UDC 534.21

MATHEMATICAL PHYSICS

M. M. POPOV

ASYMPTOTICS OF CERTAIN SUBSEQUENCES OF EIGENVALUES OF BOUNDARY-VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION IN THE MULTIDIMENSIONAL CASE

(Presented by Academician V. I. Smirnov on 10 VI 1968)

1. Let \(\Omega\) be a domain of Euclidean space \(R^{n+1}\), bounded by a sufficiently smooth closed surface \(\Sigma\). We consider problems for the Helmholtz equation

\[ [\Delta+\omega^2 C^{-2}(M)]U(M)=0,\qquad M\in\Omega; \tag{1} \]

\[ \text{I. } U|_{\Sigma}=0 \quad \left(\text{or II. } \frac{\partial}{\partial n}U|_{\Sigma}=0\right), \tag{2} \]

where \(\Delta\) is the Laplace operator in \(R^{n+1}\); \(C(M)>0\) is a sufficiently smooth function in \(\Omega\); \(\omega\) is a real parameter.

The present note is devoted to constructing, by the ray method, asymptotics of such subsequences of eigenvalues of problem (1), (2) to which there correspond eigenfunctions concentrated in a neighborhood of certain one-dimensional cycles in \(\Omega\). Asymptotically, as \(\omega\to\infty\), within the framework of the ray method the solution of equation (1) is sought in the form \(U(M)\sim [u+O(1/\omega)]\exp(i\omega\Phi)\), where \(u\) and \(\Phi\) satisfy the equations

\[ (\nabla\Phi)^2=C^{-2}(M), \tag{3} \]

\[ 2(\nabla u,\nabla\Phi)+u\Delta\Phi=0. \tag{4} \]

2. Consider in \(\Omega\) a closed curvilinear \(N\)-gon \(L_N\), whose sides are formed by smooth curves and whose vertices belong to \(\Sigma\). We shall call the polygon \(L_N\) an extremal one-dimensional cycle in \(\Omega\) if the first variation \(\delta I=0\) on \(L_N\), where \(I=\int C^{-1}(M)\,d\sigma\) is the functional of geometrical optics (\(d\sigma\) is an element of length in \(R^{n+1}\)).

In what follows it is assumed that, if the extremal cycle under consideration in \(\Omega\) is stable in the first approximation \((^{1,2})\), then there exists the desired subsequence of eigenvalues of problem (1), (2), to which there correspond eigenfunctions concentrated in a neighborhood of this cycle, i.e. essentially different from zero in some neighborhood of the cycle and decreasing sufficiently rapidly when moving away from it.

3. Let \(\mathbf r=\mathbf r(s)\), \(\mathbf r\in R^{n+1}\), be an extremal cycle in \(\Omega\) with vertices \(\mathbf r(s_1),\ldots,\mathbf r(s_N)=\mathbf r(0)\); \(s\) is arc length along the cycle. In each plane normal to the cycle, introduce the radius vector \(\mathbf q(s)\in R^n\) so that along the cycle \(\mathbf q(s)\equiv 0\). Denote the components of \(\mathbf q(s)\) by \(q_j\), \(j=1,\ldots,n\). The conjugate momenta are defined by the formula

\[ p_j=C^{-1}(s,q_j)\frac{\partial}{\partial \dot q_j}\frac{d\sigma}{ds}, \]

where \(\dot q_j=\frac{d}{ds}q_j\). We expand the Hamiltonian function \(\mathcal H(s;q_j;p_j)\), corresponding to the functional \(I\), in a neighborhood of the cycle into the series

\[ \mathcal H(s;q_j;p_j)=-\frac{1}{C(s,0)}+\mathcal H_2(s;q_j;p_j)+\mathcal H_3(s;q_j;p_j)+\ldots, \tag{5} \]

where \(\mathcal H_k(s;q_j,p_j)\) is a homogeneous polynomial of degree \(k\) in the canonical variables with coefficients depending on \(s\). In the first approximation, discarding in the expansion (5) the terms of order \(\mathcal H_k,\ k\geqslant 3\), we replace \(I\) by the functional \(I_0\):

\[ I_0=\int \left\{\sum_{j=1}^{n}p_j\,dq_j+ \left[\frac{1}{C(s,0)}-\mathcal H_2(s;q_j,p_j)\right]ds\right\}, \tag{6} \]

for which the canonical system of equations has the form

\[ \frac{d}{ds}q_j=\frac{\partial \mathcal H_2}{\partial p_j},\qquad \frac{d}{ds}p_j=-\frac{\partial \mathcal H_2}{\partial q_j},\qquad j=1,2,\ldots,n. \tag{7} \]

