UDC 517.946.9:534.121.2

ASYMPTOTIC DISTRIBUTION OF EIGENFREQUENCIES OF A PLANE MEMBRANE IN THE CASE OF SEPARABLE VARIABLES

N. V. KUZNETSOV

§ 1. INTRODUCTION

In 1912 H. Weyl [1] proved the following theorem on the asymptotic distribution of the eigenvalues of a plane membrane.

Let \(\lambda_n,\ \lambda_1 \leqslant \lambda_2 \leqslant \cdots\) be the eigenvalues of the boundary-value problem

\[ \frac{\partial^2 \varphi}{\partial x_1^2} + \frac{\partial^2 \varphi}{\partial x_2^2} + \lambda \varphi = 0, \qquad \left.\frac{\partial \varphi}{\partial n}\right|_{\Gamma} = 0 \quad \text{or} \quad \left.\varphi\right|_{\Gamma} = 0, \tag{1} \]

posed in a bounded domain \(D\), bounded by a piecewise smooth boundary \(\Gamma\). Then as \(\lambda \to \infty\)

\[ N(\lambda) = \sum_{\lambda_n \leq \lambda} 1 = \frac{\operatorname{mes} D}{4\pi}\,\lambda + o(\lambda). \tag{2} \]

H. Weyl put forward the hypothesis that in the asymptotic formula (2) it is possible to single out a second regular term [2]. It is known that, if this hypothesis is true, then equality (2) must have the form

\[ N(\lambda) = \frac{\operatorname{mes} D}{4\pi}\,\lambda \pm \frac{\operatorname{mes} \Gamma}{4\pi}\,\sqrt{\lambda} + o(\sqrt{\lambda}), \tag{3} \]

where the upper sign corresponds to the boundary condition
\[ \left.\frac{\partial \varphi}{\partial n}\right|_{\Gamma}=0, \]
and the lower sign to the condition
\[ \left.\varphi\right|_{\Gamma}=0. \]
Until now this hypothesis has been confirmed only by the example of a rectangular membrane.

In the present paper we give a confirmation of H. Weyl’s hypothesis in the case of all domains \(D\) for which the variables in equation (1) separate: a circle, ellipse, parabolic segment, circular and elliptic sector, etc. Its principal result is

Theorem 1. Let, in the domain \(D\), the eigenvalues of problem (1) be obtainable by the method of separation of variables. Then, for the distribution function of the eigenvalues of the boundary-value problem

\[ \frac{\partial^2 \varphi}{\partial x_1^2} + \frac{\partial^2 \varphi}{\partial x_2^2} + \lambda \varphi = 0, \qquad \left.\frac{\partial \varphi}{\partial n} + \sigma \varphi\right|_{\Gamma} = 0 \tag{4} \]

the asymptotic equality is valid

\[ N(\lambda) = \frac{\operatorname{mes} D}{4\pi}\,\lambda \pm \frac{\operatorname{mes} \Gamma}{4\pi}\,\sqrt{\lambda} + o(\sqrt{\lambda}), \tag{5} \]

where the upper sign is taken in the case when everywhere on \(\Gamma\), \(|\sigma|<\infty\), and the lower one in the boundary condition \(\varphi|_{\Gamma}=0\).

Remark 1. It is known (see, for example, [3]) that the coordinates in which the variables in equation (1) separate are exhausted by the following: Cartesian, polar, parabolic, and elliptic. We prove the theorem by considering all these cases in turn; for the circular membrane the theorem was proved jointly with B. V. Fedosov.

Remark 2. It can be shown that, for the case of separation of variables in polar and elliptic coordinates, the remainder term in (5) is equal to \(O(\lambda^{1/3})\), and for the case of separation of variables in parabolic coordinates, to \(O(\lambda^{2/5})\), [4].

Remark 3. Let us note that H. Weyl’s hypothesis on the possibility of isolating the second term in the asymptotic formula (2) also contains the assumption of a certain dependence of this term on the boundary conditions. The possible character of this dependence is illustrated by the following result [4]:

Let on a portion of the boundary \(\gamma_k\) of the domain \(D\), which is a coordinate line, the function \(\sigma(s)\) in (4) either be bounded and integrable, or be everywhere on \(\gamma_k\) equal to \(\infty\) (i.e., the boundary condition on \(\gamma_k\) has the form \(\varphi|_{\gamma_k}=0\)). Then the second term in (5) should be replaced by
\[ \frac{\sqrt{\lambda}}{4\pi}\oint_{\Gamma} h(\sigma(s))\,ds, \]
where \(h(\sigma)=+1\) for \(|\sigma|<\infty\), and \(h(\infty)=-1\).

§ 2. PLAN OF THE PROOF

The basic idea is that, in the case of separating variables, the investigation of the asymptotic distribution of the eigenvalues of the Laplace operator reduces to a problem in the metric theory of numbers concerning the number of integer points in plane domains.

Indeed, suppose that the variables in problem (1) separate, i.e., the transformation
\[ x_1=x_1(\xi,\eta),\qquad x_2=x_2(\xi,\eta), \]
mapping the domain \(D\) onto a rectangle \(\widetilde D\) in the \((\xi,\eta)\)-plane, brings equation (1) to the form
\[ F_1(\xi,\varphi)+F_2(\eta,\varphi)=0, \]
where \(F_1\) contains only functions of \(\xi\) and derivatives with respect to \(\xi\), and \(F_2\) only functions of \(\eta\) and derivatives with respect to \(\eta\). Then the eigenfunctions of problem (1) are exhausted by products of the form \(v_n(\xi)w_m(\eta)\), where
\[ F_1(\xi,v_n(\xi))=0,\qquad F_2(\eta,w_m(\eta))=0 \tag{2.1} \]
and the functions \(v_n,w_m\) satisfy homogeneous boundary conditions. We number them so that the \(n\)-th function \((n\geq 1)\) has \(n-1\) zeros inside the corresponding interval, and denote by \(\lambda(n,m)\) the eigenvalue corresponding to the eigenfunction \(v_n(\xi)w_m(\eta)\). Then the number of eigenvalues of problem (1) not exceeding \(\lambda\) is equal to the number of integer points \(n,m\geq 1\) in the domain \(B(\lambda)\) of the \((n,m)\)-plane, defined by the inequality
\[ \lambda(n,m)\leq \lambda. \tag{2.2} \]

The main difficulty of the problem consists in finding equations describing the boundary of the domain \(B(\lambda)\). For this purpose we have to obtain uniform asymptotic estimates for the solutions of equations (2.1), which in our case have the form
\[ \frac{d^2 v}{d\xi^2}+\lambda p_1(\xi,\alpha)v=0,\qquad \frac{d^2 w}{d\eta^2}+\lambda p_2(\eta,\alpha)w=0. \tag{2.3} \]

(here \(\alpha\) is the separation parameter). To obtain the estimates mentioned, we use the so-called “model” method, based on the Liouville–Green transformation of equations of the form (2.3). This method is as follows. In the equation

\[ \frac{d^2 w}{d \xi^2}+\lambda p(\xi,\alpha)w=0 \tag{2.4} \]

we make a change of the variable and of the function according to the formula \(w(\xi)=\dot{x}^{-1/2}(\xi)y(x)\). Then \(y(x)\) satisfies the equation

\[ y''+\lambda \dot{x}^{-2}(\xi)p(\xi,\alpha)y=f(x)y,\qquad \left(\dot{\ }=\frac{d}{d\xi},\quad {}'=\frac{d}{dx}\right), \tag{2.5} \]

where

\[ f(x)=\frac{1}{2}\frac{\dddot{x}}{\dot{x}^{3}}-\frac{3}{4}\frac{\ddot{x}^{2}}{\dot{x}^{4}}. \]

Next we choose the function \(x(\xi)\) in such a way that \(f(x)\) is continuous and equation (2.5) for \(f(x)\equiv 0\) is as simple as possible. (For example, if the function \(p(\xi,\alpha)\) has no zeros for \(a\leq \xi\leq b\), we set \(\dot{x}^{-2}p=1\); if \(p(\xi,\alpha)\) has two real zeros \(\xi_1,\xi_2\) and \(p(\xi,\alpha)>0\) for \(\xi_1<\xi<\xi_2\), we set \(\dot{x}^{-2}p=c^2-x^2\), \(x(\xi_1)=-c\), with \(c\) chosen appropriately.) After this, equation (2.5) is studied by the standard method of reduction to an integral equation and solution of the latter by the method of successive approximations.

