MATHEMATICS

I. M. GLAZMAN

ON THE NEGATIVE PART OF THE SPECTRUM OF ONE-DIMENSIONAL AND MULTIDIMENSIONAL DIFFERENTIAL OPERATORS ON VECTOR FUNCTIONS

(Presented by Academician S. N. Bernstein, 24 X 1957)

The present note is devoted to a generalization of the theorems of note \((^{1a})\), which supplements the results obtained earlier \((^{1b})\) on the spectrum of one-dimensional and multidimensional differential operators on vector functions.

Let \(\mathscr{L}_2(0,\infty)\) be the Hilbert space of vector functions \(\mathbf{y}(x)=\{y_k\}_{k=1}^m\) \((m<\infty)\) with scalar product

\[ (\mathbf{y},\mathbf{z})=\int_0^\infty \sum_{k=1}^m y_k(x)\overline{z_k(x)}\,dx, \]

and let \(l[\mathbf{y}]\) be a differential operation of the form

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

where \(Q(x)\) is a Hermitian matrix function of order \(m\). Denote the smallest and, respectively, the largest eigenvalue of the matrix \(Q(x)\) by \(\mu(x)\) and \(\nu(x)\). Let \(\widetilde{L}\) denote any self-adjoint extension of the operator with minimal domain of definition generated in \(\mathscr{L}_2(0,\infty)\) by the operation (1). The negative part of any function \(f(x)\) will be denoted by \(f^*(x)\), so that \(f^*(x)=\min\{0,f(x)\}\).

The use of Lemma 1 of note \((^{1a})\), where the functional \(\Phi_\varepsilon[y]\) should be replaced by the functional

\[ \Phi_\varepsilon[\mathbf{y}] = \int_0^\infty \sum_{k=1}^m \left|y_k^{(n)}(x)\right|^2\,dx + \int_0^\infty \sum_{j,k=1}^m Q_{jk}(x)y_j(x)\overline{y_k(x)}\,dx + \varepsilon\int_0^\infty \sum_{k=1}^m |y_k(x)|^2\,dx, \]

leads to the following results.

Theorem 1. If for every \(\delta>0\) the inequality

\[ \int_{M_\delta} |\mu^*(x)|\,dx<\infty, \]

holds, where \(M_\delta\) is the set of values \(x\) for which \(|\mu^*(x)|\ge \delta\), then the negative part of the spectrum of the operator \(\widetilde{L}\) is bounded below and discrete.

Putting, further,

\[ \alpha_n=\frac{(2n-1)!!}{2^n},\qquad A_n=(2n-1)^{-1/2}\left[\sum_{k=1}^{n}\frac{(-1)^{k-1}C_{n-1}^{k-1}}{2n-k}\right]^{-1}(n-1)!, \]

\[ B_n^2=\frac{n(4n^2-1)}{3\cdot 4^{\,n-1}} \sum_{k=1}^{n}\frac{1}{2k-1} \sum_{k=0}^{2n-2}\frac{(-1)^k C_{2n-2}^{k}}{4n-3-k} \left[\sum_{k=1}^{n}\frac{(-1)^{k-1}C_{n-1}^{k-1}}{2n-k}\right]^{-2}, \]

we note the following two theorems.

Theorem 2. The negative part of the spectrum of the operator \(\widetilde L\) consists of a finite number of eigenvalues if one of the following conditions is satisfied:

\(1^\circ.\) \(\mu(x)\ge -\alpha_n^2 x^{-2n}\) for large \(x\).

\(2^\circ.\) For every \(\delta>0\)

\[ \int_{M_\delta} x^{2n-1}\,|\mu^*(x)|\,dx<\infty, \]

where \(M_\delta\) is the set of values \(x\) for which

\[ |\mu^*(x)|\ge(\alpha_n^2-\delta)x^{-2n}. \]

\(3^\circ.\) For some \(p\ge 1\)

\[ \int_{0}^{\infty} x^{2np-1}\,|\mu^*(x)|^p\,dx<\infty . \]

Theorem 3. The negative part of the spectrum of the operator \(\widetilde L\) is an infinite set if one of the following conditions is satisfied:

\(1^\circ.\) For some \(\delta>0\) and large \(x\)

\[ \nu(x)<-(\alpha_n^2+\delta)x^{-2n}. \]

