A. G. KOSTYUCHENKO

ASYMPTOTIC DISTRIBUTION OF THE EIGENVALUES OF ELLIPTIC OPERATORS

(Presented by Academician I. G. Petrovskii on 4 IV 1964)

It is known that the Schrödinger operator \(-\Delta + q(x)\), defined in the whole space \(R_3\), has a discrete spectrum if \(q(x) \to \infty\) as \(|x| \to \infty\). Titchmarsh \((^1)\), B. M. Levitan \((^2)\), and Ray \((^3)\) established that, under certain additional conditions on \(q(x)\), the number of eigenvalues \(N(\lambda)\) less than \(\lambda\) has the following simple asymptotics:

\[ N(\lambda)\sim \frac{1}{6\pi^2}\int_{q(x)<\lambda}(\lambda-q)^{3/2}\,dx . \]

In the present work similar formulas are established for arbitrary self-adjoint elliptic operators.

1. Consider the elliptic operator

\[ L=(-1)^m \sum_{k_1+\cdots+k_n=2m} A^{k_1\cdots k_n}(x) \frac{\partial^{2m}}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} + L_1\left(x,\frac{\partial}{\partial x}\right) + q(x) = \]

\[ = L_0+L_1+q(x). \]

Here \(x=(x_1,\ldots,x_n)\), \(-\infty<x_i<\infty\), \(A^{k_1\cdots k_n}(x)\) are functions bounded in the whole space \(R_n\) and satisfying a Lipschitz condition:

\[ \left|A^{k_1\cdots k_n}(x)-A^{k_1\cdots k_n}(\xi)\right| \le K|x-\xi|^\gamma,\quad \text{if } |x-\xi|<1, \]

\(\gamma>0\), and the constant \(K\) does not depend on \(x\) and \(\xi\). In addition, it is assumed that the form of the highest terms is uniformly elliptic.

The operator \(L_1\left(x,\dfrac{\partial}{\partial x}\right)\) contains derivatives of order \(<2m\), and its coefficients are bounded in the whole space. The function \(q(x)\ge 1\) and satisfies the following conditions:

\[ \begin{aligned} 1)&\quad |q(x)-q(\xi)|\le Bq^\alpha(\xi)|x-\xi| \quad \text{for } |x-\xi|\le 1,\quad \alpha<1+\frac{1}{2m}.\\ 2)&\quad q(x)\le B\exp\left[C_0|x-\xi|q^{1/2m}(\xi)\right] \quad \text{for } |x-\xi|>1.\\ 3)&\quad \int_{|x-\xi|\le 1}\exp[-a q(\xi)]\,d\xi \le B\exp\left[-\frac{a}{k_0}q(x)\right] \quad \text{for some } k_0\ge 1.\\ 4)&\quad q(x)\le A q(\xi) \quad \text{for } |x-\xi|\le 1.\\ 5)&\quad \int_{R_n}\frac{dx}{q^\beta(x)}<\infty \quad \text{for some } \beta>0. \end{aligned} \]

Define the function

\[ M(\lambda)=\int_{q(x)<\lambda}\Phi(x)(\lambda-q)^{n/2m}\,dx, \quad \text{where } \Phi(x)= \]

\[ =\int_{-\infty}^{\infty} e^{-L_0(x,s)}\,ds, \]

and \(L_0(x,s)\) is the homogeneous form corresponding to the group of highest terms

\[ L_0\left(x,\frac{\partial}{\partial x}\right). \]

We shall assume that, for sufficiently large \(\lambda_1,\lambda_2\), the condition

\[ 6)\quad \frac{M(\lambda_1)}{M(\lambda_2)} \le C\left(\frac{\lambda_1}{\lambda_2}\right)^{\gamma_1}, \quad \gamma_1>0,\quad \lambda_1>\lambda_2. \]

Theorem 1. If the elliptic operator \(L\left(x,\dfrac{\partial}{\partial x}\right)\) satisfies the requirements formulated above, then its spectrum is discrete and the number of eigenvalues \(N(\lambda)\) less than \(\lambda\) has an asymptotic expression of the form

\[ N(\lambda)\sim \frac{1}{(2\pi)^n\Gamma(n/2m+1)} \int_{q(x)<\lambda}\Phi(x)(\lambda-q(x))^{n/2m}\,dx . \tag{1} \]

Let us note that if the coefficients of the highest derivatives do not depend on \(x\), then \(\Phi(x)\) is a constant.