As a result, we find conditions on \(\lambda\) and \(\alpha\) (for large \(\lambda\)) under which equation (2.4) has a solution \(v(\xi)\) satisfying boundary conditions of the form \(-v'(a)+\sigma_1v(a)=v'(b)+\sigma_2v(b)=0\) (\(\sigma_1,\sigma_2\) are real numbers) and having exactly \(n-1\) zeros inside the interval \((a,b)\). Under the assumption that equation (2.4) has no more than two turning points, these conditions have the form

\[ \frac{\sqrt{\lambda}}{\pi}\int_a^b \operatorname{Re}\sqrt{p(t,\alpha)}\,dt+\chi(\lambda,\alpha,\sigma_1,\sigma_2)=n, \tag{2.6} \]

where, for large \(\lambda\), the function \(\chi\) differs from a constant only by small terms. As a result, it is established that the boundary of the region \(B(\lambda)\), defined by inequality (2.2), for large \(\lambda\) is given by the equation

\[ n=\frac{\sqrt{\lambda}}{\pi}\int \operatorname{Re}\sqrt{p_1(t,\alpha)}\,dt+n_0+\cdots, \]

\[ m=\frac{\sqrt{\lambda}}{\pi}\int \operatorname{Re}\sqrt{p_2(t,\alpha)}\,dt+m_0+\cdots, \tag{2.7} \]

where \(n_0,m_0\) are constants.

The derivation of equations (2.7) and the estimate of the remainder term constitute the main difficulty; after this, to obtain Theorem 1 it suffices to use the known theorems of the metric theory of numbers of van der Corput [5] and I. M. Vinogradov [6].

It should be noted that such a method was first applied in the work of E. C. Titchmarsh [7].

§ 3. ESTIMATES OF SOLUTIONS OF AN EQUATION WITH TWO TURNING POINTS

We shall derive equations (2.6) for the eigenvalues only in the case when equation (2.4) has two turning points. The behavior of solutions of an equation of the form (2.4) with one turning point (or without turning points) is well known, and we may restrict ourselves to the formulation of the result.

Lemma 1. Let, in equation (2.4), the function \(p(\xi,\alpha)\) be twice continuously differentiable with respect to \(\xi\), continuous with respect to \(\alpha\), and have a positive lower bound for \(\xi \in [a,b]\), \(\alpha \in [\alpha_1,\alpha_2]\). Then this equation has a solution \(v(\xi)\) satisfying the boundary conditions

\[ -v'(a)+\sigma_1 v(a)=v'(b)+\sigma_2 v(b)=0 \]

and having exactly \(n-1\) zeros inside \((a,b)\), if

\[ \frac{\sqrt{\lambda}}{\pi}\int_a^b \sqrt{p(t,\alpha)}\,dt+\omega(\sigma_1,\sigma_2)+O\left(\frac{1}{\sqrt{\lambda}}\right)=n, \tag{3.1} \]

where \(\omega(\sigma_1,\sigma_2)=1\) for \(\sigma_1^2+\sigma_2^2<\infty\), \(\omega(\sigma_1,\sigma_2)=\dfrac12\) for \(\sigma_1=\infty,\ |\sigma_2|<\infty\) or \(\sigma_2=\infty,\ |\sigma_1|<\infty\), and \(\omega(\infty,\infty)=0\).

For the proof of this lemma it suffices to use the estimates of F. W. J. Olver [8] and then argue as in [9], pp. 220–222.

Lemma 2. Let, in the equation

\[ \frac{d^2v}{d\xi^2}+\lambda p(\xi,\alpha)v=\varphi(\xi)v,\qquad 0<\xi<b \tag{3.2} \]

for \(\alpha\in[\alpha_1,\alpha_2]\), the function \(p(\xi,\alpha)\) have exactly one simple zero \(\xi_0\in(a,b)\) and be subject to one of the following restrictions:

1) \(p(\xi,\alpha)\), together with its derivatives with respect to \(\xi\) up to second order, is continuous on the closed interval \([0,b]\); moreover, \(\varphi(\xi)\) is continuous on the same interval;

2) as \(\xi\to 0\), \(p(\xi,\alpha)\) is equal to

\[ -\frac{\alpha^2}{\xi^2}+O(1) \]

and the function

\[ \frac{\alpha^2}{\xi^2}+p(\xi,\alpha) \]

is twice continuously differentiable on \([0,b]\); moreover,

\[ \varphi(\xi)=-\frac{1}{4\xi^2}. \]

Then the condition for the existence of a solution \(v(\xi)\) of equation (3.2), having exactly \(n-1\) zeros inside \((0,b)\) and satisfying boundary conditions of the form

\[ -v'(0)+\sigma_1 v(0)=v'(b)+\sigma_2 v(b)=0 \]

(where the first condition in case 2) is replaced by the requirement that the solution be finite as \(\xi\to0\)), is the equality

\[ \frac{\sqrt{\lambda}}{\pi}\int_{\xi_0}^{b}\sqrt{p(t,\alpha)}\,dt+\gamma(\sigma_2)+O(\exp(-\sqrt{\lambda}R_1))+O\left(\frac{1}{\sqrt{\lambda}R_2}\right)=n, \tag{3.3} \]

where

\[ R_1=2\int_0^{\xi_0}\sqrt{-p(t,\alpha)}\,dt,\qquad R_2=\int_{\xi_0}^{b}\sqrt{p(t,\alpha)}\,dt \]

and \(\gamma(\widetilde{\sigma}_2)=3/4\) for \(|\sigma_2|<\infty\), \(\gamma(\infty)=1/4\).

To obtain Lemma 2, one must use estimates of the solutions from [10], pp. 116–126, and the asymptotic expansions for the \(n\)-th zero of the functions \(\operatorname{Ai}(z)\) and \(\operatorname{Ai}'(z)^*\) [11].

Let now the function \(p(\xi,\alpha)\) in equation (2.4), for \(\alpha\in[\alpha_0-\delta,\alpha_0+\delta]\) (by \(\delta>0\) we denote a fixed sufficiently small number), have two close zeros \(\xi_1(\alpha)\) and \(\xi_2(\alpha)\), merging into one double zero as \(\alpha\to\alpha_0\). We shall assume that \(p(\xi,\alpha)\) is an analytic function of \(\xi\) (real for real \(\xi\)) in some rectangle \(a-\widetilde{\delta}\leq \operatorname{Re}\xi\leq b+\widetilde{\delta}\), \(|\operatorname{Im}\xi|\leq\widetilde{\delta}\), and that in this rectangle there are no zeros of \(p(\xi,\alpha)\) other than \(\xi_1,\xi_2\).

We always choose the transformation of equation (2.4) to the form (2.5) to be real; therefore we consider separately the following cases of the location of the zeros \(\xi_1\) and \(\xi_2\).