\(2^\circ.\) \(\nu(x)\le 0\) for large values of \(x\) and

\[ \liminf_{\rho\to\infty}\rho^{2n-1}\int_{\rho}^{\infty}|\nu(x)|\,dx>A_n^2 . \]

\(3^\circ.\) \(\nu(x)\le -\alpha_n^2 x^{-2n}\) for large \(x\) and

\[ \liminf_{\rho\to\infty}\ln\rho\int_{\rho}^{\infty}x^{2n-1}\left|\nu(x)+\alpha_n^2x^{-2n}\right|\,dx>B_n^2 . \]

\(4^\circ.\)

\[ \int_{0}^{\infty}\nu(x)\,dx=-\infty . \]

In conditions \(2^\circ\) and \(3^\circ\), one may replace \(\liminf_{\rho\to\infty}\) by \(\lim_{\rho_k\to\infty}\).

Theorems 1–3 are connected with the oscillatory properties of the system of differential equations

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

which, for \(n=1\), were studied by Sternberg \((^2)\). Conditions \(1^\circ\) of Theorem 2 and \(1^\circ\) of Theorem 3 give a generalization of the well-known Kneser theorem on oscillation.

solutions of a second-order differential equation. For \(n=1\) one can obtain the following refinement of condition \(1^\circ\) of Theorem 3, which for the case of a second-order differential equation was given by Hill \((^3)\) (see also \((^4)\)).

Theorem 4. If, for some \(\delta>0\) and some natural number \(r\), for all sufficiently large values of \(x\) the inequality
\[ \nu(x)<-\frac{1}{4x^2}-\frac{1}{4x^2\ln^2 x}-\cdots-\frac{1+\delta}{4x^2\ln^2 x\ldots \ln_r^2 x}, \]
holds, where \(\ln_k x=\ln\ln_{k-1}x\), then the negative part of the spectrum of the operator \(\widetilde L\) consists of a finite number of eigenvalues.

The results presented extend in part to multidimensional differential operations on vector-functions of the form
\[ l[u]=-\Delta u+Q(P)u, \tag{2} \]
where \(P\) is a point of \(n\)-dimensional Euclidean space \(\mathscr E\); \(Q(P)\) is a Hermitian matrix-function of order \(m\), defined on all of \(\mathscr E\).

The operation (2) generates in the Hilbert space \(\mathfrak L_2(\mathscr E)\) of vector-functions \(\mathbf u(P)=\{u_k(P)\}_{k=1}^{m}\), with scalar product
\[ (\mathbf u,\mathbf v)=\int_{\mathscr E}\sum_{k=1}^{m}u_k(P)\overline{v_k(P)}\,d\omega_P, \]
a certain differential operator \(L\) with minimal domain of definition.

Let \(\mu(P)\) be the least eigenvalue of the matrix \(Q(P)\), and let
\(\mu^*(P)=\min\{0,\mu(P)\}\). We give, for example, the statement of the theorem corresponding to Theorem 1, and prove it for \(m=1,\ n=2\) (in this case \(\mu(P)=Q(P)\)).

Theorem 5. If, for every \(\delta>0\), the integral
\[ \int_{0}^{\infty}|\mu_\delta^*(P)|\,dr, \]
where
\[ \mu_\delta^*(P)= \begin{cases} \mu^*(P), & |\mu^*(P)|>\delta,\\ 0, & |\mu^*(P)|\leqslant \delta, \end{cases} \]
converges uniformly with respect to the angular coordinates, then:

1) the operator \(L\) with minimal domain of definition (see \((1^{\mathrm B})\)) is self-adjoint;

2) the negative part of the spectrum of the operator \(L\) is bounded below and discrete (i.e. consists of eigenvalues of finite multiplicity with the only possible limit point \(\lambda=0\)).

