Abstract
Full Text
MATHEMATICS
D. P. KOSTOMAROV
ON BOUNDARY-VALUE PROBLEMS FOR EIGENVALUES AND EIGENFUNCTIONS OF ORDINARY DIFFERENTIAL EQUATIONS CONTAINING A SMALL PARAMETER AT THE HIGHEST DERIVATIVE
(Presented by Academician I. G. Petrovskii on 1 II 1957)
Let two linear differential operators of even orders be given on some interval ([a,b]):
[
L[u]=(-1)^n \sum_{i=0}^{2n} p_i(x) u^{(i)}(x);
\qquad
l[u]=(-1)^m \sum_{i=0}^{2m} q_i(x) u^{(i)}(x).
]
[
(m>0,\ n-m=k>0).
]
Consider the boundary-value problems
[
\begin{array}{ll}
\mathrm{I} &
\begin{cases}
\mu L[u]+l[u]=\lambda \rho(x)u;\
u^{(s)}(a)=u^{(s)}(b)=0 \quad (0\leq s\leq n-1);
\end{cases}
\end{array}
\tag{1}
]
[
\begin{array}{ll}
\mathrm{II} &
\begin{cases}
l[v]=\lambda \rho(x)v;\
v^{(s)}(a)=v^{(s)}(b)=0 \quad (0\leq s\leq m-1),
\end{cases}
\end{array}
\tag{2}
]
whose eigenvalues and normalized eigenfunctions we shall denote respectively by (\lambda_j(\mu), u_j(x,\mu)) and (\lambda_j, v_j(x)). We shall regard the parameter (\mu) as small and positive; then, as was shown in paper ((^1)), under certain restrictions the relations
[
\lambda_j(\mu)=\lambda_j+O(\mu^{1/2k}), \qquad
u_j(x,\mu)=v_j(x)+O(\mu^{1/2k})
]
hold.
(We note that in paper ((^1)) problems with more general boundary conditions than ours were considered.) The purpose of the present note is to compute the eigenvalues and eigenfunctions of problem I with accuracy up to and including terms of order (\mu^{1/2k}).
- We shall assume henceforth that the functions (p_i(x), q_i(x)), and (\rho(x)) satisfy the following conditions on the interval ([a,b]): the functions (p_i(x)) ((0\leq i\leq 2n-2)) are continuous; the functions (\rho(x)) and (q_i(x)) ((0\leq i\leq 2m-2)) are continuously differentiable (2k) times; the functions (p_{2n-1}(x)) and (q_{2m-1}(x)) are ((2n-1)) times differentiable, and the functions (p_{2n}(x)) and (q_{2m}(x)) are (2n) times differentiable; moreover (p_{2n}(x)>0,\ q_{2m}(x)>0). Under the assumptions made, the following lemma is valid:
Lemma 1. For every closed domain (g) of the plane of the complex variable (\lambda), for sufficiently small positive values of (\mu) ((0<\mu\leq \mu_0(g))), equation (1) has (2n) linearly independent solutions ({u_i(x,\lambda,\mu)}) ((1\leq i\leq 2n)), which, together with their derivatives with respect to (x) up to order ((2n-1)) inclusive, can be represented in the form
[
u_i^{(s)}(x,\lambda,\mu)=v_i^{(s)}(x,\lambda)+\mu^{1/k}\varphi_{is}(x,\lambda,\mu)
]
[
(1\leq i\leq 2m,\ 0\leq s\leq 2n-1),
\tag{3}
]
[
u_i^{(s)}(x,\lambda,\mu)=\mu^{-s/2k}
\left[
\left(\varepsilon_{i-2m}^{2k}\frac{q_{2m}(x)}{p_{2n}(x)}\right)^{s/2k} h(x)+
\mu^{1/2k}\varphi_{is}(x,\lambda,\mu)
\right]
\exp\left[
\frac{\varepsilon_{i-2m}}{\mu^{1/2k}}
\int_{x_i}^{x} \sqrt[2k]{\frac{q_{2m}(t)}{p_{2n}(t)}}\,dt
\right]
\tag{4}
]
[
(2m+1\leq i\leq 2n,\quad 0\leq s\leq 2n-1),
]
where ({v_i(x,\lambda)}) ((1\leq i\leq 2m)) are linearly independent solutions of the degenerate equation (2), which are analytic functions of (\lambda) for (\lambda\in g); ({\varepsilon_\alpha}) are the roots of the equation (\varepsilon^{2k}+(-1)^k=0), and
[
\varepsilon_\alpha=e^{\,i\frac{\pi}{k}\left(\alpha-\frac{k+1}{2}\right)}
\quad (1\leq \alpha\leq 2k);
]
(x_i=b) ((2m+1\leq i\leq 2m+k)); (x_i=a) ((2m+k+1\leq i\leq 2n));
[
h(x)=
\left(\frac{p_{2n}(x)}{q_{2m}(x)}\right)^{\frac{2n+2m-1}{4k}}
\exp\int_a^x
\left(
\frac{q_{2m-1}(t)}{q_{2m}(t)}
-
\frac{p_{2n-1}(t)}{p_{2n}(t)}
\right)\,dt;
]
(\varphi_{is}(x,\lambda,\mu)) ((1\leq i\leq 2n,\; 0\leq s\leq 2n-1)) are functions, ((2n-s)) times continuously differentiable with respect to (x) on the interval ([a,b]) and analytic with respect to (\lambda) and (\mu) ((\lambda\in g,\;0<\mu\leq\mu_0(g))), and in the indicated domains of variation of the variables (x,\lambda), and (\mu),
[
|\varphi_{is}(x,\lambda,\mu)|\leq \varphi_0,
]
(\varphi_0=\varphi_0(g)) is some constant.