A). The zeros \(\xi_1,\xi_2\) of the function \(p(\xi,\alpha)\) are complex (for \(a\leq \xi\leq b,\ p>0\)). We may assume that they are purely imaginary: \(\xi_1=-\xi_2=i\xi_0(\alpha)\), \(\xi_0>0\). In this case put

\[ \dot{x}^{-2}(\xi)p(\xi,\alpha)=x^2+c^2,\qquad x(0)=0. \tag{3.4} \]

and choose \(c^2(\alpha)\) so that the equality \(x(i\xi_0)=+ic\) holds. Integrating (3.4), we obtain for \(x(\xi)\) the equation (\(x>0\) for \(\xi>0\))

\[ \zeta_1(x)=\frac{1}{2}x\sqrt{x^2+c^2} +\frac{1}{2}c^2\ln\left\{\frac{x}{c}+\sqrt{\frac{x^2}{c^2}+1}\right\} = \]

\[ =\int_0^{\xi_0}\sqrt{p(t,\alpha)}\,dt. \tag{3.5} \]

It is easy to verify that \(x(i\xi_0)=ic\) when

\[ c^2(\alpha)=\frac{4}{\pi}\int_0^{\xi_0}\sqrt{p(it,\alpha)}\,dt, \]

and then, by virtue of the oddness of \(\zeta_1(x)\), \(x(-i\xi_0)=-ic\). Thus the function \(f(x)\) in (2.5), equal to

\[ \left(\frac{x^2+c^2}{p}\right)^{1/4} \frac{d^2}{dx^2} \left(\frac{p}{x^2+c^2}\right)^{1/4}, \]

is continuous for all \(\xi\in[a,b]\) and \(\alpha\in[\alpha_0-\delta,\alpha_0+\delta]\).

B). If the zeros \(\xi_1,\xi_2\) of the function \(p(\xi,\alpha)\) are real and \(p(\xi,\alpha)<0\) for \(\xi_1<\xi<\xi_2\), we put

\[ \dot{x}^{-2}(\xi)p(\xi,\alpha)=x^2-c^2,\qquad x(\xi_1)=-c, \tag{3.6} \]

i.e. \(x(\xi)\) is determined from the equation

\[ \zeta_2(x)=\frac{1}{2}x\sqrt{x^2-c^2} -\frac{1}{2}c^2\ln\left\{\frac{x}{c}+\sqrt{\frac{x^2}{c^2}-1}\right\} = \]

\[ ^*\) By \(\operatorname{Ai}(z)\) we denote the Airy function; for its properties see, for example, [10], pp. 113–116 or [11]. \]

\[ = \int_{\xi_1}^{\xi} \sqrt{p(t,\alpha)}\,dt . \tag{3.7} \]

The equation \(x(\xi_2)=+c\) gives in this case

\[ c^2(\alpha)=\frac{2}{\pi}\int_{\xi_1}^{\xi_2}\sqrt{-p(t,\alpha)}\,dt . \]

C). Finally, if the zeros \(\xi_1\) and \(\xi_2\) are real, but \(p(\xi,\alpha)>0\) for \(\xi_1<\xi<\xi_2\), we determine \(x(\xi)\) from the equation

\[ \dot{x}^{-2}(\xi)p(\xi,\alpha)=c^2-x^2,\qquad x(\xi_1)=-c, \tag{3.8} \]

where

\[ c^2(\alpha)=\frac{2}{\pi}\int_{\xi_1}^{\xi_2}\sqrt{p(t,\alpha)}\,dt . \]

Thus, our problem is reduced to obtaining estimates for solutions of the equation (we put \(\lambda=\mu^2\))

\[ y''+\mu^2 q(x)y=f(x)y,\qquad x_1\leq x\leq x_2, \tag{3.9} \]

where \(q(x)=x^2\pm c^2\) or \(c^2-x^2\), and \(f(x)\) is a continuous function.

Let \(q(x)=x^2-c^2\). Following [12], denote by \(U(a,z)\) the solution of Weber’s equation

\[ \frac{d^2w}{dz^2}=\left(\frac{1}{4}z^2+a\right)U(a,z), \]

normalized by the condition, as \(z\to0^*\),

\[ U(a,z)= \frac{\Gamma(1/2)2^{-a/2-1/4}}{\Gamma\left(3/4+\frac{a}{2}\right)} + \frac{\Gamma\left(-\frac{1}{2}\right)2^{-a/2-3/4}}{\Gamma\left(\frac{1}{4}+\frac{a}{2}\right)}z +O(z^2). \tag{3.10} \]

For real \(a,x\), denote by \(W(a,x)\) the function connected with \(U(ia,e^{-i\pi/4}x)\) by the equality (for more detail see [12])

\[ (2k)^{-1/2}W(a,x)+i(2k)^{+1/2}W(a,-x)= \]

\[ =\exp\left(\frac{\pi a}{4}+\frac{i}{2}\arg\Gamma\left(\frac{1}{2}+ia\right)+\frac{i\pi}{8}\right) U\left(ia,e^{-i\pi/4}x\right), \tag{3.11} \]

where

\[ k=\{1+\exp(2\pi a)\}^{1/2}-\exp(\pi a), \]

and by \(\arg\Gamma\left(\frac{1}{2}+ia\right)\) is meant that branch which is equal to \(a\ln a-a+O(a^{-1})\) as \(a\to+\infty\). With these notations, equation (3.9) (for \(q(x)=x^2-c^2\)) is equivalent to the integral equation

\[ y(x)=c_1W_+(x)+c_2W_-(x)+\frac{1}{\sqrt{\mu}}\int_{x_0}^{x}B(x,t)f(t)y(t)\,dt, \tag{3.12} \]

\[ {}^*)\ \text{In the notation of [13] (pp. 176–183), this is the parabolic-cylinder function } D_{-a-1/2}(z). \]

where, for brevity, we have put

\[ W_{\pm}(x)=W\left(\frac12 \mu c^2,\ \pm x\sqrt{2\mu}\right), \]

\[ B(x,t)=W_{+}(x)W_{-}(t)-W_{+}(t)W_{-}(x). \tag{3.13} \]

First let \(c_1=1,\ c_2=0,\ x_0=x_2\). The solution of equation (3.13) for this choice of constants will be denoted by \(y_{+}(x)\). We have

\[ y_{+}(x)=\sum_{n=0}^{\infty} y_{+}^{(n)}(x),\qquad y_{+}^{(0)}(x)=W_{+}(x), \]

\[ y_{+}^{(n+1)}(x)=\frac{1}{\sqrt{\mu}}\int_{x_2}^{x} B(x,t)f(t)y_{+}^{(n)}(t)\,dt . \tag{3.14} \]

The iteration series (3.14) for \(y_{+}(x)\) certainly converges (since equation (3.12) is of Volterra type). Let us estimate the quantities \(y_{+}^{(n)}(x)\). For this we use an inequality of F. W. J. Olver [14]:

\[ |W_{\pm}(x)|\leq \psi_{\pm}(x)\equiv \gamma\left(1+(\mu c^2)^{1/12} +\mu^{1/4}|x^2-c^2|^{1/4}\right)^{-1} \left|\exp\left(\pm\frac{\pi}{4}\mu c^2+i\mu\xi_2(x)\right)\right|, \tag{3.15} \]

where \(\xi_2(x)\) is defined by equality (3.7), and \(\gamma\) is an absolute constant. It is easy to verify that for all \(x,t\) such that \(x\leq t\leq x_2\),

\[ \operatorname{Re}\{i\xi_2(x)-i\xi_2(t)\}\leq 0. \tag{3.16} \]

Using (3.15) and (3.16), we obtain for \(y_{+}^{(1)}(x)\) the estimate

\[ |y_{+}^{(1)}(x)|\leq \psi_{+}(x)\frac{2\gamma^2}{\sqrt{\mu}}\frac{F_{+}(x)}{1!}, \tag{3.17} \]

where

\[ F_{+}(x)=\int_{x}^{x_2} |f(t)| \left(1+(\mu c^2)^{1/12}+\mu^{1/4}|t^2-c^2|^{1/4}\right)^{-2}\,dt . \tag{3.18} \]

