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UDC 517.948.35
MATHEMATICS
M. G. GASIMOV
ON THE DISTRIBUTION OF EIGENVALUES OF SELF-ADJOINT ORDINARY DIFFERENTIAL OPERATORS
(Presented by Academician I. N. Vekua on 28 VI 1968)
§ 1. Statement of the problem. Consider the formally symmetric differential expression
\[ l\left(x,\frac{d}{dx}\right)y = (-1)^n y^{(2n)} + \sum_{j=0}^{n-1}(-1)^j\{p_j(x)y^{(j)}(x)\}^{(j)}, \qquad -\infty<x<\infty, \]
and suppose that its minimal extension in \(L_2(-\infty,\infty)\) generates a self-adjoint operator \(L\), bounded below by zero. Let the operator \(L\) have a discrete spectrum \(\lambda_1\leq \lambda_2\leq\cdots\) with corresponding normalized eigenfunctions \(\varphi_1(x),\varphi_2(x),\ldots\). Put
\[ N(\lambda)=\sum_{\lambda_n<\lambda}1. \]
In the present paper we shall find the principal term of the asymptotics of the function \(N(\lambda)\) as \(\lambda\to+\infty\) in the case when it is determined by one or several coefficients of the expression \(l(x,d/dx)\); here, in contrast to previous works, in our considerations the free term \(p_0(x)\) is subordinated to the other coefficients and may even have no effect on the principal term of the asymptotics.
There is a large series of works devoted to the study of the asymptotics of \(N(\lambda)\) for various elliptic operators in \(L_2(\Omega)\), where \(\Omega\) is a finite or infinite domain in \(E_n\) (for a complete bibliography see \((^1)\)).
Up to now the literature has considered the case when the principal term of the asymptotics of \(N(\lambda)\) is determined by the principal part and the free coefficient of the elliptic operator. An exception is the work of A. G. Kostyuchenko \((^2)\), where all coefficients have the same influence on the asymptotics of \(N(\lambda)\). However, in \((^2)\), besides conditions on the coefficients, conditions are also imposed on the roots of the corresponding characteristic equation.
§ 2. Conditions on the coefficients \(p_j(x)\). In what follows we assume that the functions \(p_0(x),\ldots,p_{n-1}(x)\) satisfy the following conditions (by \(c_1,c_2,\ldots\) various positive constants are denoted):
-
\[ c_1(1+|x|)^{\alpha_j}\leq p_j(x)\leq c_2(1+|x|)^{\alpha_j} \]
for large \(x\). -
For \(|u-x|\leq |u|/2\),
\[ |p_j(u)-p_j(x)| \leq C|x-u|(1+|x|)^{\alpha_j-1}. \]
- For \(|x-u|\geq |u|/2\geq 1\),
\[ |p_j(u)-p_j(x)| \leq C|x-u|^{[\alpha_j]+1}(1+|u|)^{\alpha_j-[\alpha_j]-1}. \]
- Each function \(p_j(x)\) has derivatives up to order \(j\), inclusive, and
\[ |p_j^{(k)}(x)| \leq C(1+|x|)^{\alpha_j-k}, \qquad k=1,\ldots,j. \]
- For some positive integer \(k\) the integral
\[ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{dx\,d\xi}{\{l(x,i\xi)+1\}^{2k}} \tag{1} \]
converges.
We note that condition 2 follows from condition 4, while condition 5 imposes certain restrictions on the numbers \(a_0,a_1,\ldots,a_{n-1}\). They are given in Lemma 1. In all other respects the conditions are independent.
Lemma 1. The integral (1) converges if and only if for at least one \(j_0\) the number \(a_{j_0}\) is greater than \(2j_0\); moreover, if \(j_0>0\), then one may put \(k=1\), while if \(j_0=0\), then \(k\ge [1/a_0]+1\).
II. Asymptotics of the trace of \((L+\mu)^{-2k}\).
Theorem 1. Suppose that conditions 1–5 of § 2 are satisfied. Then \((L+\mu)^{-2k}\) is a nuclear operator and, for large positive \(\mu\),
\[ \sum_{n=1}^{\infty}\frac{1}{(\lambda_n+\mu)^{2k}} \sim \frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{dx\,d\xi}{\{l(x,i\xi)+\mu\}^{2k}} . \tag{2} \]
Let us briefly outline the proof of the theorem.
Our arguments are similar to those of Titchmarsh ([3], Ch. XVII). In his case, as well as in all cases considered up to now, the kernel of the resolvent \((L_v+\mu)^{-2k}\) of the operator \(L_v\), which is constructed below, decreases exponentially as \(|v|\to\infty\), and this simplifies the arguments. However, in our case, when the free term \(p_0(x)\) is subordinate to the intermediate coefficients, this fact does not hold. More precisely: the kernel of the resolvent \((L_v+\mu)^{-2k}\) has only a power order of decrease as \(|v|\to\infty\). For simplicity, let us assume that the integral (1) converges for \(k=1\).
