UDC 517.948.35

ON THE ASYMPTOTIC DISTRIBUTION OF THE EIGENVALUES OF ELLIPTIC OPERATORS IN \(n\)-DIMENSIONAL SPACE

Yu. N. Sudarev

§ 1. FORMULATION OF THE PROBLEM

Consider in \(n\)-dimensional space \(R^n\) the operator generated by a differential expression of the form

\[ \begin{aligned} L\left(x,\frac{\partial}{\partial x}\right) &= (-1)^m \sum_{k_1+\cdots+k_n=2m} A_{2m}^{k_1\ldots k_n}(x) \frac{\partial^{2m}}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} + \\[4pt] &\quad + (-1)^{m-1} \sum_{k_1+\cdots+k_n=2m-2} A_{2m-2}^{k_1\ldots k_n}(x) \frac{\partial^{2m-2}}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} + \\[4pt] &\quad + \cdots + (-1) \sum_{k_1+\cdots+k_n=2} A_2^{k_1\ldots k_n}(x) \frac{\partial^2}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}} + L^1\left(x,\frac{\partial}{\partial x}\right) + q_0(x) = \\[4pt] &= L^0\left(x,\frac{\partial}{\partial x}\right) + L^1\left(x,\frac{\partial}{\partial x}\right) + q_0(x), \end{aligned} \tag{1.1} \]

\[ x=(x_1,\ldots,x_n),\qquad -\infty < x_i < \infty \qquad (i=1,2,\ldots,n). \]

We require continuity of all the coefficients of \(L\), symmetry and positive definiteness of \(L\) on finite functions. We also assume that \(L^1\left(x,\frac{\partial}{\partial x}\right)\) contains derivatives of order less than \(2m\), and that the coefficients of \(L^0\) and \(q_0(x)\) are positive. (The latter requirement will be weakened below.) In addition, we require that the following conditions be satisfied.

1. The \(A_{2m}^{k_1\ldots k_n}(x)\) are bounded in \(R^n\) and satisfy the uniform Lipschitz condition

\[ \left|A_{2m}^{k_1\ldots k_n}(x)-A_{2m}^{k_1\ldots k_n}(\xi)\right| \le K|x-\xi|^\gamma,\qquad K,\gamma>0. \]

2.

\[ a_2|\sigma|^{2m} \le \sum_{k_1+\cdots+k_n=2m} A_{2m}^{k_1\ldots k_n}(x)\sigma_1^{k_1}\cdots\sigma_n^{k_n} \le a_1|\sigma|^{2m}, \]

\[ a_1,a_2>0,\qquad |\sigma|=\sqrt{\sigma_1^2+\cdots+\sigma_n^2} \]

for all real \(\sigma_1,\ldots,\sigma_n\) (uniform ellipticity).

1. \[ \left|A_{2m-2k}^{k_1\ldots k_n}(x)-A_{2m-2k}^{k_1\ldots k_n}(\xi)\right| \leq C\left[A_{2m-2k}^{k_1\ldots k_n}(\xi)\right]^{\alpha_k}|x-\xi|^{*}) \]
   for \(|x-\xi|\leq 1,\quad 0<\alpha_k<1+\frac{1}{2k}\).

Introduce the functions \(q_{2(m-k)}(x)\), setting by definition
\[ q_{2(m-k)}(x)=B_k^{*}\max_{(k_1\ldots k_n)} A_{2(m-k)}^{k_1\ldots k_n}(x),\qquad B_k^{*}=\inf B_k, \]
where the infimum is taken over all \(B_k\) satisfying the conditions
\[ 1\leq B_k, \]
\[ \left| \sum_{k_1+\cdots+k_n=2m-2k} A_{2(m-k)}^{k_1\ldots k_n}(x)\sigma_1^{k_1}\cdots\sigma_n^{k_n} \right| \leq B_k\max_{(k_1\ldots k_n)}A_{2(m-k)}^{k_1\ldots k_n}(x)|\sigma|^{2m-2k}. \]
The set \(\{B_k\}\) is nonempty, since, obviously, the inequality
\[ \left| \sum_{k_1+\cdots+k_n=2m-2k} A_{2(m-k)}^{k_1\ldots k_n}(x)\sigma_1^{k_1}\cdots\sigma_n^{k_n} \right| \leq \]
\[ \leq (2m-2k+1)^n \max_{(k_1\ldots k_n)} A_{2(m-k)}^{k_1\ldots k_n}(x)|\sigma|^{2m-2k} \]
holds.

Putting \(q(x)=\max_k q_{2(m-k)}^{m/k}(x)\), we impose additional conditions.

