shows that A. M. Molchanov’s criterion, in the general case, cannot be carried over to systems in terms of the eigenvalues of the matrix \(Q(x)\) itself or of its integral.

Theorem 1. 1) The condition

\[ \lim_{|x|\to\infty}\int_x^{x+\omega}\mu Q(t)\,dt=+\infty \qquad (\omega\ \text{fixed}) \tag{8} \]

is sufficient for discreteness of the spectrum of the operator (4), but is not necessary.

2) On the other hand, the condition

\[ \lim_{|x|\to\infty}\mu\int_x^{x+\omega} Q(t)\,dt=+\infty \qquad (\omega\ \text{fixed}) \tag{9} \]

is necessary in the presence of a discrete spectrum, but is not sufficient.

The proof of the theorem is based on the following lemma.

Lemma of A. M. Molchanov (localization principle [2], § 2).
The spectrum of the operator (4) is discrete if and only if

\[ \lim_{|x|\to\infty}\inf_{y\in D_0(\omega,x)} \frac{\displaystyle \int_x^{x+\omega}\bigl[\,|y'|^2+(Qy,y)\,\bigr]\,dt} {\displaystyle \int_x^{x+\omega}|y|^2\,dt} =+\infty, \tag{10} \]

where \(D_0(\omega,x)\) denotes the class of vector-functions on the interval \((x,x+\omega)\) satisfying the condition \(y(x)=y(x+\omega)=0\). (One may take the infimum over the class \(\dot D(\omega,x)\) of all piecewise-smooth vector-functions concentrated on the interval \((x,x+\omega)\).)

We now prove the theorem. 1) Since \(Q(x)\geq \mu Q(x)\cdot 1_k\), where \(1_k\) is the identity matrix in \(E_k\), it follows from the discreteness of the spectrum of the operator (4) with potential \(\mu Q(x)1_k\) that the spectrum of the operator (4) with potential \(Q(x)\) is discrete.

A counterexample may be furnished by an example of B. V. Fedosov, and under the condition \(\mu Q(x)<\nu Q(x)\)—by Theorem 2, proved below.

2) Since the class \(D(\omega,x)\) is admissible in the lemma, putting in (10) \(y(t)=\xi\), where \(\xi\) is an arbitrary constant vector from \(E_k\), we obtain condition (9).

The latter condition, however, is not sufficient for discreteness of the spectrum. Consider the matrix

\[ Q_0(x)= \begin{pmatrix} \cos x, & -\sin x\\ \sin x, & \cos x \end{pmatrix} \cdot \begin{pmatrix} 0, & 0\\ 0, & v(x) \end{pmatrix} \cdot \begin{pmatrix} \cos x, & \sin x\\ -\sin x, & \cos x \end{pmatrix}, \]

where \(\nu(x)\to+\infty\) as \(|x|\to\infty\). The vector-function
\(k_0(x)=\cos x\cdot e_1+\sin x\cdot e_2\) is a normalized eigenvector of this matrix corresponding to the smallest eigenvalue \(\mu Q_0(x)=0\). Putting in (10) \(y(t)=k_0(t)\), we shall see that the operator \(L_0\) with potential \(Q_0(x)\) has nondiscrete spectrum. We now show that the mean value of the generated matrix \(Q_0(x)\) nevertheless increases as \(|x|\to\infty\), i.e.
\[ \lim \mu \int_x^{x+\omega} Q_0(t)\,dt=+\infty . \]
First take the scalar \(\nu(x)\) out of \(Q_0(x)\):
\[ Q_0(x)=\nu(x)\cdot A(x). \]
For the matrix \(A(x)\) we have
\[ \int_x^{x+\omega} A(t)\,dt = 0.5\,\omega\,1_2 - 0.5\int_x^{x+\omega} \begin{pmatrix} \cos 2t, & \sin 2t\\ \sin 2t, & -\cos 2t \end{pmatrix} dt, \]
therefore
\[ \mu\int_\Delta A(t)\,dt = 0.5\,\omega - 0.5\sqrt{ \left(\int_\Delta \sin 2t\,dt\right)^2 + \left(\int_\Delta \cos 2t\,dt\right)^2 } = \]
\[ =0.5(\omega-|\sin\omega|)>0 \]
for any \(\omega>0\). Since
\[ \mu\int_\Delta Q_0(t)\,dt = \mu\int_\Delta \nu(t)A(t)\,dt \ge \left(\mu\int_\Delta A(t)\,dt\right)\min_{t\in\Delta}\nu(t), \]
the theorem is proved.