Proof. Transforming the quadratic functional
\[ \Phi_\varepsilon[u]=\iint_{(\mathscr E)}|\nabla u|^2\,r\,dr\,d\varphi +\iint_{(\mathscr E)}Q|u|^2\,r\,dr\,d\varphi +\varepsilon\iint_{(\mathscr E)}|u|^2\,r\,dr\,d\varphi \]
on any finite function \(u\in D_L\) by the change of variables \(u\sqrt r=v\), we obtain
\[ \Phi_\varepsilon[u]= \iint_{(\mathscr E)}|\nabla v|^2\,dr\,d\varphi +\iint_{(\mathscr E)}\left[Q(r,\varphi)+\frac{1}{4r^2}\right]|v|^2\,dr\,d\varphi +\varepsilon\iint_{(\mathscr E)}|v|^2\,dr\,d\varphi. \]

For an arbitrarily given \(\varepsilon\) \((0<\varepsilon<1)\), choose the number \(N\) so that

\[ \int_N^\infty |Q^*(r,\varphi)|\,dr<\frac{\varepsilon}{4}, \]

and show that the functional

\[ \Phi_\varepsilon[u]=\int_0^{2\pi}d\varphi\int_N^\infty \left\{|\nabla v|^2+\left[Q^*(r,\varphi)+\frac{1}{4r^2}+\varepsilon\right]|v|^2\right\}dr \]

is nonnegative on any finite function \(v\in D_L\) equal to zero in the disk \(r<N\).

From the Cauchy—Bunyakovsky inequality it follows that

\[ \int_0^{2\pi}\int_N^\infty |\nabla v|^2\,dr\,d\varphi \geq \int_0^{2\pi}d\varphi\int_N^\infty \left|\frac{dv}{dr}\right|^2dr \geq \frac14\left\{\int_0^{2\pi}|\hat v(\varphi)|^2\,d\varphi\right\}^2, \]

where the function \(v(r,\varphi)\) is normalized by the condition

\[ \int_0^{2\pi}d\varphi\int_N^\infty |v|^2\,dr=1 \]

and

\[ \hat v(\varphi)=\max_{N<r<\infty}|v(r,\varphi)|. \]

Consider separately two cases:

\[ 1^\circ.\quad \int_0^{2\pi}|\hat v(\varphi)|^2\,d\varphi\leq \varepsilon. \qquad 2^\circ.\quad \int_0^{2\pi}|\hat v(\varphi)|^2\,d\varphi>\varepsilon. \]

In the first case

\[ \Phi_\varepsilon[u]\geq \int_0^{2\pi}\int_N^\infty Q^*(r,\varphi)|v|^2\,dr\,d\varphi +\varepsilon\int_0^{2\pi}\int_N^\infty |v|^2\,dr\,d\varphi \geq \]

\[ \geq -\int_0^{2\pi}|\hat v(\varphi)|^2 \int_N^\infty |Q^*(r,\varphi)|\,dr\,d\varphi+\varepsilon \geq \varepsilon\left(1-\frac{\varepsilon}{4}\right)>0. \]

In the second case

\[ \Phi_\varepsilon[u]\geq \frac14\left\{\int_0^{2\pi}|\hat v(\varphi)|^2\,d\varphi\right\}^2 +\int_0^{2\pi}\int_N^\infty Q^*(r,\varphi)|\hat v(\varphi)|^2\,dr\,d\varphi, \]

so that

\[ \Phi_\varepsilon[u]\geq \frac14\int_0^{2\pi}|\hat v(\varphi)|^2\,d\varphi \left[ \int_0^{2\pi}|\hat v(\varphi)|^2\,d\varphi-\varepsilon \right]>0, \]

and the inequality \(\Phi_\varepsilon[u]\geq 0\) is established.

From this inequality, first of all, follows \((1^\Gamma)\) the boundedness from below of the operator \(L\), and hence, by a theorem of A. Ya. Povzner \((^5)\), the self-adjointness of the operator \(L\) follows.

Next, from this same inequality, on the basis of \((1^\Gamma)\) and Lemma 1 \((1^a)\), we conclude that the negative part of the spectrum of the operator \(L\) is discrete. The theorem is proved.

Kharkov Polytechnic Institute
named after V. I. Lenin

Received
24 X 1957

References

\({}^1\) I. M. Glazman, a) DAN, 118, No. 3 (1958); b) Uchen. zap. Kharkov. gos. ped. inst., 18, matem. ser. (1956); c) Tr. Kharkov. politekhn. inst., 5, ser. inzh.-fiz., issue 1 (1955); d) Matem. sborn., 35 (77), 2 (1954).
\({}^2\) R. Sternberg, Duke Math. J., 19, 311 (1952).
\({}^3\) E. Hille, Trans. Am. Math. Soc., 64, 234 (1948).
\({}^4\) R. Bellman, Theory of Stability, IL, 1954.
\({}^5\) A. Ya. Povzner, Matem. sborn., 32 (74), 1 (1953).