MATHEMATICS

A. M. IL’IN

ON THE EIGENFUNCTIONS OF AN ELLIPTIC OPERATOR IN CERTAIN UNBOUNDED DOMAINS

(Presented by Academician I. G. Petrovskii, 29 X 1964)

Let, in a domain \(D\) of \(n\)-dimensional Euclidean space, there be defined an elliptic differential operator

\[ \mathcal{L}u=\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}\left(a_{ij}(x)\frac{\partial u}{\partial x_j}\right)+b(x)u, \]

where \(x=(x_1,x_2,\ldots,x_n)\); the matrix \(\|a_{ij}\|\) is symmetric and positive definite; the coefficients of the operator are real; \(a_{ij}(x)\in C_1(\overline D)\), \(b(x)\in C(\overline D)\). The domain \(D\) is the union of a bounded domain \(D_0\), for which \(x_1<0\), and the cylinder \(\{[0,\infty)\times\Omega\}\), where \(\Omega\) is a bounded domain in the space \((x_2,x_3,\ldots,x_n)\). The boundary \(S\in C_2\), and for \(x_1>0\) the operator \(\mathcal{L}u\equiv \Delta u\).

It is known \((^1)\) that, in order to single out a unique solution of the problem

\[ \mathcal{L}u+\lambda u=f, \tag{1} \]

\[ u|_S=0 \tag{2} \]

for positive real \(\lambda\), one must require the fulfillment of “partial radiation conditions”:

\[ \frac{dw_k}{dx_1}-i\sqrt{\lambda-\alpha_k}\,w_k \xrightarrow[x_1\to\infty]{}0 \quad \text{for } \alpha_k<\lambda, \tag{3} \]

\[ w_k \xrightarrow[x_1\to\infty]{}0 \quad \text{for } \alpha_k>\lambda, \]

where

\[ w_k(x_1)=\int_{\Omega}u(x)\varphi_k(x^*)\,dx^*. \]

Here \(x^*=(x_2,x_3,\ldots,x_n)\); \(\varphi_k(x^*)\) are eigenfunctions; \(\alpha_k\) are the eigenvalues of the Dirichlet problem for the operator \(-\Delta\) in the domain \(\Omega\). If the homogeneous equation

\[ \mathcal{L}u+\lambda u=0 \tag{4} \]

has no nontrivial solution in \(W_2^2(D)\) equal to zero on the boundary, then for a finite function \(f(x)\in \mathcal{L}_2\) the problem (1), (2) has a unique solution \(u(x)\in W_2^2\) in every finite part of \(D\) and satisfying conditions (3) \((^2)^*\).

In the present paper we investigate the eigenvalues of the operator \(-\mathcal{L}\), i.e., those \(\lambda\) for which equation (4) has a nontrivial solution in \(W_2^2(D)\) equal to zero on the boundary. Of special interest are those eigenvalues which are greater than \(\alpha_1\), since the remaining points of the half-line \([\alpha_1,\infty)\) are points of the continuous spectrum of the operator under consideration. Its eigenvalues are of finite multiplicity and have no finite limit points \((^3)\).

For a certain class of domains it has been proved that the Laplace operator has no eigenvalues for them \((^2)\). As an example of an operator whose point spectrum lies on the continuous spectrum, one may take the operator

\[ Mu\equiv \frac{\partial^2u}{\partial x_1^2}+\frac{\partial^2u}{\partial x_2^2}+a(x_1)u \]

in the half-strip \([0,\infty)\times[0,\pi]\) on functions equal to zero on the boundary. If \(z(x_1)\) is a solution of the equation

\[ \text{* The result given in }(^2)\text{ differs inessentially from the one stated above because of other restrictions imposed on the boundary of the domain.} \]

\(z''+a(x_1)z-\mu^2 z=0\) such that \(z(0)=z(\infty)=0\) (the corresponding real finite function \(a(x_1)\) is easy to construct), then \(k^2-\mu^2\) are eigenvalues, and \(z(x_1)\sin kx_2\) are eigenfunctions of the operator \(-M\) for all natural \(k\). One can construct domains where the operator \(-\Delta\) also has eigenvalues on the continuous spectrum. Moreover, for some domains there is, in addition, a discrete spectrum (to the left of the point \(a_1\)), while for other domains it is absent.