For the equation of the second order,

\[ \Phi(x)=(\det\|a_{ij}(x)\|)^{-n/2}. \]

Remark 1. One may also write the asymptotics of the weighted trace \(N_s(\lambda)\):

\[ N_s(\lambda)=\sum_{\lambda_n<\lambda}\int_{R_n}q^s(x)\varphi_n^2(x)\,dx, \]

where \(\varphi_n(x)\) are the eigenfunctions of the operator \(L\). This asymptotic formula is valid under the previous conditions and has the form

\[ N_s(\lambda)\sim \frac{1}{(2\pi)^n\Gamma\left(\dfrac{n}{2m}+1\right)} \int_{q(x)<\lambda}\Phi(x)q^s(x)(\lambda-q(x))^{n/2m}\,dx . \]

Remark 2. Theorem 1 is also valid for so-called quasi-elliptic operators, which are somewhat more general than elliptic ones.

For example, the function \(N(\lambda)\) of the operator

\[ \sum_{i=1}^{n}(-1)^{m_i}\frac{\partial^{2m_i}}{\partial x_i^{2m_i}}+q(x) \]

under the previous restrictions on \(q(x)\) has principal term

\[ N(\lambda)\sim \frac{M}{(2\pi)^n\Gamma\left(\sum \dfrac{1}{2m_i}+1\right)} \int_{q(x)<\lambda}(\lambda-q(x))^{\frac12\sum \frac{1}{m_i}}\,dx, \]

where the constant \(M\) is equal to

\[ \int_{-\infty}^{\infty}\exp\left[-\sum s_i^{2m_i}\right]\,ds. \]

Let us note that, as is not difficult to verify, conditions 2)—5) are satisfied if \(q(x)\) is enclosed between two paraboloids:

\[ B_1(1+|x|^k)\leq q(x)\leq B_2(1+|x|^k),\qquad k>0. \]

Condition 1) will be satisfied, for example, if \(q(x)\) has a derivative \(q'(x)\) satisfying the estimate

\[ |q'(x)|\leq Cq^\alpha(x),\qquad \alpha<1+\frac{1}{2m}. \]

2. Let us briefly outline the proof of the theorem. The Green function \(G(x,y,t)\) of the Cauchy problem for the parabolic equation

\[ \frac{\partial u}{\partial t}=-Lu \tag{2} \]

is connected with the spectral function \(\theta(x,y,\lambda)\) of the elliptic operator \(L\geq0\) by the simple relation

\[ G(x,y,t)=\int_{0}^{\infty}e^{-\lambda t}\,d\theta(x,y,\lambda). \]

If the spectrum is discrete, then from this equality it immediately follows that

\[ \int_{-\infty}^{\infty}G(x,x,t)\,dx = \int_{0}^{\infty}e^{-\lambda t}\,dN(\lambda). \]

Thus the problem reduces to obtaining the asymptotics of the Green’s function \(G(x,y,t)\) for small \(t\), uniformly in \(x\) in the whole space \(R_n\). It should be said that writing down the asymptotics for small \(t\) in each bounded domain \(S \subset R_n\) is not difficult. It depends only on the group of highest derivatives and does not take into account the decisive influence of \(q(x)\). We shall seek the Green’s function \(G(x,y,t)\) by the “parametrix” method in the form

\[ G(x,y,t)=G_0(x-y,y,t)e^{-tq(y)}+ \]

\[ +\int_0^t d\tau \int_{-\infty}^{\infty} G_0(x-\xi,\xi,t-\tau)e^{-(t-\tau)q(\xi)} \varphi(\xi,y,\tau)\,d\xi . \]