1. \(\theta q(x)\leq q_0(x)\leq Cq(x)\),

\(\theta\) is a constant depending on \(n,m,\displaystyle \sup_{(x,k_1\ldots k_n)}|A_{2m}^{k_1\ldots k_n}(x)|\); \(C\) is arbitrary. Thus we exclude from consideration the case \(q(x)\equiv 0\). This case was treated in [1].

1. \[ |q_0(x)-q_0(\xi)|\leq Cq_0^{\alpha_0}(\xi)|x-\xi|;\qquad 0<\alpha_0<1+\frac{1}{2m} \]
   for \(|x-\xi|\leq 1\).

2. The coefficients \(P_k^{k_1\ldots k_n}(x)\) of the operator \(L^1\left(x,\frac{\partial}{\partial x}\right)\) at derivatives of order \(k\) satisfy the inequality
   \[ |P_k^{k_1\ldots k_n}(x)|\leq Cq^{\nu_k}(x);\qquad 0\leq \nu_k<1-\frac{k}{2m}. \]

3. \(q(x)\geq 1\).

4. \(q(x)\leq Aq(\xi)\) for \(|x-\xi|\leq 1\).

5. \(q(x)\leq C\exp\{\rho |x-\xi|q^{1/2m}(\xi)\},\quad |x-\xi|>1\), where \(\rho\) is a sufficiently small positive constant depending on \(n,m\) and \(\sup |A_{2m}^{k_1\ldots k_n}(x)|\).

6. \[ \int_{-\infty}^{+\infty}q^{-\kappa}(x)\,dx<\infty \]
   for some \(\kappa>0\).

7. \[ \int_{-\infty}^{+\infty}\exp\{-tb_1q(x)\}\,dx = O(1)\int_{-\infty}^{+\infty}\exp\{-tq(x)\}\,dx \]
   for every \(b_1>0\).

\[ \text{*) Here and below, all constants whose exact value is immaterial for us are denoted by the same letter } C. \]

In view of the imposed restrictions, the closure of the operator \(L\) is self-adjoint (see [1]) and, as we shall see below, its spectrum is discrete. The problem consists in finding an asymptotic formula for the function \(N(\lambda)=\sum_{\lambda_n<\lambda}1\), where \(\lambda_n\) are the eigenvalues of the operator \(L\). The method of deriving this formula is as follows.

Consider the Cauchy problem for the parabolic equation

\[ \frac{\partial u}{\partial t}=-L\left(x,\frac{\partial}{\partial x}\right)u. \tag{1.2} \]

The Green function \(G(x,y,t)\) of this problem is connected with the spectral function \(\theta(x,y,\lambda)\) of the operator \(L\) by the following relation (see [1]):

\[ G(x,y,t)=\int_0^\infty \exp\{-\lambda t\}\,d\theta(x,y,\lambda). \tag{1.3} \]

In view of the discreteness of the spectrum of \(L\), we have

\[ \int_{-\infty}^{+\infty}\theta(x,x,\lambda)\,dx = \sum_{\lambda_n\leq\lambda}\int_{-\infty}^{+\infty}|\varphi_n(x)|^2\,dx = N(\lambda), \tag{1.3'} \]

where \(\{\varphi_n(x)\}\) is the proper orthonormal basis of \(L\). From (1.3) and (1.3′) it follows that

\[ \int_{-\infty}^{+\infty}G(x,x,t)\,dx = \int_0^\infty \exp\{-\lambda t\}\,dN(\lambda). \tag{1.4} \]

Therefore we seek the asymptotics of \(\int_{-\infty}^{+\infty}G(x,x,t)\,dx\) and, applying a Tauberian theorem, obtain the required formula.

§ 2. DERIVATION OF THE ASYMPTOTICS OF THE TRACE OF THE GREEN FUNCTION

We seek the Green function of problem (1.2) in the form

\[ G(x,y,t)=G_0(x-y,y,t)+ \]

\[ +\int_0^t d\tau\int_{-\infty}^{+\infty} G_0(x-\xi,\xi,t-\tau)\varphi(\xi,y,\tau)\,d\xi, \tag{2.1} \]

where \(G_0(x-y,\eta,t)\) is the Green function of the Cauchy problem

\[ \frac{\partial u}{\partial t} = -\left\{ L^0\left(\eta,\frac{\partial}{\partial x}\right)+q_0(\eta) \right\}u. \tag{2.2} \]