Next suppose

\[ |y_{+}^{(n)}(x)|\leq \psi_{+}(x)\left(\frac{2\gamma^2}{\sqrt{\mu}}\right)^n \frac{|F_{+}(x)|^n}{n!}. \]

Then

\[ |y_{+}^{(n+1)}(x)| =\frac{1}{\sqrt{\mu}} \left|\int_{x_2}^{x} B(x,t)f(t)y_{+}^{(n)}(t)\,dt\right| \leq \]

\[ \leq \psi_{+}(x)\frac{2\gamma^2}{\sqrt{\mu}} \int_{x}^{x_2}|f(t)|\left(1+(\mu c^2)^{1/12} +\mu^{1/4}|t^2-c^2|^{1/4}\right)^{-2} \left(\frac{2\gamma^2}{\sqrt{\mu}}\right)^n \frac{[F_{+}(t)]^n}{n!}\,dt = \]

\[ = \psi_+(x)\left(\frac{2\gamma^2}{\sqrt{\mu}}\right)^{n+1} \frac{[F_+(x)]^{n+1}}{(n+1)!}. \tag{3.19} \]

Therefore inequality (3.19) is valid for all \(n\), and for \(y_+(x)\) we obtain

\[ \left|y_+(x)-W_+(x)\right| \leq \psi_+(x)\left\{\exp\left(\frac{2\gamma^2}{\sqrt{\mu}}F_+(x)\right)-1\right\}. \tag{3.20} \]

Entirely analogous arguments prove the existence of a second solution \(y_-(x)\) of equation (3.9), for which the inequality

\[ \left|y_-(x)-W_-(x)\right| \leq \psi_-(x)\left\{\exp\left(\frac{2\gamma^2}{\sqrt{\mu}}F_-(x)\right)-1\right\}, \tag{3.21} \]

holds, where

\[ \psi_-(x)= \frac{\gamma\left|\exp\left(\frac{\pi\mu c^2}{4}+i\mu \xi_2(x)\right)\right|} {1+(\mu c^2)^{1/12}+\mu^{1/4}|x^2-c^2|^{1/4}}, \]

\[ F_-(x)=F_+(x_1)-F_+(x). \tag{3.22} \]

In order to obtain estimates for the derivatives, we differentiate (3.12) and substitute series of the form (3.14) into the right-hand side instead of \(y_\pm(t)\); using the estimates obtained for \(y_\pm^{(n)}(t)\) and the inequalities for the derivatives of Weber functions from [14], we arrive at the inequalities

\[ \left| \frac{dy_\pm(x)}{dx} - \frac{dW_\pm(x)}{dx} \right| \leq \sqrt{\mu}\left(1+(\mu c^2)^{1/12} +\right. \]

\[ \left. +\mu^{1/4}|x^2-c^2|^{1/4}\right)^2 \psi_\pm(x) \left\{ \exp\left(\frac{2\gamma^2}{\sqrt{\mu}}F_\pm(x)\right)-1 \right\}. \tag{3.23} \]

The arguments in the other cases (i.e., when \(q(x)=x^2+c^2\) or \(c^2-x^2\)) differ little from those just described. As a result we obtain the following inequalities.

The equation \(y''+\mu^2(x^2+c^2)y=f(x)y\) has solutions \(y_\pm(x)\) such that

\[ \left| y_\pm(x)- W\left(-\frac{1}{2}\mu c^2,\ \pm x\sqrt{2\mu}\right) \right| \leq \]

\[ \leq \gamma\left(1+\mu^{1/4}(x^2+c^2)^{1/4}\right)^{-1} \left\{ \exp\left( \frac{2\gamma^2}{\sqrt{\mu}} \left| \int_{x_0}^{x} \frac{|f(t)|\,dt} {\left(1+\mu^{1/4}|t^2+c^2|^{1/4}\right)^2} \right| \right)-1 \right\}, \tag{3.24} \]

\[ \left| \frac{dy_\pm(x)}{dx} - \frac{d}{dx} W\left(-\frac{1}{2}\mu c^2,\ \pm x\sqrt{2\mu}\right) \right| \leq \]

\[ \leq \gamma\left(1+\mu^{1/4}(x^2+c^2)^{1/4}\right)\times \]

\[ \times \left\{ \exp\left( \frac{2\gamma^2}{\sqrt{\mu}} \left| \int_{x_0}^{x} \frac{|f(t)|\,dt} {\left(1+\mu^{1/4}|t^2+c^2|^{1/4}\right)^2} \right| \right)-1 \right\}, \tag{3.25} \]

where \(x_0\) is an arbitrary point of the interval \([x_1,x_2]\).

The equation \(y''+\mu^2(c^2-x^2)y=f(x)y\) has solutions \(y_\pm(x)\) such that

\[ \left|y_\pm(x)-U\left(-\frac{1}{2}\mu c^2,\ \pm x\sqrt{2\mu}\right)\right|\leq \]
\[ \leq \varphi_\pm(x)\left\{\exp\left(\frac{2\gamma^2}{\sqrt{\mu}}\widetilde F_\pm(x)\right)-1\right\}, \tag{3.26} \]

\[ \left|\frac{dy_\pm(x)}{dx}-\frac{d}{dx}U\left(-\frac{1}{2}\mu c^2,\ \pm x\sqrt{2\mu}\right)\right|\leq \]
\[ \leq \sqrt{\mu}\left(1+(\mu c^2)^{1/12}+\mu^{1/4}|x^2-c^2|^{1/4}\right)^2\varphi_\pm(x)\times \]
\[ \times \left\{\exp\left(\frac{2\gamma^2}{\sqrt{\mu}}\widetilde F_\pm(x)\right)-1\right\}, \tag{3.27} \]

where

\[ \varphi_\pm(x)=\gamma\left(\frac{\mu c^2}{2e}\right)^{(1/4)\mu c^2}\left|\exp(\mp \mu \xi_3(x))\right|\times \]
\[ \times \left(1+(\mu c^2)^{1/12}+\mu^{1/4}|x^2-c^2|^{1/4}\right)^{-1}, \tag{3.28} \]

\[ \xi_3(x)=\frac{1}{2}x\sqrt{x^2-c^2}-\frac{1}{2}c^2\ln\left|\frac{x}{c}+\sqrt{\frac{x^2}{c^2}-1}\right|, \tag{3.29} \]

\[ \widetilde F_+(x)=\int_{x_1}^{x} \frac{|f(t)|\,dt} {\left(1+(\mu c^2)^{1/12}+\mu^{1/4}|t^2-c^2|^{1/4}\right)^2}, \]
\[ \widetilde F_-(x)=\widetilde F_+(x_2)-\widetilde F_+(x). \tag{3.30} \]

§ 4. Equations for eigenvalues

The estimates, obtained in § 3, for solutions of the equation with two turning points make it possible for us to find asymptotic equations for the eigenvalues*) of a boundary-value problem of the form

\[ \frac{d^2v}{d\xi^2}+\mu^2 p(\xi,\alpha)v=0,\qquad a\leq \xi\leq b, \tag{4.1} \]

\[ -v'(a)+\sigma_1v(a)=v'(b)+\sigma_2v(b)=0, \tag{4.2} \]

where \(\sigma_1,\sigma_2\) are real numbers (and \(p(\xi;\alpha)\) satisfies the conditions under which the estimates for solutions were obtained).

The process of finding equations for the eigenvalues reduces to substituting into the boundary conditions (4.2) the asymptotic estimates for the solutions and expanding the Weber functions and their derivatives that occur into asymptotic series (for the corresponding expansions see [12] and [13]). In doing this, it is necessary to keep track only of the proper choice of phase constants, which would ensure that the corresponding solution has the prescribed number of zeros inside \((a,b)\). As a result we arrive at the following lemmas.