Here \(G_0(x-y,\eta,t)\) is the Green’s function of the parabolic equation

\[ \frac{\partial u}{\partial t}=-L_0\left(\eta,\frac{\partial}{\partial x}\right)u . \]

From the fact that \(G(x,y,t)\) is the Green’s function of equation (2), we obtain that the kernel \(\varphi(x,y,t)\) satisfies the integral equation

\[ \varphi(x,y,t)-K_0(x,y,t)= \int_0^t d\tau \int_{-\infty}^{\infty} K_0(x,\xi,t-\tau)\varphi(\xi,y,\tau)\,d\xi, \tag{3} \]

where

\[ K_0(x,y,t)=\left\{L\left(x,\frac{\partial}{\partial x}\right)- L\left(y,\frac{\partial}{\partial x}\right)\right\} G_0(x-y,y,t)e^{-tq(y)} . \]

It is not hard to verify that, when the conditions of item 1 are fulfilled, the kernel \(K_0(x,y,t)\) satisfies the inequalities

\[ |K_0(x,y,t)|\leq \frac{ A\exp\left[-c\frac{|x-y|^r}{t^{1/(2m-1)}}\right] }{ t^{(n+2m-\varepsilon)/2m} } \exp[-c_1tq(y)] \quad \text{for } |x-y|\leq 1, \]

\[ |K_0(x,y,t)|\leq \frac{ A\exp\left[-c\frac{|x-y|^r}{t^{1/(2m-1)}}\right] }{ q^l(y) } \exp[-c_1tq(y)] \quad \text{for } |x-y|>1. \]

Here \(A,c,c_1\) are some constants \(>0\); \(l\) is an arbitrarily large integer; \(r\) is the number conjugate to \(2m\), i.e. \(\frac{1}{r}+\frac{1}{2m}=1\), \(\varepsilon>0\).

It turns out that the integral equation (3) can be solved by the method of successive approximations. Moreover, up to a certain number \(k\), the iterated kernels

\[ K_j(x,y,t)=\int_0^t d\tau \int_{-\infty}^{\infty} K_0(x,\xi,t-\tau)K_{j-1}(\xi,y,\tau)\,d\xi \]

will satisfy the estimate

\[ |K_j(x,y,t)|\leq \begin{cases} \displaystyle \frac{ A_j\exp\left[-c_j\frac{|x-y|^r}{t^{1/(2m-1)}}\right] }{ t^{n/2m-j\delta} } \exp[-c_jtq(y)] & \text{for } |x-y|\leq 1, \\[1.2em] \displaystyle \frac{ A_j\exp\left[-c_j\frac{|x-y|^r}{t^{1/(2m-1)}}\right] }{ q^l(y) } \exp[-c_jtq(y)] & \text{for } |x-y|>1; \end{cases} \tag{4} \]

where \(c_j>0\), \(\delta>0\).

It follows from inequality (4) that, for sufficiently large \(j\), \(j>n/2m\delta=j_0\), the kernel \(K_j(x,y,t)\) will have no singularity with respect to \(t\). Moreover, for \(j\) greater than some \(j_0'\), the inequality

\[ |K_j(x,y,t)|\leq A_j\frac{ \exp\left[-c_j\frac{|x-y|^r}{t^{1/(2m-1)}}\right] }{ q^l(y) } \quad \text{for all } x \text{ and } y \tag{5} \]

will hold.