Remark 1. It is not hard to show that Theorem 1 is valid for the operator
\(Ly=(-1)^n y^{(2n)}+Q(x)y,\ n>1\), where \(Q(x)\) is absolutely integrable on every finite interval and is semibounded below.

Remark 2. Conditions (8) and (9) may coincide in some cases. For example, if \(Q(x)\) varies slowly, i.e.
\(\|Q(x)-Q(t)\|\le C_0\) for all \(|x-t|\le 1\), then discreteness of the spectrum is equivalent to the requirement
\[ \lim_{|x|\to\infty}\mu Q(x)=+\infty . \]

Remark 3. If the spectrum is discrete, then the diagonal elements \(q_{ii}(x)\), and still more \(\nu Q(x)\), satisfy the condition of A. M. Molchanov. This follows from condition (9) and the inequality
\[ \mu\int_\Delta Q(t)\,dt\le \int_\Delta q_{ii}(t)\,dt . \]

Let us consider in more detail the operator (4) in the case when \(Q(x)\ge0\) is a symmetric twice differentiable matrix in two-dimensional Euclidean space. Without loss of generality, one may assume that
\[ \mu Q(x)<\nu Q(x) \tag{11} \]
everywhere on the axis. Indeed, separation of the eigenvalues of a second-order matrix \(Q(x)\) can always be achieved by adding to it a bounded nondiagonal perturbation, which does not affect the discreteness of the spectrum of the operator \(L\).

Under condition (11) we may regard the eigenvector-matrices of \(Q(x)\) as continuous. As in the construction of counterexample 2) in Theorem 1, take the normalized (in \(E_2\)) vector
\[ k(x)=\cos\varphi(x)\cdot e_1+\sin\varphi(x)\cdot e_2, \]
corresponding to the smallest eigenvalue \(\mu Q(x)\), and put

in (10) \(y(t)=k(t)\). Then we obtain that, in the presence of a discrete spectrum, the following requirement is necessarily satisfied:

\[ \lim_{|x|\to\infty}\int_x^{x+\omega}\bigl[\varphi'^2(t)+\mu Q(t)\bigr]\,dt=+\infty \qquad (\omega\ \text{fixed}). \tag{12} \]

Condition (12) immediately attracts attention, since the term

\[ \int_\Delta [\varphi'(t)]^2\,dt \]

is specific to the vector case. In particular, if the rate of rotation of the vector \(k(x)\) (and hence of the basis of eigenvectors of the matrix \(Q(x)\)) is bounded, then condition (8) is equivalent to discreteness of the spectrum. In the case of an increasing rate of rotation, the situation may change radically, as the following shows.

Theorem 2. Suppose that, as \(|x|\to\infty\), one of the following two conditions is satisfied:

\[ 1)\quad [\varphi'^2(x)-\varphi''^2(x)]\to+\infty \quad\text{and}\quad [\nu Q(x)-3\varphi'^2(x)]\to+\infty \]

or

\[ 2)\quad |\varphi'(x)|\to+\infty \quad\text{and}\quad [\nu Q(x)-3\varphi'^2(x)-\varphi''(x)]\to+\infty, \]

where \(\varphi(x)\) is a continuous branch of the angle of rotation of the basis of eigenvectors of the two-dimensional symmetric twice differentiable matrix \(Q(x)\) in Euclidean space, for which \(0\le \mu Q(x)<\nu Q(x)\) for sufficiently large \(|x|\).

Then, independently of the behavior of \(\mu Q(x)\), operator (4) has a discrete spectrum.

Proof. We reduce \(Q(x)\) to diagonal form by an orthogonal transformation \(y=Uz\), where \(z=(m,n)\), and

\[ U(x)= \begin{pmatrix} \cos\varphi(x), & -\sin\varphi(x)\\ \sin\varphi(x), & \cos\varphi(x) \end{pmatrix}. \]

On the basis of A. M. Molchanov’s lemma, discreteness of the spectrum takes place if and only if

\[ \lim_{|x|\to\infty} J(\omega,x)=+\infty, \]

where

\[ J(\omega,x)= \inf_{z\in D_0(\omega,x)} \left\{ \int_x^{x+\omega} \bigl[ m'^2+n'^2+\varphi'^2\cdot(m^2+n^2) +2\varphi'\cdot(mn'-m'n) +\mu Qm^2+\nu Qn^2 \bigr]\,dt \right\} \left\{ \int_x^{x+\omega}[m^2+n^2]\,dt \right\}^{-1}. \tag{13} \]