Applying the operator \(\dfrac{\partial}{\partial t}+L\left(x,\dfrac{\partial}{\partial x}\right)\) to the left- and right-hand sides of (2.1), we obtain the following integral equation for the unknown “density” \(\varphi(x,y,t)\):

\[ \varphi(x,y,t)-K_0(x,y,t) = \int_0^t d\tau\int_{-\infty}^{+\infty} K_0(x,\xi,t-\tau)\varphi(\xi,y,\tau)\,d\xi, \tag{2.3} \]

where

\[ K_0(x,y,t) = \left\{ L^0\left(y,\frac{\partial}{\partial x}\right)+q_0(y) - L\left(x,\frac{\partial}{\partial x}\right) \right\} G_0(x-y,y,t). \]

By means of the Fourier transform we obtain an expression for \(G_0\):

\[ G_0(x,\eta,t)=\frac{1}{(2\pi)^n}\int_{R^n} Q(\sigma,\eta,t)\exp\{i(x,\sigma)\}\,d\sigma, \tag{2.4} \]

where

\[ Q(\sigma,\eta,t)=\exp\left\{-t\left( \sum_{k_1+\cdots+k_n=2m} A_{2m}^{k_1\ldots k_n}(\eta)\sigma_1^{k_1}\cdots\sigma_n^{k_n} +\cdots+ \sum_{k_1+\cdots+k_n=2} A_2^{k_1\ldots k_n}(\eta)\sigma_1^{k_1}\cdots\sigma_n^{k_n} +q_0(\eta)\right)\right\}, \]

\[ (x,\sigma)=\sum_{i=1}^n x_i\sigma_i . \tag{2.5} \]

It follows from this that \(Q(s,\eta,t)\) is an entire function of the variables

\[ s_j=\sigma_j+i\gamma_j \qquad (j=1,2,\ldots,n). \]

Let us estimate \(Q(s,\eta,t)\) in modulus:

\[ \begin{aligned} \varphi &\equiv \operatorname{Re} t\left\{ -\sum_{k_1+\cdots+k_n=2m} A_{2m}^{k_1\ldots k_n}(\eta) (\sigma_1+i\gamma_1)^{k_1}\cdots(\sigma_n+i\gamma_n)^{k_n} \right.\\ &\qquad\left. -\cdots- \sum_{k_1+\cdots+k_n=2} A_2^{k_1\ldots k_n}(\eta) (\sigma_1+i\gamma_1)^{k_1}\cdots(\sigma_n+i\gamma_n)^{k_n} \right\}\\ &\le t\left\{ -\sum_{k_1+\cdots+k_n=2m} A_{2m}^{k_1\ldots k_n}(\eta)\sigma_1^{k_1}\cdots\sigma_n^{k_n} \right.\\ &\qquad + C_1(m,n)M_1\sum_{k=0}^{2m-1}|\sigma|^k|\gamma|^{2m-k}\\ &\qquad + C_2(m,n)q_{2(m-1)}(\eta)\sum_{k=0}^{2m-2}|\sigma|^k|\gamma|^{2m-2-k}\\ &\qquad\left. +\cdots+C_m(m,n)q_2(\eta)\sum_{k=0}^{2}|\sigma|^k|\gamma|^{2-k} \right\}; \end{aligned} \]

\[ M_1=\sup_{(x,k_1\ldots k_n)} \left|A_{2m}^{k_1\ldots k_n}(x)\right|. \]

Using condition 2 (§ 1) and Young’s inequality

\[ |ab|\le \varepsilon^p\frac{|a|^p}{p} +\varepsilon^{-q}\frac{|b|^q}{q}, \qquad \frac{1}{p}+\frac{1}{q}=1, \qquad \varepsilon>0, \]

we obtain

\[ \begin{aligned} \varphi \le t\left\{ -a_2|\sigma|^{2m} + C_1(m,n)M_1\left[ \sum_{k=1}^{2m-1} \left( \frac{k\varepsilon^{2m/k}}{2m}|\sigma|^{2m} \right.\right.\right.\\ \left.\left.\left. +\frac{(2m-k)\varepsilon^{-2m/(2m-k)}}{2m}|\gamma|^{2m} \right) +|\gamma|^{2m} \right] +\right. \end{aligned} \]

