UDC 517.544

MATHEMATICS

V. N. TULOVSKII

ASYMPTOTIC DISTRIBUTION OF EIGENVALUES FOR DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS

(Presented by Academician A. A. Dorodnitsyn, 27 IV 1970)

Let \(G\) be a bounded domain in \(R^n\). Let \(P(D)\) be a symmetric differential operator with constant coefficients; here \(D_j=-i\partial/\partial x_j\).

Suppose that the operator \(P(D)\) satisfies the following conditions:

\[ \lim_{|\xi|\to\infty} P(\xi)=+\infty, \tag{1} \]

\[ \lim_{t\to\infty}\frac{S(P(\xi)=t)}{\mu(P(\xi)<t)}=0, \tag{2} \]

where \(S(P(\xi)=t)\) is the area of the surface \(P(\xi)=t\); \(\mu(P(\xi)<t)\) is the measure of the set of those \(\xi\in R^n\) for which \(P(\xi)<t\).

Denote by \(P_0\) the operator \(P(D)\) on functions from \(C_0^\infty(G)\). Then \(P_0\) is symmetric and, by virtue of condition (1), bounded from below. Let \(\widetilde P\) be its self-adjoint Friedrichs extension. In this paper the asymptotic behavior of the eigenvalues of the operator \(\widetilde P\) is studied. Previously this result was known only for elliptic operators. For the case of the Laplace operator and some other second-order operators it was obtained by Courant by the variational method. In the present paper, instead of Courant’s variational principle, the following theorem is used \((^1)\):

Theorem 1. Let \(A\) be a self-adjoint operator bounded from below. Then the number of eigenvalues of the operator \(A\) that are less than \(t\) is equal to the maximal dimension of linear manifolds \(D\) such that

\[ D\subseteq D(A),\qquad (Au,u)<t(u,u),\qquad u\in D. \]

Since \(\widetilde P\) is the Friedrichs extension, in the present case the problem reduces to estimating the maximal dimension of linear manifolds \(D\) such that

\[ D\subseteq C_0^\infty(G),\qquad (P(D)u,u)<t(u,u),\qquad u\in D. \]

1. Estimate of the dimension of manifolds \(D\) in the case of a cube

Let

\[ Q_n=(x\in R^n;\ 0\le x_i\le 2\pi;\ i=1,2,\ldots,n). \]

Lemma 1. Let \(D\subseteq C_0^\infty(Q_n)\) be a manifold of finite functions such that

\[ (P(D)u,u)<t(u,u),\qquad u\in D. \]

Then

\[ \dim D\le \mu(P(\xi)<t). \tag{3} \]

Proof. Denote by \(Z^n\) the set of tuples \(k=(k_1,\ldots,k_n)\) of integers, and let \(\beta=(\beta_1,\ldots,\beta_n)\in I^n\), where \(I^n\) is the unit cube in \(R^n\). The set of functions \(e^{i\langle k+\beta,x\rangle}\), \(k\in Z^n\), for any fixed \(\beta\) forms an orthonormal basis in \(L_2(Q_n)\). Construct a self-adjoint extension of the operator \(P_0\) in such a way that these functions are its

eigenfunctions with eigenvalues \(P(k+\beta)\). Since \(C_0^\infty(Q_n)\) will belong to the domain of definition of this self-adjoint extension, by Theorem 1,

\[ \dim D \leq N(k+\beta;\, P(k+\beta)<t), \]

where \(N(k+\beta;\, P(k+\beta)<t)\) is the number of points \(k\in Z^n\) such that \(P(k+\beta)<t\). Integrating this inequality over \(\beta\in I^n\), we obtain

\[ \dim D \leq \mu(P(\xi)<t). \]

Lemma 2. The following relation holds

\[ N(k+\beta;\, P(k+\beta)<t)=\mu(P(\xi)<t)+O(S(P(\xi)=t)). \tag{4} \]

Proof. Denote the function \(N(k+\beta;\, P(k+\beta)<t)\) by \(N(\beta,t)\); the function \(N(\beta,t)\) satisfies the relation

\[ \int_{I^n} N(\beta,t)\,d\beta=\mu(P(\xi)<t). \tag{5} \]

Let \(\beta_1,\beta_2\in I^n\). Compare \(N(\beta_1,t)\), \(N(\beta_2,t)\). Consider a partition of the space \(R^n\) into cubes with vertices at the points \(k+\beta_1\). Let \(R(\beta_1,t)\) be the number of cubes lying entirely in the set \((\xi;\,P(\xi)<t)\), and let \(T(\beta_1,t)\) be the number of cubes intersecting the surface \(P(\xi)=t\).

