MATHEMATICS

M. I. VISHIK and Corresponding Member of the Academy of Sciences of the USSR L. A. LYUSTERNIK

PERTURBATION OF EIGENVALUES AND EIGENELEMENTS FOR CERTAIN NON-SELF-ADJOINT OPERATORS

In ((^1)) the authors investigated the perturbation of solutions of certain linear equations—algebraic and differential. In the present note, analogous methods are used to study the question of the perturbation of eigenvalues and elements (in the non-self-adjoint case)*.

1. Algebraic case. Let (A_0, A_1), and (A_\varepsilon = A_0 + \varepsilon A_1) be (n)-dimensional matrices (generally speaking, non-Hermitian), and let (\lambda = 0) be an eigenvalue of (A_0); to it corresponds an invariant space (S_0) of dimension (N)—the linear span of the eigenvectors and associated vectors corresponding to the eigenvalue (\lambda = 0).

As is known, for the matrix (A_0) there exists a Jordan basis ({x_{ij}}) corresponding to the eigenvalue (\lambda = 0), where ({x_{i0}}) ((i = 1, 2, \ldots, r)) is a complete system of eigenvectors corresponding to the eigenvalue (\lambda = 0): (A_0 x_{i0} = 0); (x_{ij}) ((j = 1, 2, \ldots, n_i - 1)) are the associated vectors corresponding to (x_{i0}) (of the (j)-th order), forming a Jordan chain of length (n_i) and satisfying the relations (A_0 x_{ij} = x_{i,j-1}) (for (j = 1, 2, \ldots, n_i - 1)).

Let (E_{i0}) be the linear span of all eigenvectors (x_{i0}) to which there correspond Jordan chains of one and the same length (n_i), (n_1 > n_2 > \cdots > n_l > 1); (p_i) is the dimension of (E_{i0}); (E_{ij}) ((j = 1, 2, \ldots, n_i - 1)) is the linear span of the eigenvectors (x_{ij}) of the (j)-th order associated with (x_{i0}). Obviously, the dimension (N) of the space (S_0) is equal to

[
N = \sum_{i=1}^{l} p_i n_i
]

and

[
S_0 = \sum_{i=1}^{l} \sum_{j=0}^{n_i-1} E_{ij}.
]

Theorem. Under the described structure of the invariant space (S_0) corresponding to the eigenvalue (\lambda = 0) of the matrix (A_0), the matrix (A_\varepsilon) has (N = \sum_{i=1}^{l} n_i p_i) eigenvalues (\lambda_\varepsilon) tending to zero as (\varepsilon \to 0); among them, (n_i p_i) eigenvalues are expressed by power series of the form

[
\lambda_\varepsilon = \sum_{k=1}^{\infty} \lambda_k \varepsilon^{k/n_i},
\tag{1}
]

* The contents of both notes were reported at the Fifth All-Union Conference on Functional Analysis in Baku in October 1959.

where (\lambda_k=\lambda_k(q)) ((q=1,2,\ldots,n_i p_i)), and they correspond to the eigenvectors

[
v_\varepsilon=\sum_{k=0}^{\infty} v_k \varepsilon^{k/n_i}.
\tag{2}
]

Here (v_0=\sum_{j=1}^{i} v_{j0}), (v_{j0}\in E_{j0}); (v_{i0}) and (\lambda_1^{n_i}) are the eigenvector and eigenvalue, respectively, of a certain operator (C_i), acting from (E_{i0}) into (E_{i0}), while (v_{j0}) are determined from (v_{i0}) by applying a fixed linear operator (B_{ij}), acting from (E_{i0}) into (E_{j0}).

In the formulation of the theorem it must be stipulated that each operator (C_i) has nonzero and distinct eigenvalues. The matrix of the operator (C_1) has the form (C_1=|(A_1 y_i,z_j)|{i,j=1,\ldots,p_1}), where (y_i) ((i=1,\ldots,p_1)) form a basis in (E^}), and (z_j) ((j=1,\ldots,p_1)) form the corresponding basis in the subspace (E_{10), which plays the analogous role for the adjoint matrix (A_0^). Similarly, on the basis of a certain recurrent process, (C_2,\ldots,C_l), as well as (B_{ij}), are constructed.

The coefficients (\lambda_k, v_k) ((k=1,2,\ldots)) of the expansions (1), (2) are obtained by successively solving the system of equations

[
A_0 v_0=0;
\tag{3}
]

[
A_0 v_k=b_k=
\begin{cases}
\displaystyle \sum_{j=1}^{k} \lambda_j v_{k-j}, & \text{for } k<n_i,\[1.2em]
\displaystyle \sum_{j=1}^{k} \lambda_j v_{k-j}-A_1 v_{k-n_i}, & \text{for } k\ge n_i,
\end{cases}
\tag{4}
]

obtained as a result of comparing coefficients in the equality ((A_\varepsilon-\lambda_\varepsilon I)v_\varepsilon=0) after substituting into it the expansions (1), (2). At the same time, since we are on the spectrum, the solvability conditions for equations (4) are used, as a result of which (v_k) and (\lambda_{k+1}) are determined at the ((n_i+k))-th step of the iteration.

