MATHEMATICS

F. G. MASLOVA

A PROBLEM IN THE SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

(Presented by Academician I. M. Vinogradov on 25 IV 1963)

Let (D) be the square in the plane ((x,y)): (0 \leq x \leq 1,\ 0 \leq y \leq 1); let (B) be its boundary. Consider the boundary-value problem:

[
\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\lambda u=0,\qquad (x,y)\in D,\qquad u(x,y)\big|_{B}=0.
]

The eigenvalues of this problem are the numbers (\lambda=\pi^{2}(n^{2}+m^{2})), where (n\geq 1,\ m\geq 1) are integers. The multiplicity of the eigenvalue (\lambda) is equal to the number of representations of the number (\lambda) in the indicated form. The normalized eigenfunctions corresponding to this value of (\lambda) are

[
\omega(x,y)=4\sin^{2}n\pi x\,\sin^{2}m\pi y.
]

One of the well-known problems in the theory of differential operators is the study of the asymptotic properties (as (T\to\infty)) of the function (\Phi(T)), representing the number of eigenvalues not exceeding (T). Each eigenvalue is counted as many times as its multiplicity. The value of (\Phi(T)) is equal to the number of lattice points in the region (x^{2}+y^{2}\leq T/\pi^{2}), (x\geq 1,\ y\geq 1).

With the aid of the estimate of En Wen-lin ({}^{1}) we obtain the number of lattice points in a circle

[
\Phi(T)=
\sum_{\substack{n^{2}+m^{2}\leq T/\pi^{2}\ n\geq 1,\ m\geq 1}} 1
=
\frac{T}{4\pi}-\frac{\sqrt{T}}{\pi}+O!\left(T^{12/37+\varepsilon}\right),
\tag{1}
]

(\varepsilon>0). Closely connected with this problem is the problem of the asymptotic behavior, as (T\to\infty), of the quantity

[
4\sum_{\substack{n^{2}+m^{2}\leq T/\pi^{2}\ n\geq 1,\ m\geq 1}}
\sin^{2} n\pi x\,\sin^{2} m\pi y,
]

where ((x,y)) is a fixed interior point of the square (D). Denote by (l(x,y)) the distance from the point ((x,y)) to the boundary of the square. The following result is obtained without difficulty:

Theorem 1. As (T\to\infty),

[
4\sum_{\substack{n^{2}+m^{2}\leq T/\pi^{2}\ n\geq 1,\ m\geq 1}}
\sin^{2} n\pi x\,\sin^{2} m\pi y
=
\frac{T}{4\pi}
+
O!\left(\frac{\sqrt{T}}{l(x,y)}\right).
\tag{2}
]

Using the method for estimating trigonometric sums of I. M. Vinogradov ({}^{2}), this theorem can be strengthened.

Theorem 2. As (T\to\infty),

[
4\sum_{\substack{n^{2}+m^{2}\leq T/\pi^{2}\ n\geq 1,\ m\geq 1}}
\sin^{2} n\pi x\,\sin^{2} m\pi y
=
]

[

\frac{T}{4\pi}
+
O!\left(\frac{T^{1/3}\ln T}{l(x,y)}\right)
+
O!\left(\frac{1}{(l(x,y))^{2}}\right)
+
O!\left(\frac{T^{1/4}}{(l(x,y))^{3/2}}\right).
\tag{3}
]

Apparently, by carrying out the calculation more economically, the exponent (1/3) in estimate (3) can be somewhat lowered.

In equalities (2), (3), the constants in the (O)-symbols are absolute.

V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
24 IV 1963

CITED LITERATURE

({}^{1}) Yin Wen-lin, Sci. Sinica, 11, No. 1, 10 (1962). ({}^{2}) I. M. Vinogradov, Selected Works, Moscow, 1952.