In each cube lying entirely in the set \((\xi;\,P(\xi)<t)\) there will be a point of the form \(k+\beta_2\). Therefore \(N(\beta_2,t)\geq R(\beta_1,t)\), while

\[ |N(\beta_1,t)-R(\beta_1,t)|\leq C T(\beta_1,t), \]

whence we obtain

\[ N(\beta_2,t)\geq N(\beta_1,t)-C T(\beta_1,t). \]

For large \(t\) the surface \(P(\xi)=t\) is a connected surface without singular points, since the set \((\xi;\operatorname{grad} P(\xi)=0)\) consists of a finite number of connected pieces (2) and, by condition (1), they are all bounded in \(R^n\). Then the number of cubes intersecting the surface \(P(\xi)=t\) is estimated in terms of the area of this surface:

\[ T(\beta_1,t)\leq C_1 S(P(\xi)=t). \]

Hence we obtain

\[ N(\beta_2,t)\geq N(\beta_1,t)-C_2 S(P(\xi)=t). \]

Interchanging \(\beta_1\) and \(\beta_2\), we obtain the estimate

\[ |N(\beta_2,t)-N(\beta_1,t)|\leq C_3 S(P(\xi)=t). \]

From this estimate and from (5) we obtain

\[ N(\beta,t)=\mu(P(\xi)<t)+O(S(P(\xi)=t)). \]

Lemma 3. The maximal dimension of subspaces \(D\subseteq C_0^\infty(Q_n)\) such that

\[ (P(D)u,u)<t(u,u),\qquad u\in D, \]

is

\[ \dim D=\mu(P(\xi)<t)+O(S(P(\xi)=t)). \tag{6} \]

Proof. The estimate of the dimension of the subspace \(D\) from above is given in Lemma 1. To prove the lemma it remains to estimate the dimension of \(D\) from below.

Consider the points with integer coordinates lying in the set \((\xi;\,P(\xi)<t)\). Take all exponentials of the form

\[ e^{i\langle k,x\rangle},\quad P(k)<t \]

and span the linear space \(L(t)\) over them. Then

\[ \dim L(t)=\mu(P(\xi)<t)+O(S(P(\xi)=t)), \]

\[ (P(D)u,u)<t(u,u),\qquad u\in L(t). \tag{7} \]

Now in \(L(t)\) we construct a subspace \(L_0(t)\) consisting of functions having a zero of order \(m\) on the boundary \(Q_n\), where \(m\) is the degree of the operator \(P(D)\). Such functions belong to the domain of definition of the operator \(\hat P\). To construct the space \(L_0(t)\), take an arbitrary point \(k_0\) such that \(P(k_0)<t\). Through the point \(k_0\) draw a straight line parallel to the \(x_1\)-axis and take all integer points \(k\) on this line satisfying the condition \(P(k)<t\). For each such point \(k\) consider the exponential \(e^{i(k,x)}\) and span the linear space \(L_1(k_0)\) over them. In the space \(L_1(k_0)\) choose a subspace \(L_1^0(k_0)\) such that

\[ D_x^\alpha u\big|_{x_1=0,\;x_1=2\pi}=0,\qquad |\alpha|\le m,\qquad u\in L_1^0(k_0). \]

The dimension of this space will be

\[ \dim L_1^0(k_0)\ge \dim L_1(k_0)-2(m+1). \]

Now take all such lines and, for each of them, construct an analogous space. The number of such lines is \(O(S(P(\xi)=t))\). Taking the sum of all these spaces, we obtain a space \(L_1(t)\) such that

\[ D_x^\alpha u\big|_{x_1=0,\;x_1=2\pi}=0,\qquad |\alpha|\le m,\qquad u\in L_1(t), \]

\[ \dim L_1(t)=\mu(P(\xi)<t)+O(S(P(\xi)=t)). \]

Construct the spaces \(L_2(t),\ldots,L_n(t)\) corresponding to the variables \(x_2,\ldots,x_n\). Put

\[ L_0(t)=L_1(t)\cap L_2(t)\cap\cdots\cap L_n(t). \]

Then

\[ D_x^\alpha u\big|_{\partial Q_n}=0,\qquad |\alpha|\le m,\qquad u\in L_0(t), \]

\[ \dim L_0(t)=\mu(P(\xi)<t)+O(S(P(\xi)=t)). \]