Starting from this point, we shall estimate the iterated kernels differently: we assume that

\[ |K_0(x,y,t)| \leq \frac{ A \exp\left[-c_0 \frac{|x-y|^r}{t^{1/(2m-1)}}\right] }{ t^{(n+2m-\varepsilon)/2m} }. \]

We shall estimate the kernels \(K_j(x,y,t)\) according to inequality (5). The iterated kernels, however, will be estimated as S. D. Eidelman did \((^4)\). As a result, we obtain that the Green’s function has the form

\[ G(x,y,t)=G_0(x-y,t)e^{-tq(y)} +\frac{e^{-ctq(y)}}{t^{n/2m-\varepsilon_0}}O(1) +\frac{O(1)}{t^{n/2m-\varepsilon_0}q^l(y)} . \tag{6} \]

From this inequality we obtain

\[ g(t)=\int_{-\infty}^{\infty} G(x,x,t)\,dx = \frac{1}{t^{n/2m}}\int_{-\infty}^{\infty}\Phi(x)e^{-tq(x)}\,dx + \frac{ O(1)\int_{-\infty}^{\infty}e^{-ctq(x)}\,dx }{ t^{n/2m-\varepsilon} }. \]

Next, note that

\[ g_0(t)= \frac{1}{t^{n/2m}}\int_{-\infty}^{\infty}\Phi(x)e^{-tq(x)}\,dx = \frac{1}{\Gamma(n/2m+1)} \int_{0}^{\infty} e^{-\lambda t}\,dM(\lambda). \]

If one assumes that

\[ \int_{-\infty}^{\infty} e^{-tq(x)}\,dx \quad\text{and}\quad \int_{-\infty}^{\infty} e^{-ctg(x)}\,dx \]

have singularities at zero in \(t\) of the same order, then \(g(t)\sim g_0(t)\). Applying now the Tauberian theorem of B. I. Korenblum \((^5)\), we obtain formula (1). We note that condition 6) of Sec. 1 is the “Tauberian condition” necessary for the application of the Tauberian theorem.

As for the discreteness of the spectrum of the operator \(L\), it can likewise be obtained from the representation of the Green’s function \(G(x,y,t)\) in the form (6), since the kernel \(H(x,y,\lambda)\) of the resolvent \(R_\lambda=(L-\lambda E)^{-1}\) is related to the function \(G(x,y,t)\) by the formula

\[ H(x,y,\lambda)=\int_{0}^{\infty}G(x,y,t)e^{-\lambda t}\,dt . \]

It will follow from estimate (6) that \(H(x,y,\lambda)\) defines an operator some power of which is a Hilbert–Schmidt operator. From the estimate obtained in (6) it also follows that the eigenfunctions decrease faster than \(c_k/q^k(x)\) for any \(k>0\). The assertions stated in Remarks 1 and 2 are proved according to the scheme indicated above. It would be possible to dispense with the requirement that the coefficients of the operator \(L_1\) be bounded and to allow a certain growth at infinity, coordinated in a definite way with the growth of \(q(x)\). Formula (1) remains unchanged in that case.

In conclusion we note that conditions 1)—5) of Sec. 1 are satisfied mainly by functions of power growth. If, however, \(q(x)\) grows at infinity faster than any power, then formula (1) is also preserved; moreover, the required asymptotics for the Green’s function \(G(x,y,t)\) can be obtained still more simply \((^6)\), if one requires the existence for the function \(q(x)\) of derivatives of order \(2m\) satisfying the inequalities:

\[ |q^{(i)}(x)| \leq c_i q^{(2m+i)/2m}(x), \qquad i=1,2,\ldots,2m. \]

Moscow State University
named after M. V. Lomonosov

Received
6 III 1964

CITED LITERATURE

\(^1\) E. Titchmarsh, Proc. London Math. Soc. (3), 3, No. 10, 153 (1953).
\(^2\) B. M. Levitan, Mat. sborn. 41 (83), 439 (1957).
\(^3\) D. Ray, Trans. Am. Math. Soc., 77, 2, 299 (1954).
\(^4\) S. D. Eidelman, Mat. sborn., 38 (80), No. 1, 51 (1953).
\(^5\) B. I. Korenblum, DAN, 104, No. 2, 173 (1955).
\(^6\) A. G. Kostyuchenko, DAN, 132, No. 1, 32 (1960).