Let us estimate \(J(\omega,x)\) from below. First integrate by parts:

\[ 2\int_x^{x+\omega}\varphi' mn'\,dt = -2\int_x^{x+\omega}\varphi'' mn\,dt -2\int_x^{x+\omega}\varphi' m'n\,dt \]

(here we have taken into account that \(m(t)\) and \(n(t)\) vanish at the ends of the interval). Now estimate the term
\(-4\varphi' m'n \ge -2|2\varphi'n|\cdot |m'| \ge -4\varphi'^2 n^2-m'^2\). Then

\[ J(\omega,x)\ge \inf_z \frac{ \displaystyle \int_x^{x+\omega} \bigl[ \varphi'^2 m^2-2\varphi'' mn+(\nu Q-3\varphi'^2)n^2 \bigr]\,dt }{ \displaystyle \int_x^{x+\omega}[m^2+n^2]\,dt } = \]

\[ = \inf_z \frac{\displaystyle \int_x^{x+\omega} (Bz,z)\,dt} {\displaystyle \int_x^{x+\omega} |z|^2\,dt} \geq \inf_{t\in (x,x+\omega)} \mu B(t), \]

where

\[ B(t)= \begin{pmatrix} \varphi'^2, & -\varphi''\\ -\varphi'', & \nu Q-3\varphi'^2 \end{pmatrix}. \]

Consequently, if as \(|t|\to\infty\)

\[ f(t)=2\mu B(t)= \left[\nu Q(t)-2\varphi'^2(t) -\sqrt{(\nu Q-4\varphi'^2)^2+4\varphi''^2}\right]\to +\infty, \]

then the spectrum is discrete. To obtain the more readily checkable conditions formulated in the theorem, let us estimate the term \(-2\varphi''mn\) in two ways:
\(-2\varphi''mn\geq -\varphi''^2m^2-n^2\) and
\(-2\varphi''mn\geq -m^2-\varphi''^2n^2\). In the first case

\[ J(\omega,x)\geq \inf_z \frac{\displaystyle \int_x^{x+\omega} \left[m^2(\varphi'^2-\varphi''^2)+n^2(\nu Q-3\varphi'^2-1)\right]\,dt} {\displaystyle \int_x^{x+\omega} [m^2+n^2]\,dt}, \]

in the second case

\[ J(\omega,x)\geq \inf_z \frac{\displaystyle \int_x^{x+\omega} \left[m^2(\varphi'^2-1)+n^2(\nu Q-3\varphi'^2-\varphi''^2)\right]\,dt} {\displaystyle \int_x^{x+\omega} [m^2+n^2]\,dt}. \]

The proof is complete.

Thus, we may conclude that the natural attempt to impose conditions on the eigenvalues does not lead to the corresponding criterion: in the vector case, discreteness of the spectrum of operator (4) is essentially connected with the rotation of the eigenvectors of the potential matrix. Incidentally, in the example of B. V. Fedosov with a diagonal matrix, we observed the phenomenon of rapid jumping of eigenvectors, which is a “discrete” analogue of the situation just considered.

§ 2. BOUNDEDNESS AND DISCRETENESS OF THE NEGATIVE PART OF THE SPECTRUM OF A BINOID VECTOR OPERATION. CONNECTION WITH THE CONTINUOUS SPECTRUM

Lemma of I. M. Glazman ([1], pp. 74, 60). In order that the negative part of the spectrum of any self-adjoint extension \(\widetilde L\) of the operator

\[ Ly=(-1)^n y^{(2n)}+Q(x)y \]

in \(L_2(0,\infty)\), with a Hermitian matrix \(Q(x)\) in \(E_k\), absolutely integrable on every finite interval of the half-line, be bounded below and discrete, it is necessary and sufficient that for every \(\varepsilon>0\) there exist an \(N_\varepsilon>0\) such that the quadratic functional

\[ \int_{N_\varepsilon}^{\infty} \left[|y^{(n)}|^2+(Qy,y)+\varepsilon |y|^2\right]dx \]

be nonnegative on every \(N_\varepsilon\)-finite

\(y(x)\) from the domain of definition \(D(L)\) of the operator \(L\) (i.e., the compact support of the vector-function \(y(x)\) is located to the right of the point \(N_\varepsilon\)).