\[ + C_2(m,n) q_{2(m-1)}(\eta)\left[\sum_{k=1}^{2m-2}\left(\frac{k}{2m-2}|\sigma|^{2m-2}+\frac{2m-2-k}{2m}|\gamma|^{2m-2}\right)+|\gamma|^{2m-2}\right] \]
\[ +\cdots+C_m(m,n)q_2(\eta)\left[\sum_{k=1}^{2}\left(\frac{k}{2}|\sigma|^2+\frac{2-k}{2}|\gamma|^2\right)+|\gamma|^2\right]\Biggr\}\leq \]
\[ \leq t\Biggl\{\left[-a_2+C_1(m,n)M_1\sum_{k=1}^{2m-1}\frac{k}{2m}\varepsilon^{2m/k}+\right. \]
\[ +\left(C_2(m,n)\sum_{k=1}^{2m-2}\frac{k}{2m-2}\right)^{m/m-1}\frac{(m-1)}{m}\varepsilon^{m/m-1} \]
\[ +\cdots+\left(C_m(m,n)\sum_{k=1}^{2}\frac{k}{2}\right)^m\frac{\varepsilon^m}{m}\Biggr]|\sigma|^{2m}+ \]
\[ +\left[C_1(m,n)M_1\sum_{k=1}^{2m-1}\frac{(2m-k)}{2m}\varepsilon^{-2m/2m-k}+\right. \]
\[ +\cdots+C_m^m(m,n)\left(\sum_{k=1}^{2}\frac{2-k}{2}+1\right)^m\frac{\varepsilon^{-m}}{m}\Biggr]|\gamma|^{2m} +q_{2(m-1)}^m(\eta)\frac{\varepsilon^{-m}}{m}+ \]
\[ +\cdots+q_2^{m/m-1}(\eta)\frac{(m-1)}{m}\varepsilon^{-m/m-1}+ \]
\[ +\frac{\varepsilon^m}{m}q_{2(m-1)}^m(\eta)+\cdots+\frac{\varepsilon^{m/m-1}(m-1)}{m}q_2^{m/m-1}(\eta)\Biggr\}. \]

Choosing \(\varepsilon\) so small that the expression in the bracket multiplying \(|\sigma|^{2m}\) becomes negative, we obtain

\[ \varphi \leq -t a_3|\sigma|^{2m}+tF|\gamma|^{2m} +t\sum_{k=1}^{m-1}\gamma_k q_{2(m-k)}^{m/k}(\eta), \tag{2.6} \]

\[ a_3,\ F>0,\qquad \gamma_k=\frac{k}{m}\left(\varepsilon^{-m/k}+\varepsilon^{m/k}\right). \]

The constant \(\theta\) in condition 4 (§ 1) must satisfy the equality

\[ \theta=\sum_{k=1}^{m-1}\gamma_k+\varepsilon_1,\qquad \varepsilon_1>0. \tag{2.6'} \]

From (2.4), (2.5), and (2.6) there follows the estimate

\[ |Q(s,\eta,t)|\leq \exp\{-a_3t|\sigma|^{2m}+Ft|\gamma|^{2m}-\varepsilon_1tq(\eta)\}. \tag{2.7} \]

Now the estimate of \(G_0(x-y,y,t)\) and of its derivatives with respect to \(x\) is obtained in the usual way. In view of (2.6), the integral (2.4) may be taken not over the real axes, but over the straight lines

\[ \beta_j+i\xi_j,\qquad -\infty<\beta_j<\infty\qquad (j=1,2,\ldots,n) \]

for arbitrary fixed \(\xi_j\). We have

\[ G_0(x-y,y,t)=\frac{1}{(2\pi)^n t^{n/2m}}\times \]

\[ {}\times \int_{R^n} Q\left(\frac{\beta+i\xi}{t^{1/2m}},\,y,\,t\right) \exp\left\{i\left(\frac{x-y}{t^{1/2m}},\,\beta+i\xi\right)\right\}\,d\beta . \]

Choosing
\[ \xi_k=\xi_0[\operatorname{sgn}(x_k-y_k)]^{2m/2m-1}|x_k-y_k|^{1/2m-1}t^{-1/2m(2m-1)}, \]
we obtain the required estimates (see [2])

\[ \left|D_x^k G_0(x-y,y,t)\right| \le \frac{C}{t^{n+k/2m}} \exp\left\{ -c_0\frac{|x-y|^{2m/2m-1}}{t^{1/2m-1}}-\varepsilon_1tq(y) \right\} \tag{2.8} \]

\[ (k=0,1,\ldots,2m). \]