*) Let us note that the spectral parameter in problem (4.1), (4.2) is not fixed; by “secular equations” for the eigenvalues we mean the asymptotic form of conditions on \(\mu\) and \(\alpha\) (for large \(\mu\)), under which there exists a solution \(v(\xi)\) of equation (4.1) satisfying conditions (4.2) and having exactly \(n-1\) zeros inside \((a,b)\).

Lemma 3. Let \(p(\xi,\alpha)\), nonnegative for \(a\le \xi \le b\) and for \(\alpha \in [\alpha_0-\delta,\alpha_0+\delta]\), have purely imaginary zeros \(\pm i\xi_0(\alpha)\), coalescing into one double zero as \(\alpha \to \alpha_0\); suppose that \(p(\xi,\alpha)\) is an analytic function of \(\xi\), real for real \(\xi\), and has no other zeros in some rectangle
\[ a-\tilde\delta \le \operatorname{Re}\xi \le b+\tilde\delta,\quad |\operatorname{Im}\xi|\le \tilde\delta; \]
let also \(a<-\tilde\delta,\ b>+\tilde\delta\). Then equation (4.1) has a solution \(v(\xi)\), satisfying the boundary conditions (4.2) and having exactly \(n-1\) zeros inside \((a,b)\), if
\[ \frac{\mu}{\pi}\int_a^b \sqrt{p(t,\alpha)}\,dt +\frac{1}{\pi}\arg \Gamma\left(\frac12-\frac{i\mu c^2}{2}\right) + \frac{\mu c^2}{2\pi}\ln \frac{\mu c^2}{2e} +\omega(\sigma_1,\sigma_2) +O\left(\frac{1}{\sqrt{\mu}}\right) =n, \tag{4.3} \]
where
\[ c^2(\alpha)=\frac{4}{\pi}\int_0^{\xi_0}\sqrt{p(it,\alpha)}\,dt \]
and
\[ \omega(\sigma_1,\sigma_2)=1 \]
for \(|\sigma_1|,|\sigma_2|<\infty\),
\[ \omega(\sigma_1,\infty)=\omega(\infty,\sigma_2)=\frac12 \]
for \(|\sigma_1|,|\sigma_2|<\infty\), and
\[ \omega(\infty,\infty)=0. \]

Lemma 4. Suppose that in the preceding lemma \(a=0,\ b>\tilde\delta\); then the equations for the eigenvalues have the form
\[ \frac{\mu}{\pi}\int_0^b \sqrt{p(t,\alpha)}\,dt +\frac{1}{2\pi}\arg \Gamma\left(\frac12-\frac{i\mu c^2}{2}\right) + \frac{\mu c^2}{4\pi}\ln \frac{\mu c^2}{2e} +\chi(\mu,\alpha,\sigma_1,\sigma_2) +O\left(\frac{1}{\sqrt{\mu}}\right) =n, \tag{4.4} \]
where
\[ \chi(\mu,\alpha,\sigma_1,\sigma_2)= \begin{cases} -\dfrac{1}{\pi}\arctg k+\dfrac14, & \sigma_1=\sigma_2=\infty,\\[6pt] -\dfrac{1}{\pi}\arctg k+\dfrac34, & \sigma_1=\infty,\quad |\sigma_2|<\infty,\\[6pt] \dfrac{1}{\pi}\arctg k+\dfrac14, & |\sigma_1|<\infty,\quad \sigma_2=\infty,\\[6pt] \dfrac{1}{\pi}\arctg k+\dfrac34, & |\sigma_1|,\ |\sigma_2|<\infty, \end{cases} \tag{4.5} \]
and
\[ k=\left(1+\exp(-\pi\mu c^2)\right)^{1/2}-\exp\left(-\frac{\pi}{2}\mu c^2\right). \]

Lemma 5. Let the zeros \(\xi_1\) and \(\xi_2\) of the function \(p(\xi,\alpha)\) be real and let
\[ p(\xi,\alpha)<0 \]
for \(\xi_1<\xi<\xi_2\), while otherwise \(p(\xi,\alpha)\) satisfies the conditions of Lemma 3. Let the real function \(x(\xi)\) be defined by the equation
\[ \frac12 x\sqrt{x^2-c^2} -\frac12 c^2\ln\left\{\frac{x}{c}+\sqrt{\frac{x^2}{c^2}-1}\right\} = \int_{\xi_2}^{\xi}\sqrt{p(t,\alpha)}\,dt, \tag{4.6} \]

where

\[ c^{2}(\alpha)=\frac{2}{\pi}\int_{\xi_{1}}^{\xi_{2}}\sqrt{-p(t,\alpha)}\,dt. \]

Then the equations for the eigenvalues of problem (4.1), (4.2) have the form

1) for \(x(a)\leq -\widetilde{\delta}\), \(x(b)\geq \widetilde{\delta}\) \((\gg c)\)

\[ \frac{\mu}{\pi}\int_{a}^{\xi_{1}}\sqrt{p(t,\alpha)}\,dt+ \frac{\mu}{\pi}\int_{\xi_{2}}^{b}\sqrt{p(t,\alpha)}\,dt+ \frac{1}{\pi}\arg \Gamma\left(\frac{1}{2}+\frac{i\mu c^{2}}{2}\right) -\frac{\mu c^{2}}{2\pi}\ln\frac{\mu c^{2}}{2e} +\omega(\sigma_{1},\sigma_{2})+O\left(\frac{1}{\sqrt{\mu}}\right)=n, \tag{4.7} \]

2) for \(x(a)=0\), \(x(b)\geq \widetilde{\delta}\) \((\gg c)\)

\[ \frac{\mu}{\pi}\int_{\xi_{2}}^{b}\sqrt{p(t,\alpha)}\,dt+ \frac{1}{2\pi}\arg \Gamma\left(\frac{1}{2}+\frac{i\mu c^{2}}{2}\right) -\frac{\mu c^{2}}{4\pi}\ln\frac{\mu c^{2}}{2e} +\widetilde{\chi}(\mu,\alpha,\sigma_{1},\sigma_{2}) +O\left(\frac{1}{\sqrt{\mu}}\right)=n, \tag{4.8} \]

where \(\omega(\sigma_{1},\sigma_{2})\) is defined as in Lemma 3, and \(\widetilde{\chi}\) is obtained from \(\chi\), defined in Lemma 4, by replacing \(\widetilde{k}\) by

\[ \widetilde{k}=\left(1+\exp(\pi\mu c^{2})\right)^{1/2}-\exp\left(\frac{\pi\mu c^{2}}{2}\right). \]

Lemma 6. Let the zeros \(\xi_{1}\) and \(\xi_{2}\) of the function \(p(\xi,\alpha)\) be real and let \(p(\xi,\alpha)>0\) for \(\xi_{1}<\xi<\xi_{2}\), while otherwise \(p(\xi,\alpha)\) satisfies the conditions of Lemma 3. Let the real function \(x(\xi)\) be defined by the equation

\[ \frac{1}{2}x\sqrt{x^{2}-c^{2}} -\frac{1}{2}c^{2}\ln\left\{\frac{x}{c}+\sqrt{\frac{x^{2}}{c^{2}}-1}\right\} =\int_{\xi_{2}}^{\xi}\sqrt{-p(t,\alpha)}\,dt, \]

where

\[ c^{2}(\alpha)=\frac{2}{\pi}\int_{\xi_{1}}^{\xi_{2}}\sqrt{p(t,\alpha)}\,dt. \]

Then the equations for the eigenvalues of problem (4.1), (4.2) have the form

1) for \(x(a)\leq -\widetilde{\delta}\), \(x(b)\geq \widetilde{\delta}\) \((\gg c)\)

\[ \frac{\mu}{\pi}\int_{\xi_{1}}^{\xi_{2}}\sqrt{p(t,\alpha)}\,dt+ \frac{1}{2} +O\left(\frac{1}{\sqrt{\mu}}\right)=n, \tag{4.9} \]

2) for \(x(a)=0\), \(x(b)\geq \widetilde{\delta}\) \((\gg c)\)

\[ \frac{\mu}{2\pi}\int_{\xi_{1}}^{\xi_{2}}\sqrt{p(t,\alpha)}\,dt+ \gamma(\sigma_{2})+ O\left(\frac{1}{\sqrt{\mu}}\right)=n, \tag{4.10} \]

where \(\gamma(\sigma_{2})=3/4\) for \(|\sigma_{2}|<\infty\) and \(\gamma(\infty)=1/4\).