It is said ([1], p. 131) that a certain property of the spectrum, independent of the choice of the self-adjoint extension \(\tilde L\), is \(h\)-stable if it is preserved when the matrix \(Q(x)\) is replaced by the matrix \(\dfrac{1}{h}Q(x)\), whatever \(h>0\) may be. This replacement of the potential is obviously equivalent to passing to the operation

\[ L_h y=(-1)^n h y^{(2n)}+Q(x)y. \]

Theorem 3. In order that the negative part of the spectrum of the operator
\(\tilde L y=(-1)^n y^{(2n)}+Q(x)y\) be \(h\)-stably bounded below and discrete, it is sufficient, and for \(Q(x)\leqslant 0\) also necessary, that the condition

\[ \lim_{x\to+\infty}\left\|\int_x^{x+\omega} Q^{-}(t)\,dt\right\|=0 \tag{14} \]

hold for some (and hence for every) \(\omega>0\).

Here the Euclidean norm of the integrated matrix is involved. We also recall that \(Q^{-}(t)\leqslant 0\) denotes the negative part of the matrix \(Q(t)\).

The proof of necessity in the case \(Q(x)\leqslant 0\) is by contradiction. Suppose (14) is not fulfilled, i.e., for some sequence \(x_p\to+\infty\), with some \(\omega_0>0\), \(\rho_0>0\),

\[ \left\|\int_{x_p}^{x_p+\omega_0} Q(t)\,dt\right\|>\rho_0. \]

Consequently, there is a sequence of constant vectors \(\xi_p\in E_k\), with \(|\xi_p|=1\), such that

\[ \int_{x_p}^{x_p+\omega_0} (Q(t)\xi_p,\xi_p)\,dt<-\rho_0. \]

By the lemma we must show that for every \(N>0\) there exists an \(N\)-finite and “trial” function \(y_N(x)\), i.e., such that

\[ J_N=\int_N^\infty\left[\left|y_N^{(n)}\right|^2+\frac{1}{h_0}(Qy_N,y_N)+\varepsilon_0|y_N|^2\right]dx<0 \]

for some \(h_0>0\), \(\varepsilon_0>0\). We take the sequence of real “trial” functions constructed in the scalar case (see [1], p. 140): \(\varphi_p(x)=\varphi(x-x_p)\), where the smooth function

\[ \varphi(x)= \begin{cases} 1, & \text{for } 0\leqslant x\leqslant \omega_0,\\ 0, & \text{for } x\leqslant -\delta,\ x\geqslant \omega_0+\delta, \end{cases} \]

and define \(y_p(x)=\varphi_p(x)\xi_p\). Then

\[ \int_0^\infty\left[\left|y_p^{(n)}\right|^2+|y_p|^2\right]dx = \int_0^\infty\left[\varphi_p^{(n)2}+\varphi_p^2\right]dx=C, \quad \text{and } \quad J_p=C+ \]

\[ +\int \frac{(Qy_p,y_p)}{h_0}\,dx < C+\frac{1}{h_0}\int_{x_p}^{x_p+\omega_0}(Q(x)\xi_p,\xi_p)<0 \]

for \(\varepsilon_0=1\), \(h_0<\rho_0/C\). Let us prove sufficiency.

Suppose we are now given: for every \(\eta>0\) there exists \(N=N_\eta\) such that

\[ \left\|\int_x^{x+1} Q^-(t)\,dt\right\|<\eta \]

as soon as \(x\geqslant N\). Divide the half-axis \((N,\infty)\) into intervals
\(\Omega_p=(a_p,a_p+1)\). Put

\[ \Phi(x)=\Phi_p(x)=-\int_x^{a_p+1} Q^-(t)\,dt =\int_x^{a_p+1}|Q^-(t)|\,dt\geqslant 0. \]

Since everywhere on \(\Omega_p\) \(\Phi'(x)\leqslant 0\), i.e., for every \(\xi\in E_k\)

\[ (\Phi'(x)\xi,\xi)=\frac{d}{dx}(\Phi(x)\xi,\xi)\leqslant 0, \]

we have \((\Phi(x)\xi,\xi)\leqslant(\Phi(a_p)\xi,\xi)\) for every \(x\in\Omega_p\). But then

\[ \|\Phi(x)\|=\nu\Phi(x)\leqslant \nu\Phi(a_p) =\left\|\int_{a_p}^{a_p+1}Q^-(t)\,dt\right\| \]

for every \(x\in\Omega_p\).