Exactly as in [1], with the help of the estimates (2.8) and using conditions 1–10 (§ 1), it is shown that equation (2.3) has a solution, continuous for \(t>0\), satisfying estimates from which there follows the legitimacy of the passage from (2.1) to (2.3), as well as the following asymptotic formula:

\[ G(x,y,t)=G_0(x-y,y,t)+ \]

\[ +\frac{O(1)}{t^{\frac{n}{2m}-\omega}} \exp\left\{ -c_1\frac{|x-y|^{2m/2m-1}}{t^{1/2m-1}}-c_2tq(y) \right\} + \]

\[ +\frac{O(1)}{q^k(y)} \exp\left\{ -c_1\frac{|x-y|^{2m/2m-1}}{t^{1/2m-1}} \right\}, \qquad c_1,c_2,\omega>0, \tag{2.9} \]

where the quantities \(O(1)\) are uniformly bounded with respect to \(x,y\) for small \(t\). From (2.9) we obtain

\[ \int_{-\infty}^{+\infty}G(x,x,t)\,dx = \int_{-\infty}^{+\infty}G_0(0,x,t)\,dx+ \]

\[ +\frac{O(1)}{t^{\frac{n}{2m}-\omega}} \int_{-\infty}^{+\infty}\exp\{-c_2tq(x)\}\,dx+O(1). \tag{2.10} \]

But

\[ G_0(0,x,t)=\frac{\exp\{-tq_0(x)\}}{(2\pi)^n} \int_{-\infty}^{+\infty} \exp\left\{ -t\left( \sum_{k_1+\cdots+k_n=2m} A_{2m}^{k_1\ldots k_n}(x)\sigma_1^{k_1}\cdots\sigma_n^{k_n} +\right.\right. \]

\[ \left.\left. +\cdots+ \sum_{k_1+\cdots+k_n=2} A_2^{k_1\ldots k_n}(x)\sigma_1^{k_1}\cdots\sigma_n^{k_n} \right) \right\}\,d\sigma . \]

For the expression in parentheses the estimate

\[ \sum_{k_1+\cdots+k_n=2m} A_{2m}^{k_1\ldots k_n}(x)\sigma_1^{k_1}\cdots\sigma_n^{k_n} + \]

\[ + \cdots + \sum_{k_1+\cdots+k_n=2} A_{2}^{k_1\ldots k_n}(x)\sigma_1^{k_1}\cdots\sigma_n^{k_n} \leq a_1|\sigma|^{2m}+ \]

\[ + q_{2(m-1)}(x)|\sigma|^{2m-2}+\cdots+q_2(x)|\sigma|^2 \leq \]

\[ \leq a_1|\sigma|^{2m}+\frac{1}{m}q_{2(m-1)}^m(x)+\frac{m-1}{m}|\sigma|^{2m}+ \]

\[ +\cdots+\frac{m-1}{m}q_2^{m/m-1}(x)+\frac{1}{m}|\sigma|^{2m}= \]

\[ =\left(a_1+\sum_{k=1}^{m-1}\frac{m-k}{m}\right)|\sigma|^{2m} +\sum_{k=1}^{m-1}\frac{(m-k)}{m}q(x). \]

Hence

\[ G_0(0,x,t)\geq \frac{ \exp\left\{-tq_0(x)-t\sum_{k=1}^{m-1}\frac{(m-k)}{m}q(x)\right\} }{(2\pi)^n} \times \]

\[ \times\int_{-\infty}^{+\infty} \exp\left\{-t\left(a_1+\sum_{k=1}^{m-1}\frac{(m-k)}{m}\right)|\sigma|^{2m}\right\}\,d\sigma \geq \]

\[ \geq \frac{C}{t^{n/2m}} \exp\left\{-tq_0(x)-t\sum_{k=1}^{m-1}\frac{(m-k)}{m}q(x)\right\}. \]

Taking into account condition 4 (§ 1), we obtain

\[ G_0(0,x,t)\geq \frac{C}{t^{n/2m}}\exp\{-btq(x)\},\quad b>0. \tag{2.11} \]

From (2.10) and (2.11), in view of condition 11 (§ 1), it follows that

\[ \int_{-\infty}^{+\infty}G(x,x,t)\,dx \sim \int_{-\infty}^{+\infty}G_0(0,x,t)\,dx \quad \text{as } t\to 0. \tag{2.12} \]

From (2.9) there follows, as in [1], the discreteness of the spectrum.

Let now the coefficients \(L^0\left(x,\dfrac{\partial}{\partial x}\right)\) change sign, but \(L\geq 0\).