We note that the remainder terms are everywhere uniform with respect to \(\alpha\), and the phase constants have been chosen so that the equations of Lemma 3, for sufficiently large

values of \(\mu c^2\) pass into the equations of Lemma 1, and for \(\mu c^2=0\) coincide with the equations of Lemma 5 when \(\xi_1=\xi_2\).

§ 5. DISTRIBUTION OF THE EIGENFREQUENCIES OF A PLANE ELLIPTIC MEMBRANE

Let us apply the results obtained in § 4 to the solution of the problem on the asymptotic distribution of the eigenvalues of the boundary-value problem

\[ \frac{\partial^2 \varphi}{\partial x_1^2} + \frac{\partial^2 \varphi}{\partial x_2^2} +\lambda \varphi=0 \quad \text{in } D, \]

\[ \left. \frac{\partial \varphi}{\partial n} +\sigma \varphi \right|_{\Gamma}=0 \quad (\sigma=\mathrm{const}), \tag{5.1} \]

where \(D\) is an ellipse with semiaxes \(a\operatorname{ch}\xi_0,\ a\operatorname{sh}\xi_0\). The variables in problem (5.1) are separated if one sets \(x_1+i x_2=a\operatorname{ch}(\xi+i\eta)\). We write the separated equations in the form

\[ \frac{d^2 v}{d\xi^2} + \mu^2(\operatorname{ch}^2\xi-\alpha^2)v=0, \qquad 0\le \xi\le \xi_0, \tag{5.2} \]

\[ \frac{d^2 w}{d\eta^2} + \mu^2(\alpha^2-\cos^2\eta)w=0, \qquad -\pi\le \eta\le \pi . \tag{5.3} \]

Here \(\mu^2\alpha^2\) is the separation parameter, and \(\mu\) is related to the spectral parameter \(\lambda\) by the equality \(\lambda=\mu^2a^2\). The function \(\varphi=v(\xi)w(\eta)\), where \(v(\xi)\) satisfies equation (5.2), and \(w(\eta)\) equation (5.3), is an eigenfunction of problem (5.1) if \(v(\xi)\) and \(w(\eta)\) have the same parity with respect to zero and satisfy the conditions

\[ v'(\xi_0)+\sigma v(\xi_0)=0, \qquad w(\eta+\pi)\equiv w(\eta-\pi). \tag{5.4} \]

It is not difficult to verify that the periodicity condition (with period \(2\pi\)) for a solution of equation (5.3) having a definite parity with respect to zero is equivalent to the equation

\[ \left. \frac{d}{d\eta}\,w^2(\eta) \right|_{\eta=\pi/2} =0, \]

i.e., it must also have a definite parity with respect to \(\eta=\pi/2\).

Denote by \(N_{gg}\) the number of eigenvalues \(\lambda\) of problem (5.1), not exceeding \(\lambda\), corresponding to eigenfunctions \(v(\xi)w(\eta)\) with \(v(\xi)\) even with respect to zero, if

\[ w\left(\frac{\pi}{2}-\eta\right) = w\left(\frac{\pi}{2}+\eta\right), \]

and by \(N_{gu}\), if

\[ w\left(\frac{\pi}{2}-\eta\right) = -w\left(\frac{\pi}{2}+\eta\right); \]

for odd \(v(\xi)\) we use the analogous notation \(N_{ug}\) and \(N_{uu}\). Obviously, for the distribution function of the eigenvalues of problem (5.1) the equality holds

\[ N(\lambda)=N_{gg}+N_{gu}+N_{ug}+N_{uu}. \]

Let us write out the equations for the eigenvalues of each of these types; we begin with those which enter into \(N_{uu}\).

Suppose first that \(0\le \alpha\le \delta\). Then equation (5.2) has no turning points, and therefore, by Lemma 1, the condition for the existence of a solution \(v(\xi)\) having \(n-1\) zeros inside \((0,\xi_0)\) and satisfying the conditions

\[ v(0)=v'(\xi_0)+\sigma v(\xi_0)=0 \]

is the equality

\[ \frac{\mu}{\pi}\int_0^{\xi_0}\sqrt{\operatorname{ch}^2 t-\alpha^2}\,dt+\gamma(\sigma)+O\left(\frac{1}{\sqrt{\mu}}\right)=n, \tag{5.5} \]

where \(\gamma(\sigma)=1\) for \(|\sigma|<\infty\) and \(\gamma(\infty)=\dfrac12\). For equation (5.3) one must use the result of Lemma 6, which leads to the equation

\[ \frac{\mu}{\pi}\cdot\int_{\arccos\alpha}^{\pi/2}\sqrt{\alpha^2-\cos^2 t}\,dt+\frac34+O\left(\frac{1}{\sqrt{\mu}}\right)=m. \tag{5.6} \]

Next, for \(\delta \leq \alpha \leq 1-\delta\) the type of equation (5.2) does not change, and therefore equation (5.5) remains valid. For equation (5.3), for these \(\alpha\) we are under the conditions of Lemma 2; it is easy to verify that its application leads to (5.6). For \(1-\delta \leq \alpha \leq 1\), for equation (5.2) we are under the conditions of Lemma 3, and for equation (5.3), under the conditions of Lemma 5. Therefore the equations for the eigenvalues have the form

\[ \frac{\mu}{\pi}\int_0^{\xi_0}\sqrt{\operatorname{ch}^2 t-\alpha^2}\,dt +\frac{1}{2\pi}\arg\Gamma\left(\frac12-\frac{i\mu c^2}{2}\right)+ \]

\[ +\frac{\mu c^2}{4\pi}\ln\frac{\mu c^2}{2e} +\frac{1}{\pi}\operatorname{arctg} k+\tilde{\gamma}(\sigma) +O\left(\frac{1}{\sqrt{\mu}}\right)=n, \tag{5.7} \]

where

\[ c^2(\alpha)=\frac{4}{\pi}\int_0^{\arccos\alpha}\sqrt{\cos^2 t-\alpha^2}\,dt, \qquad k=\left(1+e^{-\pi\mu c^2}\right)^{1/2}-e^{-\frac{\pi}{2}\mu c^2}, \tag{5.8} \]

and \(\tilde{\gamma}(\sigma)=3/4\) for \(|\sigma|<\infty\), \(\gamma(\infty)=1/4\), and

\[ \frac{\mu}{\pi}\int_{\arccos\alpha}^{\pi/2}\sqrt{\alpha^2-\cos^2 t}\,dt +\frac{1}{2\pi}\arg\Gamma\left(\frac12+\frac{i\mu c^2}{2}\right)- \]

\[ -\frac{\mu c^2}{4\pi}\ln\frac{\mu c^2}{2e} +\frac{1}{\pi}\operatorname{arctg}\tilde{k} +\frac34 +O\left(\frac{1}{\sqrt{\mu}}\right)=m, \tag{5.9} \]

where

\[ \tilde{k}=\left(1+e^{\pi\mu c^2}\right)^{1/2}-e^{\frac{\pi}{2}\mu c^2}. \]

We note that for sufficiently large \(\mu c^2\), equation (5.7) passes into (5.5), and equation (5.9) into (5.6). Thus, these equations are valid uniformly in \(\alpha\), \(0\leq \alpha \leq 1\).