Let us estimate from above \(\displaystyle \int_{\Omega_p}(|Q^-|y,y)\,dx\). First we integrate the quadratic form by parts:

\[ \int_{\Omega_p}(|Q^-|y,y)\,dx =-\int_{\Omega_p}(\Phi'y,y)\,dx =-(\Phi y,y)\big|_{\Omega_p} +2\operatorname{Re}\int_{\Omega_p}(\Phi y,y')\,dx\leqslant \]

\[ \leqslant \|\Phi(a_p)\|\,|y(a_p)|^2 +2\int_{\Omega_p}\|\Phi(x)\|\,|y|\,|y'|\,dx \leqslant \|\Phi(a_p)\|\,|y(a_p)|^2+ \]

\[ +2\|\Phi(a_p)\|\int_{\Omega_p}|y|\,|y'|\,dx \leqslant 3\|\Phi(a_p)\|\int_{\Omega_p}\bigl[|y'|^2+|y|^2\bigr]\,dx. \]

In the last estimate we used the elementary inequality

\[ |y(a_p)|^2\leqslant 2\int_{\Omega_p}\bigl[|y'|^2+|y|^2\bigr]\,dx. \]

Thus, we have proved the basic inequality:

\[ \int_{\Omega_p}(|Q^-|y,y)\,dx \leqslant 3\left\|\int_{\Omega_p}Q^-(t)\,dt\right\| \int_{\Omega_p}\bigl[|y'|^2+|y|^2\bigr]\,dt \leqslant \]

\[ \leqslant 3\eta\int_{\Omega_p}\bigl[|y'|^2+|y|^2\bigr]\,dt. \tag{15} \]

Summing over all \(\Omega_p\), we obtain the estimate

\[ \int_N^\infty (|Q^-|y,y)\,dx\leqslant \]

\[ \leqslant 3\eta \int_N^\infty \left[\, |y'|^2 + |y|^2 \,\right] dx. \]
Now let us use the finiteness of \(y(x)\): as in the scalar case, the inequality
\[ \int |v'|^2 dx \leqslant C_n \int \left[\, |v^{(n)}|^2 + |v|^2 \,\right] dx \]
is valid for any smooth finite vector function \(v(x)\). Therefore
\[ \int_N^\infty (|Q^-|y,y)dx \leqslant 3\eta(1+C_n)\int_N^\infty \left[\, |y^{(n)}|^2+|y|^2 \,\right]dx. \tag{16} \]

Let now \(h>0\) and \(\varepsilon>0\) be given. By the condition there will be found
\(N_\eta=N(h,\varepsilon)\), with
\[ \eta=\eta(h,\varepsilon)=\min \{h/3(1+C_n);\ \varepsilon h/3(1+C_n)\}, \]
such that
\[ \left\| \int_x^{x+1} Q^-(t)\,dt \right\|<\eta(h,\varepsilon) \]
for every \(x\geqslant N(h,\varepsilon)\). Then for any \(N(h,\varepsilon)\)-finite \(y(x)\) we have
\[ \int_N^\infty \left[\, |y^{(n)}|^2+\frac1h(Qy,y)+\varepsilon |y|^2 \,\right]dx \geqslant \int_N^\infty \left[\, |y^{(n)}|^2+\frac1h(Q^-y,y)+ \varepsilon |y|^2 \,\right]dx>0 \]
by (16). Consequently, the theorem is proved in both directions.

Remark. The following assertion follows from Hausdorff’s theorem on finite \(\varepsilon\)-nets: in order that a set \(U\) of vector functions \(y(x)\) be compact in the metric of the integral
\[ \int_0^\infty (Ty,y)\,dx \]
(here the matrix “weight” \(T(x)\geqslant 0\)), it is necessary and sufficient that the set \(U\) be compact in the metric
\[ \int_0^a (Ty,y)\,dx \]
for every \(a>0\), and
\[ \lim \left[\sup_{y\in U}\int_a^\infty (Ty,y)\,dx\right]=0 \quad \text{as } a\to +\infty . \tag{17} \]

This fact makes it possible to reformulate Theorem 3 (for \(Q(x)\leqslant 0\)) in terms of compactness ([1], p. 171, Lemma 2).

In order that a set of finite vector functions, bounded in the metric
\[ \int_0^\infty \left[\, |y^{(n)}|^2+|y|^2 \,\right]dx, \]
be compact in the metric
\[ \int_0^\infty (Ty,y)\,dx, \]
it is necessary and sufficient that the “weight” \(T(x)\geqslant 0\) satisfy the condition
\[ \lim_{x\to+\infty} \left\| \int_x^{x+1} T(t)\,dt \right\|=0. \tag{18} \]

With the aid of the sufficient criterion of this embedding theorem, it is proved ([1], p. 173)

Theorem 4. Let

\[ \lim_{x\to+\infty}\left\|\int_x^{x+1}|Q(t)|\,dt\right\|=0. \tag{19} \]

Then the operator \(\widetilde L_h y=(-1)^n h\,y^{(2n)}+Q(x)y\), for every \(h>0\), is bounded below, and its continuous spectrum for every \(h>0\) coincides with \([0,\infty)\)*.