Set

\[ A_{2m-2k}^{k_1\ldots k_n}(x) = {}_{+}A_{2m-2k}^{k_1\ldots k_n}(x) - {}_{-}A_{2m-2k}^{k_1\ldots k_n}(x), \quad {}_{+}A,\ -A\geq 0. \]

Accordingly,

\[ L^0\left(x,\frac{\partial}{\partial x}\right) = L_+^0\left(x,\frac{\partial}{\partial x}\right) - L_-^0\left(x,\frac{\partial}{\partial x}\right). \]

Construct, as usual, the function \(q(x)\) for the operator \(L_+^0\), and require that

\[ {}_{-}A_{2m-2k}^{k_1\ldots k_n}(x) \leq C q^{\nu_{2m-2k}}(x), \quad 0\leq \nu_{2m-2k}<\frac{k}{m}. \]

Then \(L_-^0\left(x,\dfrac{\partial}{\partial x}\right)\) will be subordinated to the operator \(L_+^0\left(x,\dfrac{\partial}{\partial x}\right)\), and it can be combined with \(L^1\left(x,\dfrac{\partial}{\partial x}\right)\), reducing everything to the preceding case. It is enough to require that \(q_0(x)\) be bounded below.

§ 3. ASYMPTOTIC FORMULA FOR \(N(\lambda)\)

Thus, we have obtained relation (2.12), where

\[ G_0(0,x,t)=\frac{1}{(2\pi)^n}\int_{-\infty}^{+\infty} \exp\{-t(L_1^0(x,\sigma)+q_0(x))\}\,d\sigma, \tag{3.1} \]

\[ L_1^0(x,\sigma)=L^0(x,i\sigma)=L_{2m}(x,\sigma)+L_{2m-2}(x,\sigma)+\cdots+L_2(x,\sigma), \]

\(L_{2j}(x,\sigma)\) is a form of order \(2j\) in \(\sigma\).

Consider the function

\[ \varphi(x,\sigma,\tau)=L_1^0(x,\tau,\sigma), \qquad 0\leq \tau<\infty . \tag{3.2} \]

We require that for all \(x\), \(\sigma\ne(0,\ldots,0)\), and \(\tau_1>\tau_2\geq 0\), the condition

\[ \varphi(x,\sigma,\tau_1)>\varphi(x,\sigma,\tau_2) \tag{3.3} \]

be satisfied.

It is also clear that \(\varphi(x,\sigma,0)=0\) and \(\varphi(x,\sigma,\tau)\to\infty\) as \(\tau\to\infty\).

Take an arbitrary constant \(0<c<\infty\). On each ray \((\tau\sigma_1,\ldots,\tau\sigma_n)\), \(0\leq\tau<\infty\), by virtue of (3.3) there is one and only one point \((\tau_0\sigma_1,\ldots,\tau_0\sigma_n)\) at which \(L_1^0(x,\tau_0\sigma)=c\). Moreover, it is clear that the surface \(L_1^0(x,\sigma)=c\) is smooth in \(\sigma\) for every \(x\). We therefore obtain the following result.

The surface \(L_1^0(x,\sigma)=c\), for arbitrary \(x\) and \(c\), is a closed smooth surface homeomorphic to the unit sphere, containing the origin in its interior; and for any point of an arbitrary ray emanating from the origin there is one and only one such surface passing through this point.

Transform the expression for \(G_0(0,x,t)\). From (3.1) we have

\[ G_0(0,x,t)=\frac{1}{(2\pi)^n}\int_{q_0(x)}^{\infty} \exp\{-\lambda t\}S_n(x,\lambda-q_0(x))\,d_\lambda\sigma_1(x,\lambda-q_0(x)), \tag{3.4} \]

where \(S_n(x,\lambda-q_0(x))\) is the area of the surface defined by the equation

\[ L_1^0(x,\sigma)=\lambda-q_0(x), \tag{3.5} \]

and \(\sigma_1(x,\lambda-q_0(x))\) is the first coordinate of the point of intersection of the positive semiaxis \((\sigma_1,0,\ldots,0)\), \(0\leq\sigma_1<\infty\), with this surface. In other words, \(\sigma_1(x,\lambda-q_0(x))\) is the positive root of the equation

\[ A_{2m}^{2m,0,\ldots,0}(x)\sigma_1^{2m} + A_{2m-2}^{2m-2,0,\ldots,0}(x)\sigma_1^{2m-2} + \cdots + A_2^{2,0,\ldots,0}(x)\sigma_1^2 = \lambda-q_0(x). \tag{3.6} \]

Set

\[ \omega(x,\lambda)=\frac{1}{(2\pi)^n}\int_{q_0(x)}^{\lambda} S_n\bigl(x,\xi-q_0(x)\bigr)\,d_\xi\sigma_1\bigl(x,\xi-q_0(x)\bigr). \tag{3.7} \]

Then

\[ G_0(0,x,t)=\int_{q_0(x)}^{\infty}\exp\{-\lambda t\}\,d_\lambda\omega(x,\lambda). \tag{3.8} \]

Introduce the new function

\[ F(x,\lambda)= \begin{cases} \omega(x,\lambda), & \lambda\geqslant q_0(x),\\ 0, & \lambda<q_0(x). \end{cases} \tag{3.9} \]

With its aid, (3.8) is rewritten in the form

\[ G_0(0,x,t)=\int_0^{\infty}\exp\{-\lambda t\}\,d_\lambda F(x,\lambda). \tag{3.10} \]

The measure \(F(x,\lambda)\), as is clear from (3.7) and (3.9), is continuously differentiable with respect to \(\lambda\), is identically zero for \(\lambda\leqslant q_0(x)\), and increases monotonically from \(0\) to \(\infty\).

