A. A. ARSEN'EV

ASYMPTOTIC PROPERTIES OF THE TRACE OF THE SPECTRAL FUNCTION OF A SELF-ADJOINT ELLIPTIC DIFFERENTIAL OPERATOR OF SECOND ORDER

(Presented by Academician S. L. Sobolev, March 11, 1964)

I. In an \(N\)-dimensional Euclidean space consider the differential equation

\[ Lu=\frac{1}{2}\sum_{i,j=1}^{N}\frac{\partial}{\partial x_i} \left(a_{ij}(x)\frac{\partial u}{\partial x_j}\right)-V(x)u=0, \qquad x=(x_1,x_2,\ldots,x_N),\quad N\geq 2. \tag{1} \]

With respect to the coefficients \(a_{ij}(x)\) and \(V(x)\) we make the following assumptions:

1. \(a_{ij}(x)\in C^{(1,\mu)}\); \(a_{ij}(x)\) are bounded in the whole space \(R(N)\).

2. \[ \sum_{i,j=1}^{N}a_{ij}(x)\alpha_i\alpha_j\geq c\sum_{i=1}^{N}\alpha_i^2; \]
   \(c>0\) and does not depend on \(x\).

3. \(V(x)\geq 0\), and for any point \(x\) there is a neighborhood \(O(x)\), a constant \(c>0\), and \(\gamma>0\), such that for \(x_1\in O(x)\) and \(x_2\in O(x)\) the inequality

\[ |V(x_1)-V(x_2)|<c|x_1-x_2|^\gamma \]

is satisfied.

Let \(\theta(x,y,\lambda)\) be the spectral function of the operator (1). Its construction is described in the work of L. Gårding \((^1)\).

In the case of a purely discrete spectrum,

\[ \theta(x,y,\lambda)=\sum_{\lambda_n\leq \lambda}\psi_n(x)\psi_n(y), \]

where \(\psi_n(x)\) and \(\lambda_n\) are the eigenfunctions and eigenvalues of the problem

\[ L\psi_n+\lambda_n\psi_n=0,\qquad \int \psi_n^2(x)\,dx=1. \]

It is known that

\[ \theta^2(x,y,\lambda)\leq \theta(x,x,\lambda)\theta(y,y,\lambda). \]

In \((^2,^3)\) the asymptotic behavior of the function \(\theta(x,y,\lambda)\) as \(\lambda\to\infty\) was studied. For the case of the Laplace and Schrödinger equations the function \(\theta(x,y,\lambda)\) was studied in detail by B. M. Levitan \((^5,^6)\).

In \((^3)\) it is proved that

\[ \theta(x,x,\lambda)\sim \frac{\lambda^{N/2}} {(2\pi)^{N/2}\Gamma(N/2+1)\sqrt{\det\|a_{ij}(x)\|}}. \tag{2} \]

The asymptotic formula (2) is uniform in \(x\) for \(x\) belonging to any bounded set; however, it is not uniform in \(x\) for \(x\) belonging to the whole domain of variation \(R(N)\) of the independent variables. Therefore the problem arises of describing in some way the behavior of the function \(\theta(x,x,\lambda)\) as a whole.

II. Let

\[ \theta(f,\lambda)=\int_{-\infty}^{\infty}\theta(x,x,\lambda)f(x)\,dx. \]

If \(f(x)\in L_1\), then, as is not difficult to show,

\[ \theta(f,\lambda)\sim \frac{\lambda^{N/2}}{(2\pi)^{N/2}\Gamma(N/2+1)} \int_{V(x)<\lambda}\frac{f(x)\,dx}{\sqrt{\det\|a_{ij}(x)\|}}. \]

If \(f(x)\notin L_1\), it may turn out that \(\theta(f,\lambda)=\infty\) for \(\lambda<\infty\). However, imposing certain restrictions on the asymptotic behavior of the functions \(f(x)\) and \(V(x)\), one can prove the formula

\[ \theta(f,\lambda)\sim \frac{\lambda^{N/2}}{(2\pi)^{N/2}\Gamma(N/2+1)} \int_{V(x)<\lambda} f(x)\left(1-\frac{V(x)}{\lambda}\right)^{N/2}\,dx. \]

For the Schrödinger equation, when \(f(x)=1\), this formula has long been known \((^4)\). For \(f(x)=V^\sigma(x)\) it was obtained by B. M. Levitan in connection with the problem of summing Fourier series by Riesz means \((^6)\).

III. We formulate the main result of the present paper. Let \(f(x)\geq 0\). Put, by definition,

\[ B(f,\lambda)= \frac{\lambda^{N/2}}{(2\pi)^{N/2}\Gamma(N/2+1)} \int_{V(x)<\lambda} \frac{f(x)}{\sqrt{\det\|a_{ij}(x)\|}} \left(1-\frac{V(x)}{\lambda}\right)^{N/2}\,dx. \]

Theorem 1. 1. Let the function \(B(f,\lambda)\) satisfy the condition of Korenblum’s Tauberian theorem \((^7)\): there exist \(\lambda_0\) and \(\gamma\) such that for all \(\lambda'>\lambda''>\lambda_0\) the inequality

\[ \frac{B(f,\lambda')}{B(f,\lambda'')} \leq \left(\frac{\lambda'}{\lambda''}\right)^\gamma \]

holds.