Moreover, Theorem 4 is a special case of a more general result.

Theorem 5. Let \(\widetilde L\) denote the self-adjoint operator generated in \(L_2(0,\infty)\) by the quasi-differential expression

\[ l=\sum_{r=0}^n(-1)^{\,n-r}\frac{d^{\,n-r}}{dx^{\,n-r}}P_r(x)\frac{d^{\,n-r}}{dx^{\,n-r}}, \tag{20} \]

where \(P_r(x)\) are continuous Hermitian matrices in the unitary space \(E_k\). If the conditions

\[ 0<\alpha 1_k \leq P_0(x)\leq \beta 1_k<+\infty, \tag{21} \]

\[ \lim_{x\to+\infty}\left\|\int_x^{x+1}|P_0(t)-1_k|\,dt\right\|=0, \tag{22} \]

\[ \lim_{x\to+\infty}\left\|\int_x^{x+1}|P_r(t)|\,dt\right\|=0 \quad (r=1,2,\ldots,n), \tag{23} \]

are satisfied, then the operator (20) is bounded below and the continuous part of its spectrum coincides with the half-line \(\lambda\geq 0\).

The proof completely repeats the argument of the scalar variant (see [3], p. 153, or [1], p. 211), with the use of criterion (18) and the inequalities
\(|(P_r\xi,\xi)|\leq (|P_r|\xi,\xi)\) and \(|P_0-1_k|\leq P_0+1_k\).

From Theorems 3, 4 there follows directly

Corollary. In order that the operator \(\widetilde L y=(-1)^n y^{(2n)}+Q(x)y\) be \(h\)-stably bounded below and that its continuous spectrum \(h\)-stably coincide with \([0,\infty)\), it is sufficient, and for \(Q(x)\leq 0\) also necessary, that the condition

\[ \lim_{x\to+\infty}\left\|\int_x^{x+1}|Q(t)|\,dt\right\|=0 \tag{24} \]

be satisfied.

Remark. Analogous problems can be considered on the whole axis, and also for systems of equations with partial derivatives. For example, embedding theorem 2.3 and remark 2.3 to it in [3] (pp. 145, 146) are automatically transferred to vector-functions. We were able to generalize the theorem of M. Sh. Birman and B. S. Pavlov ([4], Theorem 2) only under the condition \(n\geq m\) instead of \(2n>m\), where \(2n\) is the order of the differential operator, and \(m\) is the dimension of the space.

\[ \text{* Recall that } |Q(t)| \text{ denotes the modulus-matrix of the Hermitian matrix } Q(t). \]

§ 3. FINITENESS AND INFINITENESS OF THE NEGATIVE SPECTRUM OF A TWO-TERM VECTOR OPERATION.
A CRITERION FOR DISCRETENESS OF THE SPECTRUM OF A POLAR VECTOR OPERATION

As in the scalar case, between the oscillatory properties of the solutions of the system of differential equations

\[ (-1)^n y^{(2n)}+Q(x)y=\lambda y,\qquad 0\leq x<+\infty \tag{25} \]

and the spectrum of any self-adjoint extension \(\widetilde L\), generated by the operator \(Ly=(-1)^n y^{(2n)}+Qy\) in \(L_2(0,\infty)\), there is a connection, by virtue of which the set of spectral points of the operator \(\widetilde L\) situated to the left of \(\lambda_0\) will be finite or infinite according as, for \(\lambda=\lambda_0\), system (25) is oscillatory or non-oscillatory (definition of I. M. Glazman: the system of equations (25) is called oscillatory if, for every \(t\), there is a solution having to the right of the point \(t\) more than one \(n\)-fold zero. In the contrary case the system is called non-oscillatory ([1], pp. 74, 69)). Therefore the criteria for non-oscillation or oscillation of the system

\[ (-1)^n y^{(2n)}+Q(x)y=0,\qquad 0\leq x<+\infty \tag{26} \]

are simultaneously criteria for the finiteness or, respectively, infiniteness of the set of points of the negative part of the spectrum of the operator \(\widetilde L\). And to obtain the latter criteria we may use the splitting principle.