Now, taking (1.4) and (3.10) into account, we can rewrite relation (2.12) in the form

\[ \int_0^\infty \exp\{-\lambda t\}\,dN(\lambda)\sim \int_0^\infty \exp\{-\lambda t\}\,d\Phi(\lambda), \tag{3.11} \]

where

\[ \Phi(\lambda)=\int_{-\infty}^{+\infty} F(x,\lambda)\,dx. \]

We require the fulfillment of the Tauberian condition

\[ \frac{\Phi(\lambda_1)}{\Phi(\lambda_2)} \leqslant C\left(\frac{\lambda_1}{\lambda_2}\right)^\gamma, \qquad \gamma=\gamma(c)>0 \tag{3.12} \]

for every \(C>1\) and sufficiently large \(\lambda_1>\lambda_2\).

By virtue of Korenblum’s Tauberian theorem [3], we obtain the asymptotic formula for \(N(\lambda)\):

\[ N(\lambda)\sim \int_{-\infty}^{+\infty} F(x,\lambda)\,dx = \int_{q_0(x)<\lambda}\omega(x,\lambda)\,dx. \tag{3.13} \]

Consider a special case of this formula, when all the surfaces \(L_1^0(x,\sigma)=c\), for different \(0<c<\infty\), are obtained from one another by a similar dilation. By similarity,

\[ S_n\bigl(x,\lambda-q_0(x)\bigr) = S_n(x)\,\sigma_1^{\,n-1}\bigl(x,\lambda-q_0(x)\bigr), \tag{3.14} \]

where \(S_n(x)\) is equal to the area of the surface (3.5) that intersects the semiaxis \(\sigma_1\) at the point \((1,0,\ldots,0)\). In this case

\[ \omega(x,\lambda)= \frac{S_n(x)}{(2\pi)^n n}\, \sigma_1^n\bigl(x,\lambda-q_0(x)\bigr). \tag{3.15} \]

and

\[ N(\lambda)\sim \frac{1}{(2\pi)^n n}\int_{q_0(x)<\lambda} S_n(x)\,\sigma_1^n\bigl(x,\lambda-q_0(x)\bigr)\,dx . \tag{3.16} \]

Let us now transform the following integral:

\[ \int_{-\infty}^{+\infty}\exp\{-L_1^0(x,\sigma)\}\,d\sigma = \int_{q_0(x)}^{\infty}\exp\{-(\lambda-q_0(x))\}\, \]

\[ {}\times \frac{S_n(x)}{n}\,d_\lambda \sigma_1^n\bigl(x,\lambda-q_0(x)\bigr) = \frac{S_n(x)}{n}\int_0^\infty \exp\{-\xi\}\,d_\xi \sigma_1^n(x,\xi). \]

Substituting from this the expression for \(\dfrac{S_n(x)}{n}\) into (3.16), we obtain

\[ N(\lambda)\sim \frac{1}{(2\pi)^n}\int_{q_0(x)<\lambda} \psi(x)\,\sigma_1^n\bigl(x,\lambda-q_0(x)\bigr)\,dx, \tag{3.17} \]

where

\[ \psi(x)= \frac{\displaystyle \int_{-\infty}^{+\infty}\exp\{-L_1^0(x,\sigma)\}\,d\sigma} {\displaystyle \int_0^\infty \exp\{-\xi\}\,d_\xi \sigma_1^n(x,\xi)} . \tag{3.18} \]

This includes the case when \(L_1^0(x,\sigma)\) is spherically symmetric, i.e.

\[ L_1^0(x,\sigma)=P(x,r^2)=P(x,\sigma_1^2+\cdots+\sigma_n^2) \tag{3.19} \]

and satisfies the condition \(P(x,r_1^2)>P(x,r_2^2)\), \(r_1>r_2\).

In this case the surfaces \(L_1^0(x,\sigma)=c\) will be spheres.