Let \(\lambda \ne a_k\) for any \(k\), and let \(\beta>0\) be a number smaller than
\[ \min_k \sqrt{|\lambda-a_k|}. \]
Denote by \(H_{2,\beta}\) the Hilbert space of functions with scalar product
\[ (uv)_{2,\beta}=\int_D e^{2\beta x_1} \left( u\overline{v} +\sum_{i=1}^n \frac{\partial u}{\partial x_i}\frac{\partial \overline{v}}{\partial x_i} +\sum_{i,j=1}^n \frac{\partial^2 u}{\partial x_i\partial x_j} \frac{\partial^2 \overline{v}}{\partial x_i\partial x_j} \right)\,dx, \]
and \(H_{0,\beta}\) the space with scalar product
\[ (u,v)_{0,\beta}=\int_D e^{2\beta x_1}u\overline{v}\,dx. \]

Theorem 1. Let the function \(f(x)\in H_{0,\beta}\). In order that problem (1), (2) have a solution
\[ u(x)=\widetilde u(x)+\sum_{a_k<\lambda} c_k \exp\left[i\sqrt{\lambda-a_k}\,x_1\right]\varphi_k(x^*), \tag{5} \]
where \(\widetilde u(x)\in H_{2,\beta}\), it is necessary and sufficient that
\[ \int_D f(x)v(x)\,dx=0 \tag{6} \]
for every solution \(v(x)\in H_{2,\beta}\) of problem (4), (2).

For the proof, consider the domains \(D_\rho=D\cap(x_1<\rho)\) for \(\rho>0\). Denote \(S_\rho=S\cap(x_1<\rho)\). Choose \(R>0\) such that in the domains \(D_R\) and \(D_R\setminus D_0\) there exists a unique solution of the Dirichlet problem for equation (1). Denote by \(B\) the space of functions \(\varphi(x^*)\in W_2^{3/2}(\Omega)\) that are zero on the boundary of \(\Omega\) (see \((^4)\)). Let \(\varphi(x^*)\in B\). Then in the domain \(D\setminus D_0\) there exists a unique solution of the equation \(\Delta u+\lambda u=0\) of the form (5) and such that \(u|_{S\setminus S_0}=0\), \(u(0,x^*)=\varphi(x^*)\). In this case \(u(R,x^*)\in B\), and the operator \(A_1\), taking the function \(\varphi(x^*)\) to \(u(R,x^*)\), is completely continuous. If \(f(x)\in H_{0,\beta}\), then in the domain \(D\setminus D_0\) there exists a unique solution of the equation \(\Delta u+\lambda u=0\) of the form (5), equal to zero on the boundary of \(D\setminus D_0\). In this case the operator \(A_2\), taking \(f(x)\in H_{0,\beta}\) to \(u(R,x^*)\in B\), is bounded. If \(u(x)\) is a solution of equation (1) in the domain \(D_R\), equal to zero on the boundary, then \(u(0,x^*)\in B\), and the operator \(A_3\), taking \(f(x)\in H_{0,\beta}\) to \(u(0,x^*)\), is bounded. Finally, if \(u(x)\) is a solution of equation (5) in the domain \(D_R\), which is equal to zero on \(S_R\) and equal to \(\psi(x^*)\) for \(x=R\), then \(u(0,x^*)\in B\), and the operator \(A_4\), taking \(\psi(x^*)\) to \(u(0,x^*)\), is bounded.

Thus, let \(u(x)\) be a solution of problem (1), (2) of the form (5). Denote \(u(R,x^*)\) by \(\psi(x^*)\in B\). Then from the preceding it follows that the function \(\psi(x^*)\) satisfies the equation
\[ \psi=A_2 f+A_1(A_3 f+A_4\psi) \tag{7} \]
or
\[ \psi=A_5\psi+A_6 f, \]
where the operator \(A_5=A_1A_4\) is completely continuous in \(B\), and the operator \(A_6=A_2+A_1A_3\) takes \(H_{0,\beta}\) into \(B\) and is bounded.

Equation (7) and problem (1), (2) are equivalent. The Fredholm theorems are valid for equation (7). On the other hand, if problem (1), (2) has a solution, then relation (6) is fulfilled. From this it is easy to derive the conclusion of Theorem 1.

Theorem 2. Let the function
\[ f(x)=\widetilde f(x)+\sum_{\lambda>a_k} P_{m,k}(x_1) \]
\[ \exp\left[i\sqrt{\lambda-a_k}\,x_1\right]\varphi_k(x^*), \]
where \(\widetilde f(x)\in H_{0,\beta}\), and \(P_{m,k}(x_1)\) are polynomials of degree \(m\).