A solution of the system (7)
\[ \vec\chi^{\,k}(s)=\bigl(q_1^{(k)}(s),\ldots,q_n^{(k)}(s),p_1^{(k)}(s),\ldots,p_n^{(k)}(s)\bigr), \]
defined on the \(k\)-th \((k=1,2,\ldots,N)\) side \(L_N\), i.e. for \(s_{k-1}\leqslant s\leqslant s_k\), will be called a ray. Let the \(2n\) vectors \(Z_t^{(k)}(s)\), \(t=1,\ldots,2n\), form on the \(k\)-th side \(L_N\) a fundamental system of solutions of the equations (7). With the aid of the matrix
\[ W^{(k)}(s)=\|Z_1^{(k)}(s),\ldots,Z_{2n}^{(k)}(s)\| \]
the ray is defined by the formula
\[ \vec\chi^{(k)}(s)=W^{(k)}(s)\mathbf A^{(k)} \]
through the specification of a \(2n\)-dimensional vector \(\mathbf A^{(k)}\) independent of \(s\).

Upon reflection of the ray from the boundary \(\Sigma\) in a neighborhood of the \(k\)-th vertex \(L_N\) we have:

\[ \mathbf A^{(k+1)}= \bigl[W^{(k+1)}(s_k)\bigr]^{-1}\cdot \gamma_k\cdot W^{(k)}(s_k)\cdot \mathbf A^{(k)} +O\bigl(\|\vec\chi\|^2\bigr), \tag{8} \]

where the reflection matrices \(\gamma_k\) of order \(2n\) are determined by the local properties of \(\Sigma\) at the point \(\mathbf r(s_k)\), \(\det\gamma_k=1\). Let \(\mathbf A_0^{(k)}\) specify some initial ray, and let \(\mathbf A_1^{(k)}\) be the ray arising after a single “traversal” of the initial ray along the cycle; then, in the first approximation, discarding the correction terms in formula (8), we obtain
\[ \mathbf A_1^{(k)}=\Gamma_k\mathbf A_0^{(k)}, \]
and after \(m\) traversals
\[ \mathbf A_m^{(k)}=\Gamma_k^{\,m}\mathbf A_0^{(k)}, \]
with \(\det\Gamma_k=1\) by virtue of the existence of the integral invariant
\[ \int \delta q_1\ldots\delta q_n\delta p_1\ldots\delta p_n \]
for the canonical system (7).

An extremal cycle is called stable in the first approximation (1) if
\[ \|\mathbf A_m^{(k)}\|<M<\infty \]
as \(m\to\infty\). Let \(\lambda_t\), \(t=1,\ldots,2n\), be the eigenvalues of the matrix \(\Gamma_k\).* The conditions for stability of the cycle in the first approximation are that \(|\lambda_t|=1\), and the elementary divisors of \(\Gamma_k\) are simple.