Let us make the following remark concerning the obtained equations (5.7) and (5.9). For the time being, they have the meaning that, with the aid of one of them, say (5.7), we find the function \(\alpha_n(\mu)\); then the second makes it possible to find a series of eigenvalues \(\lambda_{n,m}=a^2\mu_{n,m}^2\). Thus, the quantities \(\alpha\) and \(\mu\) in these equations should have been provided with indices. We do not do this for the following reason. It is easy to verify that the function \(\lambda(n,m)^*\) which arises in this way satisfies the conditions

*) That is, the eigenvalue of problem (5.1) corresponding to an eigenfunction of the form \(v_n(\xi)w_m(\eta)\), where \(v_n(\xi)\) has \(n-1\) zeros inside \((0,\xi_0)\), and \(w_m(\eta)\) has \(m-1\) zeros inside the interval \((0,\pi/2)\).

\[ \lambda(n,m)\leqslant \lambda(n+1,m),\qquad \lambda(n,m)\leqslant \lambda(n,m+1). \tag{5.10} \]

Consequently, the boundary of the region \(\lambda(n,m)\leqslant \lambda\) in the \((n,m)\)-plane is given by the equation \(\lambda(n,m)=\lambda\), or, in parametric form, by equations of the form (5.7—5.9) without indices.

The reasoning for \(\alpha\gg 1\) (and for other types of eigenvalues) is carried out in exactly the same way. We formulate the results as a separate theorem, using the following notation:

\[ E(\alpha)=\int_{\operatorname{arch}\alpha}^{\xi_0}\sqrt{\operatorname{ch}^2 t-\alpha^2}\,dt,\qquad E_0(\alpha)=\int_0^{\xi_0}\sqrt{\operatorname{ch}^2 t-\alpha^2}\,dt, \tag{5.11} \]

\[ \overline{E}(\alpha)=\int_{\arccos\alpha}^{\pi/2}\sqrt{\alpha^2-\cos^2 t}\,dt,\qquad \overline{E}_0(\alpha)=\int_0^{\pi/2}\sqrt{\alpha^2-\cos^2 t}\,dt, \tag{5.12} \]

where \(E(\alpha)\) and \(\overline{E}_0(\alpha)\) are defined for \(\alpha\geqslant 1\), and \(E_0(\alpha)\) and \(\overline{E}(\alpha)\) for \(0\leqslant \alpha\leqslant 1\). Let also

\[ c^2(\alpha)=\frac{4}{\pi}\int_0^{\arccos\alpha}\sqrt{\cos^2 t-\alpha^2}\,dt \qquad(\alpha\leqslant 1), \]

\[ \tilde c^{\,2}=\frac{4}{\pi}\int_0^{\operatorname{arch}\alpha}\sqrt{\alpha^2-\operatorname{ch}^2 t}\,dt \qquad(\alpha>1). \tag{5.13} \]

and

\[ \Omega^{(\pm)}(x)=\frac{1}{2\pi}\arg\Gamma\left(\frac12+\frac{ix}{2}\right)- \]

\[ -\frac{x}{4\pi}\ln\frac{|x|}{2e}\pm\frac{1}{\pi}\operatorname{arctg}\left\{(1+e^{\pi x})^{1/2}-e^{\pi x/2}\right\}+\frac14 . \tag{5.14} \]

Theorem. Let \(N_{uu}\) be the number of integral points \(n,m\gg 1\) under the curve given by the parametric equations

\[ n= \begin{cases} \dfrac{\mu}{\pi}E_0(\alpha)+\Omega^{(-)}(-\mu c^2)+O\left(\dfrac{1}{\sqrt{\mu}}\right), & 0\leqslant \alpha\leqslant 1,\\[1.2em] \dfrac{\mu}{\pi}E(\alpha)+\Omega^{(-)}(\mu \tilde c^{\,2})+O\left(\dfrac{1}{\sqrt{\mu}}\right)+O\left(\dfrac{1}{\mu E}\right), & 1\leqslant \alpha<\operatorname{ch}\xi_0, \end{cases} \tag{5.15} \]

\[ m= \begin{cases} \dfrac{\mu}{\pi}\overline{E}(\alpha)+\Omega^{(-)}(\mu c^2)+O\left(\dfrac{1}{\sqrt{\mu}}\right), & 0\leqslant \alpha\leqslant 1,\\[1.2em] \dfrac{\mu}{\pi}\overline{E}_0(\alpha)+\Omega^{(-)}(-\mu \tilde c^{\,2})+O\left(\dfrac{1}{\sqrt{\mu}}\right), & 1\leqslant \alpha\leqslant \operatorname{ch}\xi_0, \end{cases} \tag{5.16} \]

\(N_{ug}\) is the number of integral points \(n,m\gg 1\) under the curve whose equations are obtained by replacing in these equations \(m\) by \(m-\dfrac12\), \(N_{gu}\) is the number of integral points \(n,m\gg 1\) under the curve

\[ n= \begin{cases} \dfrac{\mu}{\pi} E_0(\alpha)+\Omega^{(+)}(-\mu c^2)+ O\!\left(\dfrac{1}{\sqrt{\mu}}\right), & 0\leqslant \alpha \leqslant 1,\\[6pt] \dfrac{\mu}{\pi} E(\alpha)+\Omega^{(-)}(\mu \tilde c^{\,2})+\dfrac{1}{2}+ O\!\left(\dfrac{1}{\sqrt{\mu}}\right)+O\!\left(\dfrac{1}{\mu E}\right), & 1\leqslant \alpha < \operatorname{ch}\xi_0, \end{cases} \tag{5.17} \]

\[ m= \begin{cases} \dfrac{\mu}{\pi}\,\overline E(\alpha)+\Omega^{(+)}(\mu c^2)+ O\!\left(\dfrac{1}{\sqrt{\mu}}\right), & 0\leqslant \alpha \leqslant 1,\\[6pt] \dfrac{\mu}{\pi}\,\overline E_0(\alpha)+\Omega^{(+)}(-\mu \tilde c^{\,2})+ O\!\left(\dfrac{1}{\sqrt{\mu}}\right), & 1\leqslant \alpha \leqslant \operatorname{ch}\xi_0. \end{cases} \tag{5.18} \]

and \(N_{gg}\) is the number of integral points \(n,m\geqslant 1\) under the curve whose equations are obtained by replacing \(m\) in these equations by \(m-1/2\). Then the number of eigenvalues of problem (5.1), not exceeding \(\lambda\), under the boundary condition \(\varphi|_{\Gamma}=0\), is equal to the sum
\[ N_{uu}+N_{ug}+N_{gu}+N_{gg}. \]

The distribution function of the eigenvalues of problem (5.1) under the boundary condition
\[ \left.\frac{\partial \varphi}{\partial n}+\sigma\varphi\right|_{\Gamma}=0 \]
is equal to the analogous sum, provided only that in the definition of \(N_{uu}\) and \(N_{ug}\) one replaces \(n\) by \(n-1/2\), and in the definition of \(N_{gu}\) and \(N_{gg}\), in the equations for \(n(\alpha)\), for \(\alpha\leqslant 1\) one replaces \(n\) by \(n-1/2\), while for \(\alpha\geqslant 1\) one replaces \(\Omega^{(-)}(\mu \tilde c^{\,2})\) by \(\Omega^{(+)}(\mu \tilde c^{\,2})\).