For problem (1), (2) to have a solution
\[ u(x)=\tilde u(x)+\sum_{\lambda>\alpha_k} Q_{m+1,k}(x_1)\exp\,[i\sqrt{\lambda-\alpha_k}\,x_1]\varphi_k(x^*), \]
where \(\tilde u(x)\in H_{2,\beta}\), and \(Q_{m+1,k}(x_1)\) are polynomials, it is necessary and sufficient that relation (6) be satisfied. Moreover, if the coefficients of the polynomials \(P_{m,k}\) at \(x_1^l\) do not exceed in modulus \(c_k\mu_k^l/l!\), where \(\mu_k=\sqrt{\lambda-\alpha_k}\), then the coefficients of the polynomials \(Q_{m+1,k}\) at \(x_1^l\) do not exceed in modulus \(c_k\mu_k^{\,l-2}/l!\).

The proof of this theorem is analogous to the proof of Theorem 1. With its aid one can study perturbations of the eigenvalues and eigenfunctions of the operator \(\mathcal L\).

Let \(\mathcal L_1\) be the differential operator
\[ \mathcal L_1 u \equiv \sum_{i,j=1}^n \frac{\partial}{\partial x_i}\left(a'_{ij}(x)\frac{\partial u}{\partial x_j}\right)+b'(x)u, \]
where the coefficients \(a'_{ij}(x)\in C_1(\overline D)\) and \(b'(x)\in C(\overline D)\) are real and are different from zero only in the domain \(D_0\).

Theorem 3. Suppose that there exist \(r\) linearly independent solutions of the equation
\[ \mathcal L u+\lambda_0 u=0, \tag{8} \]
vanishing on \(S\) and belonging to \(W_2^2(D)\), and that \(\lambda_0\ne \alpha_k\). Then, for sufficiently small \(\varepsilon\), there exist \(r\) linearly independent functions \(u_j(x,\varepsilon)\in W_2^2(D)\) which satisfy the equation
\[ (\mathcal L+\varepsilon\mathcal L_1)u_j(x,\varepsilon)+\lambda_j(\varepsilon)u_j(x,\varepsilon)=0,\qquad j=1,2,\ldots,r, \tag{9} \]
where \(u_j(x,\varepsilon)\) and \(\lambda_j(\varepsilon)\) are analytic functions of the parameter \(\varepsilon\), and \(\lambda_j(0)=\lambda_0\).

We note that if \(u(x)\in W_2^2(D)\) and satisfies equation (8), then \(u(x)\in H_{2,\beta}\) for some \(\beta>0\). Expanding \(\lambda_j(\varepsilon)\) and \(u_j(x,\varepsilon)\) in series in \(\varepsilon\), one can obtain equations for the coefficients of the expansion and indicate a method for constructing them. Having constructed majorant series, one can verify that the power series for \(\lambda_j(\varepsilon)\) and \(u_j(x,\varepsilon)\) converge. The function \(u_j(x,\varepsilon)\) is a solution of equation (9), and
\[ u_j(x,\varepsilon)=\tilde u_j(x,\varepsilon)+\sum_{\alpha_k<\lambda} v_{j,k}(x_1,\varepsilon)\exp[i\sqrt{\lambda-\alpha_k}\,x_1]\varphi_k(x^*), \]
where \(\tilde u_j(x,\varepsilon)\in H_{2,\beta}\), and \(v_{j,k}(x_1,\varepsilon)\) are entire functions of the variable \(x_1\). Estimates of the order and type of these functions lead to the relation \(v_{j,k}(x_1)\equiv 0\), which completes the proof of the theorem.

Thus, the eigenvalues of the operator are stable with respect to small finite perturbations of the operator if \(\lambda\ne \alpha_k\) for every \(k\).

If the eigenvalue \(\lambda_0\) is located in the interval \(\sigma_k=(\alpha_{k-1},\alpha_k)\), then the eigenvalue \(\lambda_j(\varepsilon)\) is an analytic function in the interval \((-\varepsilon_0,\varepsilon_0)\) under the condition that, for these values, the operator \(\mathcal L+\varepsilon\mathcal L_1\) is elliptic and \(\lambda_j(\varepsilon)\) does not leave the interval \(\sigma_k\). The only points at which the point spectrum is transformed into the continuous spectrum are the points \(\alpha_k\). If
\[ \lambda_j(\varepsilon)\xrightarrow[\varepsilon\to\varepsilon_1]{}\alpha_k, \]
then one can choose a subsequence of eigenfunctions \(u_j(x,\varepsilon_m)\) which, in every finite domain, converge in \(W_2^2\), as \(\varepsilon_m\to\varepsilon_1\), to a solution \(u(x)\ne 0\) of the equation \(\mathcal L u+\alpha_k u=0\). The limiting eigenfunction, generally speaking, does not belong to \(\mathcal L_2(D)\), but is bounded.

The theorems formulated above are obviously generalized also to the case in which the domain \(D\) outside a sufficiently large ball consists of several semi-infinite cylinders.

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

Received
27 X 1964

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