1. For cycles stable in the first approximation there exist \(2n\) linearly independent Floquet solutions of the equations (7), \(X_t(s)\), such that
   \[ X_j(s+s_N)=e^{i\varphi_j}X_j(s),\qquad X_{n+j}(s+s_N)=e^{-i\varphi_j}X_{n+j}(s),^{**}\qquad j=1,\ldots,n. \]
   We divide the Floquet solutions into two groups of \(n\) each so that, for the solutions of the first group,

\[ W(X_j,\overline{X}_t)\equiv \sum_{k=1}^{n}\bigl(q_k^{(j)}(s)\cdot\overline{p}_k^{(t)}(s) -p_k^{(j)}(s)\cdot\overline{q}_k^{(t)}(s)\bigr) =-i\delta_{jt},\qquad j,t=1,\ldots,n, \]

where \(\delta_{jt}\) is the Kronecker symbol; \(\overline{X}_t\) is the vector complex-conjugate to the vector \(X_t\). At the same time, for the Floquet solutions of the second group
\[ W(X_{n+j},\overline{X}_{n+t})=+i\delta_{jt}. \]
Introduce normal coordinates \(^{(4)}\) \(Q_j,P_j\) by the formula
\[ \vec\chi= \sum_{j=1}^{n}\bigl(\mathbf B_j(s)Q_j+\mathbf B_{n+j}(s)P_j\bigr). \]
The functions \(\mathbf B_t(s)\), periodic on the cycle, have the form
\[ \mathbf B_j(s)=e^{-i\mu_j s}X_j(s),\qquad \mathbf B_{n+j}(s)=e^{i\mu_j s}X_{n+j}(s),\qquad \mu_j=\varphi_j/s_N. \]
In nor-

* The characteristic equation of the matrix \(\Gamma_k\) is reciprocal (see, for example, \((3)\)) and therefore reduces to an equation of degree \(n\).

** By these formulas the phases \(\varphi_j\) of the eigenvalues of the matrix \(\Gamma_k\) are determined uniquely as the increment of \(\arg X_j\) on the cycle.

normal coordinates

\[ I_0=\int\left\{-i\sum_{j=1}^{n}P_j\,dQ_j+\left[\frac{1}{C(s,0)}-\sum_{j=1}^{n}\mu_jP_jQ_j\right]\,ds\right\}, \]

and the ray is described by the functions \(Q_j(s)=Q_j(0)e^{i\mu_js}\), \(P_j(s)=P_j(0)e^{-i\mu_js}\), continuous on the cycle. Here \(Q_j(0)=\overline{P_j(0)}\) are constants of integration.

1. The set of rays whose constants of integration satisfy the conditions \(Q_j(0)P_j(0)=\varepsilon_j^2\), \(j=1,\ldots,n\), where \(\varepsilon_j\) are fixed real parameters, forms in the space \(R^{2n}\times S^1\)* an invariant manifold \(T_\varepsilon\), since for all \(s\)

\[ Q_j(s)P_j(s)=Q_j(0)P_j(0)=\varepsilon_j^2,\qquad j=1,\ldots,n. \tag{9} \]

The section of \(T_\varepsilon\) by the Euclidean plane \(R^{2n}\), \(s=\mathrm{const}\), is the direct product of \(n\) circles (9); therefore topologically \(T_\varepsilon\) is described as the direct product of \((n+1)\) circles \(S^1\times S^1\times\ldots\times S^1\). The manifolds \(T_\varepsilon\) are Lagrangian; therefore the solution of the eikonal equation (3) is found by integrating over \(T_\varepsilon\) the total differential

\[ d\Phi=-i\sum_{j=1}^{n}P_j\,dQ_j+\left[\frac{1}{C(s,0)}-\sum_{j=1}^{n}\mu_jP_jQ_j\right]\,ds. \]

1. The asymptotics of the eigenvalues of the problem (1), (2) within the framework of the ray method is found from the “quantization conditions” \((^{5,6})\) following from the requirement that the function \(ue^{i\omega\Phi}\) be single-valued on the invariant manifold \(T_\varepsilon\). The one-dimensional homology group \(H_1(T_\varepsilon)\) of the manifold \(T_\varepsilon\) is the direct product of \((n+1)\) free cyclic groups and has \((n+1)\) generators \(l_k\), \(k=0,1,\ldots,n\). In the case of boundary-value problem I, the “quantization conditions,” which constitute a system of \((n+1)\) equations with respect to \(\omega\) and the \(n\) parameters \(\varepsilon_j\), have the form

\[ \omega\oint_{l_k}d\Phi=2\pi m_k+\pi m_k'+\frac{\pi}{2}m_k'',\qquad k=0,1,\ldots,n, \tag{10} \]