Denote by \(L_v\) the operator generated in \(L_2(-\infty,\infty)\) by the differential expression \(l(v,d/dx)\), which is obtained from \(l(x,d/dx)\) by replacing \(x\) by \(v\) in the coefficients. Denote by \(G(\mu;x,u;v)\) the kernel of the resolvent \((L_v+\mu)^{-1}\). It is known that
\[ G(\mu;x,u;v)=\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{e^{i\xi(x-u)}}{l(v,i\xi)+\mu}\,d\xi . \tag{3} \]
Then (see [3], Ch. XVII)
\[ \frac{\varphi_n(u)}{\lambda_n+\mu} = \int_{-\infty}^{\infty} G(\mu;x,u;v)\varphi_n(x)\,dx - \int_{-\infty}^{\infty} \frac{\varphi_n(x)}{\lambda_n+\mu} \left\{ l\left(x,\frac{d}{dx}\right)-l\left(v,\frac{d}{dx}\right) \right\} G(\mu;x,u;v)\,dx . \tag{4} \]
Put \(v=u\) in the last identity; denote the first term on the right-hand side by \(a_n(u,\mu)\), and the second by \(b_n(u,\mu)\). Squaring both sides and integrating with respect to \(u\) over the interval \((-\infty,\infty)\), we obtain as a result
\[ \frac{1}{(\lambda_n+\mu)^2} = \int_{-\infty}^{\infty} a_n^2(u,\mu)\,du + 2\int_{-\infty}^{\infty} a_n(u,\mu)b_n(u,\mu) + \int_{-\infty}^{\infty} b_n^2(u,\mu)\,du . \]
From Parseval’s equality it follows that
\[ A(\mu)= \sum_{n=1}^{\infty}\int_{-\infty}^{\infty} a_n^2(u,\mu)\,du = \]
\[ = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} |G^2(\mu;x,u;v)|\,du\,dx = \frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{d\xi\,dx}{\{l(x,i\xi)+\mu\}^2}, \tag{5} \]
and the last integral exists. Put
\[ B(\mu)=\sum_{n=1}^{\infty}\int_{-\infty}^{\infty} b_n^2(u,\mu)\,du . \tag{6} \]
Further, using conditions 1–5, we obtain that \(B(\mu)=o[A(\mu)]\) as \(\mu\to+\infty\). To prove this, the integral for \(b_n(u,\mu)\) is split into two integrals \(b_{n,1}(u,\mu)\) and \(b_{n,2}(u,\mu)\) over the intervals \(|x-u|\le |u|/2+1\), \(|x-u|\ge |u|/2+1\), and then each is estimated separately.
We indicate the method for estimating one term in the expression for \(b_{n,1}(u,\mu)\), for example, the integral
\[ \begin{aligned} C_{n,j}^{2}(u,\mu) &= \left| \int_{|x-u|\le \frac{|u|}{2}+1} \frac{\varphi_n(x)}{\lambda_n+\mu}\{p_j(x)-p_j(u)\} \frac{\partial^{2j}}{\partial x^{2j}}G(\mu;x,u;u)\,dx \right|^{2} \\ &\le C\,\frac{(1+|u|)^{2\alpha_j-2}}{(\lambda_n+\mu)^2} \int_{|x-u|\le \frac{|u|}{2}+1} \left\{ |x-u|\frac{\partial^{2j}}{\partial x^{2j}}G(\mu;x,u;u) \right\}^{2}dx . \end{aligned} \tag{7} \]
From formula (3) it follows directly that
\[ \frac{\partial^{2j}}{\partial x^{2j}}G(\mu;x,u;v) = \frac{(-1)^j}{2\pi} \int_{-\infty}^{\infty} \frac{\xi^{2j}e^{i\xi(x-u)}}{l(v,i\xi)+\mu}\,d\xi . \]
Performing in the right-hand side of the last formula one integration by parts, we find
\[ (x-u)\frac{\partial^{2j}}{\partial x^{2j}}G(\mu;x,u;v) = \frac{(-1)^{j+1}}{2\pi i} \int_{-\infty}^{\infty} e^{i\xi(x-u)} \frac{\partial}{\partial \xi} \left[ \frac{\xi^{2j}}{l(v,i\xi)+\mu} \right]\,d\xi . \]
Therefore, from (7) and Parseval’s identity it follows that
\[ C_{n,j}^{2}(u,\mu) \le C\,\frac{(1+|u|)^{2\alpha_j-2}}{(\lambda_n+\mu)^2} \int_{-\infty}^{\infty} \left| \frac{\partial}{\partial \xi} \left[ \frac{\xi^{2j}}{l(u,i\xi)+\mu} \right] \right|^{2}d\xi . \]
Hence, using conditions 1–5, one can show that
\[ \sum_{n=1}^{\infty}\int_{-\infty}^{\infty} C_{n,j}^{2}(u,\mu)\,du = A(\mu)o(1) \quad \text{as } \mu\to\infty . \]
The function \(b_{n,2}(u,\mu)\) is equal to the sum of many terms. To estimate each term, one must first integrate by parts several times the expression for \(G(\mu;x,u;v)\) (depending on the term and on the value of \(\alpha_j\)), and then use Parseval’s identity and conditions 1–5.