Lemma of I. M. Glazman ([1], pp. 74, 61). In order that system (26) be non-oscillatory, it is necessary and sufficient that, for some \(N>0\), on all \(N\)-finite vector-functions from \(D(L)\), the functional

\[ \int_N^\infty \bigl[\, |y^{(n)}|^2+(Qy,y)\,\bigr]\,dx \]

be nonnegative.

Theorem 6. If

\[ \limsup_{x\to+\infty} x^{2n-1} \left\| \int_x^\infty Q^{-}(t)\,dt \right\| < \frac{\alpha_n^2}{2n-1}, \tag{27} \]

where

\[ \alpha_n=\frac{(2n-1)!!}{2^n}, \]

then system (26) is non-oscillatory (cf. [1], p. 149).

Proof. Put

\[ \Phi(x)=\int_x^\infty |Q^{-}(t)|\,dt\geq 0. \]

Then \(\Phi'(x)=-|Q^{-}(x)|\leq 0\). Estimate from above

\[ \int_N^\infty (\,|Q^{-}|y,y\,)\,dx. \]

Integrate the quadratic form by parts, regarding \(y(x)\) as \(N\)-finite,

\[ \int_N^\infty (\,|Q^{-}|y,y\,)\,dx = -\int_N^\infty (\Phi' y,y)\,dx = -(\Phi y,y)\Big|_N^\infty + 2\operatorname{Re}\int_N^\infty (\Phi y,y')\,dx \leq \]

\[ \leq 2\int_N^\infty \|\Phi(x)\|\,|y|\,|y'|\,dx = 2\int_N^\infty \|x^{2n-1}\Phi(x)\|\,x^{1-2n}|y|\,|y'|\,dx \leq \]

\[ \leqslant 2 \max_{x\geqslant N}\left[x^{2n-1}\|\Phi(x)\|\right]\int_N^\infty x^{1-2n}|y|\cdot |y'|\,dx \leqslant \frac{2a_n^2}{2n-1}\int_N^\infty \frac{|y|\cdot |y'|}{x^{2n-1}}\,dx \]

by the condition of the theorem. But, as in the scalar case, it is easy to prove the inequality

\[ \int_N^\infty x^{1-2n}|y|\cdot |y'|\,dx \leqslant \frac{2^{\,n-1}}{(2n-3)!!}\, \frac{2^n}{(2n-1)!!} \int_N^\infty |y^{(n)}|^2\,dx . \]

Consequently,

\[ \int_N^\infty (Q^-y,y)\,dx \geqslant -\int_N^\infty |y^{(n)}|^2\,dx . \]

Thus,

\[ \int_N^\infty \left[|y^{(n)}|^2+(Qy,y)\right]\,dx \geqslant \int_N^\infty \left[|y^{(n)}|^2+(Q^-y,y)\right]\,dx \geqslant 0 . \]

The validity of the theorem now follows from the lemma of I. M. Glazman.

Theorem 7. If \(Q(x)\leqslant 0\) and

\[ \limsup_{x\to+\infty} x^{2n-1} \left\|\int_x^\infty Q^-(t)\,dt\right\| > A_n^2, \tag{28} \]

where

\[ A_n^{-1}= \frac{\sqrt{2n-1}}{(n-1)!} \sum_{k=1}^n \frac{(-1)^{k-1}\binom{\,n-1\,}{\,k-1\,}}{2n-k}, \]

then the system of equations (26) is oscillatory (cf. [1], p. 156).

Proof. By the lemma it is sufficient, for any \(N>0\), to construct an \(N\)-finite vector function \(u_N(x)\in D(L)\) that is “trial,” i.e. such that

\[ \int_N^\infty \left[|u_N^{(n)}|^2+(Qu_N,u_N)\right]\,dx<0 . \]

As in the proof of necessity in Theorem 3, define \(u_N(x)=y_N(x)\xi_N\), where \(y_N(x)\) is the “trial” real function constructed in the scalar case, and \(\xi_N\) are constant vectors with \(|\xi_N|=1\) and

\[ \rho^{2n-1}\int_\rho^R (|Q(t)|\xi_N,\xi_N)\,dt > A_n^2+\frac{\alpha}{2}, \]

where \(\rho=\rho_N,\ R=R_N\) (see [1], p. 157).

These two theorems are of independent interest as conditions for the ordinary, non-\(h\)-stable finiteness or infiniteness of the negative part of the spectrum. In addition, conditions (27) and (28) are so close that they give criteria not for ordinary, but for “strong,” i.e. \(h\)-stable ([1], pp. 162–164), nonoscillation and oscillation. We shall formulate them immediately in spectral terms.