where \(m_k\) is an integer; \(m_k'\) and \(m_k''\) are, respectively, the “reflection” and “caustic” intersection indices, i.e. the Kronecker intersection indices (see also \((^{6,10})\)) of the oriented basis cycles \(l_k\), respectively, with the boundary \(\Sigma\) and the caustic. As \(n\) basis cycles \(l_j\), \(j=1,\ldots,n\), we take the circles (9) in the plane \(s=\mathrm{const}\); for them \(m_j'=0\), \(m_j''=2\). The cycle \(l_0\) is fixed by the conditions \(Q_j=\mathrm{const}\), \(0\leq s\leq s_N\); for it \(m_0'=N\), \(m_0''=0\).

From equations (10) there follows a formula for the eigenvalues \(\omega_{[m]}\)

\[ \omega_{[m]}\int_{0}^{s_N}\frac{ds}{C(s,0)} = 2\pi(m_0+{}^1\!/_{2}N)+\sum_{j=1}^{n}(m_j+{}^1\!/_{2})\varphi_j, \tag{11} \]

where \(m_c\gg1\); \(\varphi_j\) are the Floquet exponents of the first group of Floquet solutions.

In the case of boundary-value problem II, in formula (11) the term \({}^1\!/_{2}N\) should be omitted, since in this case the term \(\pi m_k'\) is absent in equations (10).

Formula (11), obtained on the basis of the linear system (7), coincides with the formula found by the parabolic-equation method \((^{7,8})\).

* The one-dimensional cycles under consideration are homeomorphic to the circle \(S^1\), since the points of self-intersection \(L_N\), if there are any, should, in view of the linearity of the problem (1), (2), be regarded as lying on different copies of the domain \(\Omega\).

1. Using “Birkhoff series” \({}^{9}\), one can, within the framework of the ray method, successively take into account further terms in the expansion (5), provided that \(\varphi_j\) and \(2\pi\) are linearly independent over the ring of integers. In this case, at each step, in a neighborhood of a cycle stable in the first approximation, we shall have a continuous family of invariant Lagrangian manifolds with the same topology as in the first approximation. The correction terms to formula (11) that arise in this way are homogeneous polynomials in \((m_j + 1/2)\) of the second, etc., degrees. There are grounds, however, for believing (see \({}^{11}\), where, in a special case, the next terms in the formula for the eigenvalues were obtained) that the corrections obtained by the ray method are meaningful only for \(m_0 \gg m_j \gg 1\).

The proposed scheme admits a generalization to the case of more general elliptic operators.

The author expresses his deep gratitude to his scientific adviser V. S. Buldyrev for his help in this work.

Moscow State University
named after M. V. Lomonosov

Received
2 VI 1968

REFERENCES

\({}^{1}\) V. S. Buldyrev, Vestn. LGU, No. 22, issue 4, 38 (1965).
\({}^{2}\) V. S. Buldyrev, M. M. Popov, Optics and Spectroscopy, 20, 905 (1966).
\({}^{3}\) A. M. Lyapunov, The General Problem of the Stability of Motion, Ch. III, Moscow—Leningrad, 1950.
\({}^{4}\) E. T. Whittaker, Analytical Dynamics, Moscow—Leningrad, 1937, § 192.
\({}^{5}\) J. Keller, S. Rubinow, Ann. Phys., 9, No. 1, 24 (1960).
\({}^{6}\) V. P. Maslov, Perturbation Theory and Asymptotic Methods, Moscow, 1965.
\({}^{7}\) V. M. Babich, Proceedings of the LOMI Seminars (in press).
\({}^{8}\) M. M. Popov, Vestn. LGU (in press).
\({}^{9}\) G. D. Birkhoff, Dynamical Systems, Moscow—Leningrad, 1941.
\({}^{10}\) V. I. Arnold, Functional Analysis and Its Applications, 1, issue 1, 1 (1967).
\({}^{11}\) V. F. Lazutkin, Optics and Spectroscopy, 24, 453 (1968).