1. Suppose that as \(\delta\to 0\), uniformly with respect to \(\lambda>\lambda_0\), the equality

\[ \lim_{\delta\to 0} \frac{ \displaystyle \int_{\substack{\max\limits_{|u|<\delta}V(x+u)<\lambda}} f(x)\,dx - \displaystyle \int_{\substack{\min\limits_{|u|<\delta}V(x+u)<\lambda}} f(x)\,dx }{ \displaystyle \int_{V(x)<\lambda} f(x)\,dx } =0. \]

1. There exists an \(\alpha\in(0,4)\) such that for sufficiently large \(|x|\geq R_0\) the inequality

\[ V(x)>|x|^\alpha \]

holds.

1. There exist constants \(A>0\) and \(\varepsilon>0\) such that for some \(q\)

\[ 1<q<\frac{1}{1-2/N}, \]

\[ \int f^q(x)e^{-A|x|^{\frac{\alpha-\varepsilon}{2}}}\,dx<\infty. \]

Then, as \(\lambda\to\infty\),

\[ \theta(f,\lambda)\sim B(f,\lambda). \]

IV. For the case of the Schrödinger equation, i.e. if

\[ Lu=\frac{1}{2}\Delta u - V(x)u, \tag{3} \]

the result obtained can be somewhat sharpened.

Theorem 2. Let \(\theta(x,y,\lambda)\) be the spectral function of equation (3),

\[ B(f,\lambda)= \frac{\lambda^{N/2}}{(2\pi)^{N/2}\Gamma(N/2+1)} \int_{V(x)<\lambda} f(x)\left(1-\frac{V(x)}{\lambda}\right)^{N/2}\,dx. \]

Suppose:

1. \(B(f,\lambda)\) satisfies condition 1 of Theorem 1.

2. Uniformly in \(\lambda\),

\[ \lim_{\delta\to 0} \int_{\max_{|u|<\delta} V(x+u)<\lambda} f(x)\,dx \Big/ \int_{V(x)<\lambda} f(x)\,dx =1. \]

1. For some \(p>N/2\),

\[ \lim_{t\to 0} \int\left[ 1- \int f(x-\sqrt{t}\,u)e^{-tV(x)}\,dx \Big/ \int f(x)e^{-tV(x)}\,dx \right]^p e^{-(1-\varepsilon)p u^2}\,du=0. \]

Then

\[ \theta(f,\lambda)\sim B(f,\lambda). \]

The proof of this theorem is carried out by Ray’s method, based on exact estimates of the Green’s function of the Cauchy problem for the heat-conduction equation

\[ \frac{\partial u}{\partial t}=Lu,\qquad u(x,0)=u_0(x) \]

by means of continual integrals. It is rather cumbersome, and we omit it.

V. Let us say a few words concerning conditions 1–4. Condition 1 means, roughly speaking, that the function \(B(f,\lambda)\) must grow no faster than some power and must not have excessively large jumps.

Let

\[ \varphi(\lambda)=\int_{V(x)<\lambda} f(x)\,dx. \]

Conditions 1 and 2 are fulfilled if the following requirements are fulfilled:

1. There exists a constant \(C>0\) such that for all \(\theta\) from some interval \(\theta\in[1-\delta,\,1+\delta]\) and \(\lambda>\lambda_0\),

\[ \lambda\varphi'(\theta\lambda)>C\varphi(\lambda). \]

2.

\[ \max_{|u|<\delta} V(x+u)=V(x)(1+\rho(\delta)), \]

\[ \min_{|u|<\delta} V(x+u)=V(x)(1+\rho(\delta)), \qquad \rho(\delta)\to 0,\ \delta\to 0, \]

and these relations are fulfilled uniformly in \(x\).

I express my gratitude to my scientific adviser Prof. A. A. Samarskii and to Prof. B. M. Levitan for valuable discussion of the results of the work, and to I. A. Shishmarev for reading the manuscript.

Moscow State University
named after M. V. Lomonosov

Received
1 III 1964

REFERENCES

1. L. Gårding, C. r. II Congr. Math. Scand., 1953, p. 44; Collected translations, Mathematics, 1, 3 (1957).
2. L. Gårding, Fähr. Kenigl. Fys. Sälskapets i Lund., 24, No. 21 (1954); Collected translations, Mathematics, 1, 3 (1957).
3. M. F. Bureau, C. R., 249, No. 13 (1961).
4. J. S. de Wet, F. Mandl, Proc. Roy. Soc. A, 200, 572 (1950).
5. B. M. Levitan, Tr. Moscow Math. Soc., 4, 237 (1955).
6. B. M. Levitan, Mat. Sb., 41 (83), no. 4, 439 (1957).
7. B. I. Korenblum, DAN, 88, 743 (1953).
8. D. Ray, Trans. Amer. Math. Soc., 77, 299 (1951).