In the simplest case where there is one Jordan block, the expansions of (\lambda_\varepsilon) and (v_\varepsilon) will be the same as those given below in § 2. The theorem also remains valid if the matrix (A_\varepsilon) is expanded in (\varepsilon) into a convergent power series.

2. Let us now consider the case of differential operators, restricting ourselves, for simplicity, to operators of second order and to their linear dependence on (\varepsilon). Let (L_0) be an elliptic (generally speaking, non-self-adjoint) operator of second order; (L_1), an operator of order not higher than the second; (L_\varepsilon=L_0+\varepsilon L_1), and suppose these operators act on functions (u) defined in a domain (D) with boundary (\Gamma) and vanishing on (\Gamma): (u|_\Gamma=0). The coefficients of (L_0) and (L_1) are continuous in (D+\Gamma), and the boundary (\Gamma) is smooth.

Suppose that, under these boundary conditions, the operator (L_0) has (\lambda=0) as an eigenvalue. As is known, by a Jordan chain (u_0,u_1,\ldots,u_{n-1}), corresponding to the eigenvalue (\lambda=0), one understands successive solutions of the problems

[
\begin{gathered}
L_0 u_0=0 \qquad u_0|\Gamma=0;\
L_0 u_i=u, \qquad u_i|_\Gamma=0 \quad (i=1,\ldots,n-1).
\end{gathered}
\tag{5}
]

The number of elements of such a chain ({u_i}) is finite.

The theorem formulated above for matrices turns out to be valid in our case as well. We shall present the construction of the expansion of the eigenvalues (\lambda_\varepsilon) of the operator (L_\varepsilon) and of its eigenfunctions (v_\varepsilon) in the case when there is only one Jordan chain and its length is equal to (n). As

and above, we shall seek the expansions of (\lambda_\varepsilon) and (v_\varepsilon) in the form

[
\lambda_\varepsilon=\sum_{k=1}^{\infty}\lambda_k\varepsilon^{k/n},\qquad
v_\varepsilon(x)=\sum_{k=0}^{\infty}v_k(x)\varepsilon^{k/n}.
\tag{6}
]

Substituting these expansions into the equation (((L_0+\varepsilon L_1)-\lambda_\varepsilon I)v_\varepsilon=0), we obtain the system of equations

[
L_0v_0=0;\qquad v_0|_{\Gamma}=0;
\tag{7}
]

[
L_0v_k=b_k,\qquad
b_k=
\begin{cases}
\displaystyle \sum_{j=1}^{k}\lambda_j v_{k-j}, & \text{for } 0<k<n,\[6pt]
\displaystyle \sum_{j=1}^{k}\lambda_j v_{k-j}-L_1v_{k-n}, & \text{for } k\ge n,
\end{cases}
\tag{8}
]

[
v_k|_{\Gamma}=0\qquad (k=1,2,\ldots).
]

Denote by (u_0) and (z_0) the eigenfunctions of the given and adjoint operators:

[
L_0u_0=0,\qquad u_0|{\Gamma}=0;\qquad
L_0^{*}z_0=0,\qquad z_0|=0.
\tag{9}
]

Then, as is known, the solvability condition for problem (8) is the fulfillment of the equality

[
\iint_D b_k z_0\,dx=(b_k,z_0)=0.
\tag{10}
]

It is seen from (7) that (v_0=u_0) (or (Cu_0,\ C=\mathrm{const})); the condition (10) for (k=n) (if (u_0) is normalized accordingly and it is taken into account that ((v_{k-j},z_0)=0) for (n-2\ge j>0)) leads to the equation

[
\lambda_1^{\,n}-(L_1u_0,z_0)=0.
\tag{11}
]

We make the assumption

[
(L_1u_0,z_0)=C\ne 0.
]

Thus from (11) one determines (n) distinct values (\lambda_1=\sqrt[n]{C}). Choose one of them and denote it by (\lambda_1). Then from (8), for (k=1), we find (v_1). Further, using successively (10) and (8), we find
(\lambda_2,v_2,\ldots,\lambda_p,v_p,\lambda_{p+1},\ldots)

The proof of convergence, as in the algebraic case, reduces to the construction of a corresponding majorant. Under the corresponding conditions, the iterative process also works in the general case.

1. In ((^{2})), § 9, for the self-adjoint case, the perturbation of eigenvalues and eigenfunctions is studied when the perturbed operator is of higher order than the unperturbed one. In this case boundary-layer phenomena arise. In the non-self-adjoint case, when the perturbed operator is of higher order than the unperturbed one, the expansion for the eigenvalues is analogous to formula (6), while in the expansion for the eigenfunctions additional terms of boundary-layer type will appear, as in ((^{2})).

Received
26 X 1959

CITED LITERATURE

({}^{1}) M. I. Vishik, L. A. Lyusternik, DAN, 129, No. 6 (1959). ({}^{2}) M. I. Vishik, L. A. Lyusternik, Uspekhi matem. nauk, 12, issue 5 (77), 3 (1957).