This proves the lemma. In the case of a cube \(Q\) with side of arbitrary length, by a change of variables we obtain the formula

\[ \dim \mathcal D=(2\pi)^{-n}\mu(Q)\mu(P(\xi)<t)+O(S(\partial Q)\cdot S(P(\xi)=t)). \]

2. Asymptotic distribution in the case of an arbitrary domain \(G\). Let \(G\) be a bounded domain in \(R^n\). Denote by \(N(t)\) the number of eigenvalues of the operator \(\hat P\) less than \(t\). We estimate \(N(t)\) from below. Construct pairwise nonintersecting cubes \(Q^j\) contained in \(G\), and for each cube construct a manifold \(L_0^j(t)\) such that

\[ L_0^j(t)\subseteq C_0^\infty(Q^j);\qquad (P(D)u,u)<t(u,u),\qquad u\in L_0^j(t), \]

\[ \dim L_0^j(t)=(2\pi)^{-n}\mu(Q^j)\mu(P(\xi)<t)+O(S(\partial Q^j)S(P(\xi)=t)). \]

The sum of all these manifolds gives a manifold \(L_0(t)\subseteq C_0^\infty(G)\), on which

\[ (P(D)u,u)<t(u,u),\qquad u\in L_0(t). \tag{8} \]

From Theorem 1 it follows that

\[ N(t)\ge \dim L_0(t)=\sum_j \dim L_0^j(t)\ge \]

\[ \ge (2\pi)^{-n}\mu(P(\xi)<t)\sum_j \mu(Q^j)-C S(P(\xi)=t)\sum_j S(\partial Q^j). \tag{9} \]

From (9), taking (2) into account, we obtain that

\[ \liminf_{t\to\infty} \frac{N(t)}{(2\pi)^{-n}\mu(G)\mu(P(\xi)<t)} \ge \frac{\sum_j \mu(Q^j)}{\mu(G)} \]

and hence

\[ \liminf_{t\to\infty} \frac{N(t)}{(2\pi)^{-n}\mu(G)\mu(P(\xi)<t)} \ge 1. \tag{10} \]

Now let us estimate \(N(t)\) from above. Take a cube \(Q \supseteq G\) and construct cubes \(Q^j\) so that

\[ Q^j \subseteq Q\setminus G,\qquad Q^j\cap Q^i=\varnothing,\quad i\ne j . \]

In each \(Q^j\) construct a manifold \(L_0^j(t)\) such that

\[ L_0^j(t)\subseteq C_0^\infty(Q^j),\qquad (P(D)u,u)<t(u,u),\qquad u\in L_0^j(t), \]

\[ \dim L_0^j(t)=(2\pi)^{-n}\mu(Q^j)\mu(P(\xi)<t) +O\bigl(S(\partial Q^j)S(P(\xi)=t)\bigr). \]

Let now \(D\subseteq C_0^\infty(G)\)

\[ (P(D)u,u)<t(u,u),\qquad u\in D;\qquad \dim D=N(t). \]

Take the sum of the manifolds \(D\) and \(L_0^j(t)\). On this sum the estimate

\[ (P(D)u,u)<t(u,u) \]

holds. From Lemma 1 we have

\[ N(t)+\sum_j \dim L_0^j(t) \le (2\pi)^{-n}\mu(Q)\mu(P(\xi)<t) \]

or

\[ N(t)\le (2\pi)^{-n}\mu(P(\xi)<t) \left(\mu(Q)-\sum_j \mu(Q^j)\right) + C S(P(\xi)=t)\sum_j S(\partial Q^j). \tag{11} \]

From (11) we obtain

\[ \limsup_{t\to\infty} \frac{N(t)}{(2\pi)^{-n}\mu(P(\xi)<t)\mu(G)} \le \frac{\mu(Q)-\sum_j \mu(Q^j)}{\mu(G)} \]

and hence, when \(\mu(\partial G)=0\),

\[ \limsup_{t\to\infty} \frac{N(t)}{(2\pi)^{-n}\mu(P(\xi)<t)\mu(G)} \le 1. \tag{12} \]

From (10), (12) it follows:

Theorem 2. If conditions (1), (2) are satisfied and \(\mu(\partial G)=0\), then the function \(N(t)\) has the asymptotic

\[ N(t)\sim (2\pi)^{-n}\mu(G)\mu(P(\xi)<t). \]

The author expresses gratitude to Prof. B. M. Levitan for posing the problem and to Prof. A. G. Kostyuchenko for attention to the work.

Moscow State University
named after M. V. Lomonosov

Received
22 IV 1970

REFERENCES

¹ I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, 1963, p. 31. ² Properties of Differentiable Mappings, collection of articles, 1968, p. 166.