Now Theorem 1 of § 1 is obtained by a simple application of van der Corput’s theorem to counting the number of integral points in the indicated domains. We shall use this theorem in the following form ([5], Chap. 11).

Let \(v_0-1/2,\ v_1-1/2,\ u_0-1/2\) be integers, \(v_0<v_1\). Suppose that on the interval \(v_0\leqslant v\leqslant v_1\) the function \(f(v)\) is twice continuously differentiable and satisfies the conditions

\[ f(v_0)>u_0,\qquad 0<\rho\leqslant f'(v)\leqslant \tau, \]

\[ |f''(v)|\geqslant \frac{1}{\mu}, \tag{5.19} \]

where \(\mu>1,\ \mu>\rho^{-3}\). Let \(N\) be the number of integral points in the domain
\[ v_0\leqslant v\leqslant v_1,\qquad u_0\leqslant u\leqslant f(v), \]
and let \(S\) be the area of this domain. Then

\[ N-S=O(\mu^{2/3}\tau). \tag{5.20} \]

We shall also use the following proposition of M. V. Yarnik [15].

Let \(B\) be a Jordan domain whose area is \(S\) and perimeter \(L\). Then the number of integral points \(N(B)\) in the domain \(B\) satisfies the inequality

\[ |N(B)-S|<L. \tag{5.21} \]

Division of the domain

Proceeding to the computation of \(N_{uu}\), we divide the domain \(B(\lambda)\), whose boundary is defined by equations (5.15), (5.16), into eight parts \(B_k(\lambda)\) (see the figure), choosing the division points \(n_k\) and \(m_k\) to be half-integers so that the inequalities are satisfied.

\[ c\mu^{2/3}\leq n(0)-n_1,\quad m(1)-m_1,\quad m_2-m(1), \]
\[ m(\operatorname{ch}\xi_0)-m_3\leq c_2\mu^{2/3}. \]

The half-integer point \(n_2\) can be chosen so that \(n_2\geq n(m_1)\) and the number of integer points \(N(B_3)\) in the region \(B_3\) is of order \(O(\mu^{2/3})\). It is easy to verify that, by virtue of the choice of the points \(n_k\) and \(m_k\), Jarník’s theorem gives

\[ N(B_k)=\operatorname{mes}B_k+O(\mu^{2/3}),\qquad k=1,\,5,\,8. \tag{5.22} \]

To count the number of integer points in the regions \(B_2\) and \(B_7\), we apply van der Corput’s theorem. First of all, let us note that the equations describing the boundary of the region \(B_2\) can be written in the form

\[ n=\frac{\mu}{\pi}E_0(\alpha)+\frac{1}{2} +O\!\left(\frac{\ln\mu}{\mu^{2/3}}\right), \]

\[ m=\frac{\mu}{\pi}\overline{E}(\alpha)+\frac{3}{4} +O\!\left(\frac{\ln\mu}{\mu^{2/3}}\right), \tag{5.23} \]

and for the boundary of \(B_7\),

\[ n=\frac{\mu}{\pi}E(\alpha)+\frac{3}{4} +O(\mu^{-2/3}\ln\mu), \]

\[ m=\frac{\mu}{\pi}\overline{E}_0(\alpha) +O(\mu^{-2/3}\ln\mu). \tag{5.24} \]

For this it is necessary first to establish that, for \(m\leq m_1\) and \(m\geq m_2\), the inequalities \(\mu c^2,\ \mu\tilde c^{\,2}>\operatorname{const}\,\dfrac{\mu^{2/3}}{\ln\mu}\) hold, and then to use the asymptotic expansion of the \(T\)-function. The remaining terms appearing in (5.23), (5.24), as is easy to verify, contribute to \(N(B_2)\) and \(N(B_7)\) an amount of order \(O(\mu^{1/3}\ln\mu)\), and therefore may be discarded. We can now write (for \(m\leq m_1\)):

\[ \frac{dn}{d\alpha} = -\frac{\mu}{\pi}\int_0^{\xi_0} \frac{\alpha\,dt}{\sqrt{\operatorname{ch}^2t-\alpha^2}} = -\frac{\mu}{\pi}\int_{1/\operatorname{ch}\xi_0}^{1} \frac{\alpha\,dt}{\sqrt{1-t^2}\sqrt{1-\alpha^2t^2}}, \]

\[ \frac{dm}{d\alpha} = \frac{\mu}{\pi}\int_{\arccos\alpha}^{\pi/2} \frac{\alpha\,dt}{\sqrt{\alpha^2-\cos^2t}} = \frac{\mu}{\pi}\int_0^1 \frac{\alpha\,dx}{\sqrt{1-x^2}\sqrt{1-\alpha^2x^2}}. \]

It follows that

\[ 1\leq -\frac{dm}{dn}\leq 1+ \left\{ \int_0^{1/\operatorname{ch}\xi_0}\frac{dt}{1-t^2} \right\} \left\{ \int_{1/\operatorname{ch}\xi_0}^{1}\frac{dx}{\sqrt{1-x^2}} \right\}^{-1}. \tag{5.25} \]

Further,

\[ \frac{\pi^2}{\mu^2} \left\{ m_{\alpha^2}^{\prime\prime}n_\alpha' - n_{\alpha^2}^{\prime\prime}m_\alpha' \right\} = \]

\[ = \alpha^2 \int_0^1\int_{1/\operatorname{ch}\xi_0}^{1} (t^2-x^2) \left[ (1-t^2)(1-x^2)(1-\alpha^2t^2)(1-\alpha^2x^2) \right]^{-1/2}\,dt\,dx. \tag{5.26} \]

The integral obtained is strictly positive. Indeed, denoting the integrand by \(\Phi(x,t)\), we have \(\Phi(x,t)=-\Phi(t,x)\). Therefore the integral of \(\Phi(x,t)\) over any domain symmetric with respect to the line \(x=t\) vanishes. Hence,

\[ \int\limits_0^1 \int\limits_{\frac{1}{\operatorname{ch}\xi_0}}^1 \Phi(x,t)\,dt\,dx = \int\limits_0^{\frac{1}{\operatorname{ch}\xi_0}} \int\limits_{\frac{1}{\operatorname{ch}\xi_0}}^1 \Phi(x,t)\,dt\,dx, \]

and for \(x<\dfrac{1}{\operatorname{ch}\xi_0}\) and \(t>\dfrac{1}{\operatorname{ch}\xi_0}\), \(\Phi(x,t)>0\).

Now it is easy to obtain

\[ -\frac{d^2 m}{dn^2}>\frac{c(\xi_0)}{\mu}, \tag{5.27} \]

where \(c(\xi_0)\) is a constant depending only on \(\xi_0\). Consequently, in the domain \(B_2\) the conditions of van der Corput’s theorem are satisfied and

\[ N(B_2)=\operatorname{mes} B_2+O(\mu^{2/3}). \tag{5.28} \]

In exactly the same way we obtain that the same estimate is valid for the domain \(B_7\):

\[ N(B_7)=\operatorname{mes} B_7+O(\mu^{2/3}). \]

Finally, for the domains \(B_4\) and \(B_6\) we have directly
\(N(B_k)=\operatorname{mes} B_k+O(\mu^{2/3})\). Now summing all the estimates obtained and computing the corresponding areas leads to the assertion of Theorem 1 of § 1 for the case of the elliptic membrane with remainder term \(O(\lambda^{1/3})\).

The arguments in the other cases of separation of variables proceed according to the same plan and are almost completely analogous to those already carried out. Some differences occur only in that part where the possibility of applying van der Corput’s theorem is verified.

In conclusion the author takes the opportunity to express sincere gratitude to Professor V. B. Lidskii for his constant attention to this work.

References

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Received by the editors
December 10, 1965

V. A. Steklov Mathematical Institute