The same applies when the family of surfaces of constant level are ellipsoids, i.e. when

\[ L_1^0(x,\sigma)=P(x,\xi)=P(x,a_1\sigma_1^2+\cdots+a_n\sigma_n^2), \tag{3.20} \]

\[ a_i>0\qquad (i=1,2,\ldots,n) \]

and \(P(x,\xi_1)>P(x,\xi_2)\) for \(\xi_1>\xi_2\). This also includes the case when \(L_1^0(x,\sigma)=L_{2m}(x,\sigma)\). For it, as already mentioned, relation (2.12) was proved in [1]. By virtue of the homogeneity of the form \(L_{2m}(x,\sigma)\), surfaces of constant level are obtained from one another by a similarity dilation. Equation (3.6) is rewritten in the form

\[ A_{2m}^{2m,0,\ldots,0}(x)\,\sigma_1^{2m}=\lambda-q_0(x), \]

\[ A_{2m}^{2m,0,\ldots,0}(x)\equiv A(x)>0 \]

by condition 2 (§ 1). Therefore

\[ \sigma_1\bigl(x,\lambda-q_0(x)\bigr) = A^{-1/2m}(x)\bigl(\lambda-q_0(x)\bigr)^{1/2m}, \]

\[ \sigma_1(x,\xi)=A^{-1/2m}(x)\xi^{1/2m}. \tag{3.21} \]

The integral in the denominator of (3.18) is evaluated as follows:

\[ \int_0^\infty \exp\{-\xi\}\,d_\xi \sigma_1^n(x,\xi) = \frac{n}{2m}\,A(x)^{-n/2m} \int_0^\infty \exp\{-\xi\}\,\xi^{\frac{n}{2m}-1}\,d\xi = \]

\[ = \frac{n}{2m} A^{-n/2m}(x)\Gamma\left(\frac{n}{2m}\right) = A^{-n/2m}(x)\Gamma\left(\frac{n}{2m}+1\right). \]

Hence, substituting (3.21) into (3.18) and then into (3.17), we obtain the well-known formula of A. G. Kostyuchenko

\[ N(\lambda)\sim \frac{1}{(2\pi)^n \Gamma\left(\dfrac{n}{2m}+1\right)} \int_{q_0(x)<\lambda} Q(x)\bigl(\lambda-q_0(x)\bigr)^{n/2m}\,dx, \tag{3.22} \]

where

\[ Q(x)=\int_{-\infty}^{+\infty}\exp\{-L_{2m}(x,\sigma)\}\,d\sigma . \]

The same also includes the case when the dimension of the space is equal to 1 and
\(L_1^0(x,\sigma_1)>L_1^0(x,\sigma_2)\), \(\sigma_1>\sigma_2\geq 0\). In this case one must put \(S_1(x)=2\), and (3.16) is rewritten in the form

\[ N(\lambda)\sim \frac{1}{\pi} \int_{q_0(x)<\lambda} \sigma_1\bigl(x,\lambda-q_0(x)\bigr)\,dx . \tag{3.23} \]

For example, if

\[ L^0\left(x,\frac{d}{dx}\right) = \frac{d^4}{dx^4}-2q_2(x)\frac{d^2}{dx^2}, \]

then

\[ N(\lambda)\sim \frac{1}{\pi} \int_{q_0(x)<\lambda} \sqrt{\,q_2(x)+\sqrt{q_2^2(x)+\lambda-q_0(x)}\,}\,dx . \tag{3.24} \]

Thus we have obtained an asymptotic formula for \(N(\lambda)\) in the case when the coefficients of the operator \(L^0\) grow “to the same degree” as the free term \(q_0(x)\), and condition (3.3) is satisfied. Moreover, (3.3) was needed by us only in order to obtain the final formula. And since the asymptotics of the Green’s function was obtained without this condition, the question arises of obtaining formulas for \(N(\lambda)\) in the general case.

In addition, the question of the asymptotics of \(N(\lambda)\) remains completely unexplored for the case when \(q_0(x)\) is subordinate to the coefficients of \(L^0\), for example, \(q_0(x)\equiv 0\). Here the method of estimating the Green’s function \(G_0(x,\eta,t)\) used in this paper does not work, and considerable difficulties arise even in the one-dimensional case.

In conclusion, I wish to express my sincere gratitude to my scientific adviser A. G. Kostyuchenko for the attention he gave during the writing of this work.

References

1. Kostyuchenko A. G. Dokl. Akad. Nauk SSSR, 158, No. 1, 41–44, 1964.
2. Eidelman S. D. Parabolic Systems. Moscow, “Nauka,” 1964.
3. Korenblyum B. M. Dokl. Akad. Nauk SSSR, 104, No. 2, 173–175, 1955.

Received by the editors
May 10, 1966

Moscow State University
named after M. V. Lomonosov