4. Asymptotic formulas for \(N(\lambda)\). From the preceding arguments it follows that, under conditions 1–5, the operator \((L+\mu)^{-2k}\) is nuclear and, consequently, the operator \(L\) has a discrete spectrum \(\lambda_1\le \lambda_2\le\cdots\). Incidentally, we note that if conditions 1–4 are satisfied, but the integral diverges for all \(k>0\), then the operator \(L\) does not have a purely discrete spectrum. Let us find the asymptotics for \(N(\lambda)\) as \(\lambda\to+\infty\).
From formula (2) it follows that
\[ \int_{0}^{\infty}\frac{dN(\lambda)}{(\lambda+\mu)^{2k}} \sim \frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{dx\,d\xi}{\{l(x,i\xi)+\mu\}^{2k}} = \int_{0}^{\infty}\frac{d\sigma(\lambda)}{(\lambda+\mu)^{2k}} \quad \text{as } \mu\to+\infty, \tag{8} \]
where
\[ \sigma(\lambda) = \frac{1}{2\pi} \iint_{l(x,i\xi)<\lambda} dx\,d\xi . \tag{9} \]
Theorem 2. Suppose conditions 1–5 are satisfied. Then
\[ N(\lambda)\sim \sigma(\lambda)\quad \text{as } \lambda\to +\infty . \tag{10} \]
Proof. It is not difficult to verify that the monotone function \(\sigma(\lambda)\) satisfies the conditions of B. I. Korenblyum’s refinement\({}^{5}\) of M. V. Keldysh’s Tauberian theorem\({}^{4}\). Therefore (10) follows from (8). Formula (10) is general, but inconvenient for computations. We therefore indicate particular cases in which the asymptotic formula for \(N(\lambda)\) can be written more explicitly.
Corollary 1. Suppose conditions 1–5 hold and the limits exist
\[ \lim_{|x|\to\infty}\frac{p_j(x)}{|x|^{\alpha_j}}=A_j,\qquad j=0,1,\ldots,n-1 . \]
Suppose there exists a fixed \(j_0\) such that
\[ \nu_{j_0}=(n-j_0)/n\alpha_{j_0}\le (n-j)/n\alpha_j=\nu_j,\qquad j=0,1,\ldots,n-1 . \]
Some of the numbers \(\nu_j\) may be equal to \(\nu_{j_0}\). Denote them by \(\nu_{j_1},\ldots,\nu_{j_s}\); it is assumed here that the corresponding \(A_{j_0},\ldots,A_{j_s}\) are nonzero.
Then
\[ N(\lambda)\sim A\lambda^{1/2n+\nu_{j_0}}, \tag{11} \]
where
\[ A=\frac{1}{2\pi}\operatorname{mes}\left\{(x,\xi):\ \xi^{2n}+\sum_{k=0}^{s}A_{j_k}\xi^{2j_k}|x|^{\alpha_{j_k}}\le 1\right\}. \tag{12} \]
Remark. From condition 1 it follows that all \(p_j(x)\) are positive for large \(x\). This restriction can be removed. In Corollary 1 one may require that \(p_{j_0}(x),\ldots,p_{j_s}(x)\) satisfy condition 1, while the remaining coefficients instead of this condition satisfy
\[ |p_j(x)|\le C(1+|x|)^{\alpha_j}. \]
Corollary 2. Suppose
\[ l\left(x,\frac{d}{dx}\right)y = y^{(4n)}+(-1)^n\bigl(p_n(x)y^{(n)}\bigr)^{(n)}+p_0(x)y; \tag{13} \]
\(p_n(x)\), \(p_0(x)\) satisfy conditions 1–5. Then
\[ N(\lambda)\sim \frac{1}{2\pi} \int_{p_0(x)\le \lambda} \left\{ -p_n(x)+\left[p_n^2(x)-4\bigl(p_0(x)-\lambda\bigr)\right]^{1/2} \right\}^{1/2}\,dx . \tag{14} \]
In particular, if \(p_0(x)\) is bounded, then
\[ N(\lambda)\sim \frac{1}{2\pi} \int_{-\infty}^{\infty} \left\{ -p_n(x)+\left[p_n^2(x)+4\lambda\right]^{1/2} \right\}^{1/2}\,dx . \tag{15} \]
In conclusion it should be noted that the results of the paper generalize to certain elliptic partial differential operators.
Received
25 VI 1968
REFERENCES
\({}^{1}\) Colin Clark, Slam Rev., 9, No. 4, October (1967).
\({}^{2}\) A. G. Kostyuchenko, DAN, 168, No. 1, 21 (1966).
\({}^{3}\) E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, 2, IL, 1961.
\({}^{4}\) M. V. Keldysh, Tr. Matem. Inst. im. V. A. Steklova AN SSSR, 38, 77 (1951).
\({}^{5}\) B. I. Korenblyum, DAN, 88, 745 (1953).