Theorem 8. For the \(h\)-stable infiniteness of the negative spectrum of the operator \(\widetilde L y=(-1)^n y^{(2n)}+Q(x)y\), it is necessary, and in the case when \(Q(x)\leqslant 0\), also sufficient, that the limiting relation be satisfied

\[ \lim_{x\to+\infty}\sup x^{2n-1}\left\|\int_x^\infty Q^-(t)\,dt\right\|=+\infty . \tag{29} \]

Theorem 9. For the \(h\)-stability of the negative end of the spectrum of the operator
\[ \widetilde L y=(-1)^n y^{(2n)}+Q(x)y \]
it is sufficient, and for \(Q(x)\leq 0\) also necessary, that
\[ \lim_{x\to+\infty} x^{2n-1}\left\|\int_x^\infty Q^-(t)\,dt\right\|=0 . \tag{30} \]

In conclusion we consider the question of the discreteness of the spectrum of the equation
\[ (-1)^n y^{(2n)}=\lambda R(x)y,\qquad 0\leq x<+\infty, \tag{31} \]
where \(R(x)>0\). Let \(\widetilde L(R)\) be the self-adjoint operator generated in the space with metric
\[ \int_0^\infty (Ra,a)\,dx \]
by the operation \((-1)^n R^{-1}(x)y^{(2n)}\) (the “polar” operation [1], p. 145). With the aid of Theorem 9 we obtain a criterion for the discreteness of the spectrum of equation (31).

Theorem 10. In order that the spectrum of the operator \(\widetilde L(R)\) be discrete, it is necessary and sufficient that
\[ \lim_{x\to+\infty} x^{2n-1}\left\|\int_x^\infty R(t)\,dt\right\|=0 . \tag{32} \]

Proof. Without loss of generality, for equation (31) we introduce the boundary condition
\[ y(0)=y'(0)=\cdots=y^{(n-1)}(0)=0. \]
We note that the number \(M_\alpha\) of eigenvalues of this problem lying to the left of \(\alpha>0\) is equal to the number of eigenvalues of the quadratic form
\[ \int_0^\infty (Ry,y)\,dx \]
lying to the right of \(\alpha^{-1}\) in the space with metric
\[ \int_0^\infty |y^{(n)}|^2\,dx . \]
By a lemma of M. Sh. Birman ([3], pp. 134, 166), the number \(M_\alpha\) coincides with the number of negative eigenvalues of the operator
\[ (-1)^n y^{(2n)}-\alpha Ry,\qquad y(0)=y'(0)=\cdots=y^{(n-1)}(0)=0 \]
in \(L_2(0,\infty)\), i.e., we have arrived at condition (30) for \(Q^-=Q=-R<0\). Theorem 10 is the vector analogue of a theorem of I. S. Kac and M. G. Krein [5], generalized to the case \(n>1\) by M. Sh. Birman.

Thus, we see that if the spectral parameter enters the differential equation multiplicatively (the case of the polar operation), then the discreteness criterion for the spectrum carries over to systems; but if the spectral parameter enters the equation additively (the case of the binomial operation), then the discreteness criterion does not carry over to vector-functions (at least in terms of the eigenvalues of the potential matrix or of its integral).

The simple method used in §§ 2, 3 for extending known scalar integral sufficient and necessary criteria to vector-functions can also be applied in other problems. For example, Theorem 12 ([1], p. 159) and embedding theorems 2, 5 carry over to systems.

2.7, Theorems 4.7, 4.10, and Remark 4.2 from [3]. The theorem of M. Sh. Birman and B. S. Pavlov ([4], Theorem 3) is partially carried over.

In conclusion I express my sincere gratitude to Professor M. A. Naimark and Professor V. B. Lidskii for valuable discussions and advice.

References

1. I. M. Glazman. Direct methods of qualitative spectral analysis of singular differential operators. Fizmatgiz, 1963.
2. A. M. Molchanov. Proceedings of the Moscow Mathematical Society, 2, 1953.
3. M. Sh. Birman. Mathematical Collection, 55 (97): 2, 1961.
4. M. Sh. Birman, B. S. Pavlov. Vestnik of Leningrad State University, No. 1, series Mathematics, Mechanics and Astronomy, issue 1, 1961.
5. I. S. Kats, M. G. Krein. Izvestiya vuzov MVO SSSR, Mathematics, No. 3, 1958.

Received by the editors
April 19, 1965

Moscow Institute of Physics and Technology