- The eigenvalues of the degenerate problem II are determined as the roots of the equation
[
F(\lambda)=\det
\left|
\begin{matrix}
v_i^{(s)}(a,\lambda)\
v_i^{(s)}(b,\lambda)
\end{matrix}
\right|=0
\quad
(1\leq i\leq 2m,\;0\leq s\leq m-1),
\tag{5}
]
where ({v_i(x,\lambda)}) is a fundamental system of solutions of equation (2). We shall assume that these solutions are entire functions of (\lambda).
Suppose that in some closed domain (g) one of the minors of order ((2n-1)) of determinant (5), for example the minor (\Delta(\lambda)), which is obtained by deleting the first column and the row with the quantities (v_i^{(s_0)}(a,\lambda)) ((0\leq s_0\leq m-1)), has no zeros; then the boundary-value problem
[
\text{II a}\quad
\left{
\begin{aligned}
&l[v]=\lambda\rho(x)v;\
&v^{(s)}(a,\lambda)=0\quad (0\leq s\leq m-1,\;s\ne s_0),\quad
v^{(s)}(b,\lambda)=0\quad (0\leq s\leq m-1);\
&\int_a^b \rho(x)v^2(x,\lambda)\,dx=1
\end{aligned}
\right.
]
has in the given domain a solution (v_0(x,\lambda)) unique up to a factor (\pm 1). This solution is an analytic function of (\lambda) for (\lambda\in g) and, by virtue of the condition (\Delta(\lambda)\ne 0), is linearly independent of the solutions of equation (2) (v_2(x,\lambda),\ldots,v_{2m}(x,\lambda)). We now turn to the analogous problem for equation (1):
[
\text{I a}\quad
\left{
\begin{aligned}
&\mu L[u]+l[u]=\lambda\rho(x)u;\
&u^{(s)}(a,\lambda,\mu)=0\quad (0\leq s\leq n-1,\;s\ne s_0);\
&u^{(s)}(b,\lambda,\mu)=0\quad (0\leq s\leq n-1);\
&\int_a^b \rho(x)u^2(x,\lambda,\mu)\,dx=1.
\end{aligned}
\right.
]
Lemma 2. For sufficiently small values of (\mu) ((0<\mu\leqslant \mu_0(g))), problem Ia has in the domain (g) a solution (u_0(x,\lambda,\mu)), unique up to a factor (\pm 1). This solution depends analytically on (\lambda) and (\mu) ((\lambda\in g,\ 0<\mu\leqslant \mu_0(g))) and can be represented in the form
[
\begin{aligned}
u_0(x,\lambda,\mu)
= c(\lambda,\mu){v_0(x,\lambda)
&+ \mu^{1/2k}V_0(x,\lambda)
+ \mu^{1/k}V(x,\lambda,\mu) \
&+ \mu^{m/2k}(W_1(x,\lambda,\mu)+W_2(x,\lambda,\mu))}.
\end{aligned}
\tag{6}
]
Here (v_0(x,\lambda)) is the solution of problem IIa;
[
\begin{aligned}
V_0(x,\lambda)=A_k\Bigg[
&-v_0^{(m)}(a,\lambda)
\left(\frac{p_{2n}(a)}{q_{2m}(a)}\right)^{1/2k}
\frac{\Delta_1(x,\lambda)}{\Delta(\lambda)} \
&+v_0^{(m)}(b,\lambda)
\left(\frac{p_{2n}(b)}{q_{2m}(b)}\right)^{1/2k}
\frac{\Delta_2(x,\lambda)}{\Delta(\lambda)}
\Bigg],
\end{aligned}
]
where (\Delta_1(x,\lambda)) and (\Delta_2(x,\lambda)) are determinants obtained from the determinant (\Delta(\lambda)): the first by replacing the row with the quantities (v_i^{(m-1)}(a,\lambda)), the second—the row with the quantities (v_i^{(m-1)}(b,\lambda)), by the functions (v_i(x,\lambda)) ((2\leqslant i\leqslant 2m)); in the case (s_0=m-1), (\Delta_1(x,\lambda)\equiv 0),
[
A_k=\sum_{\alpha=1}^{k}\cos\frac{\pi}{k}\left(\alpha-\frac{k+1}{2}\right);
]
(V(x,\lambda,\mu)) is a function (2n) times continuously differentiable with respect to (x) on the interval ([a,b]) and analytic with respect to (\lambda) and (\mu) ((\lambda\in g,\ 0<\mu\leqslant \mu_0(g))), and in the indicated domains of variation of the variables (x,\lambda,\mu),
[
|V^{(s)}(x,\lambda,\mu)|\leqslant C
\qquad (0\leqslant s\leqslant 2n-1),
]
(C=C(g)) is some constant;
[
W_1(x,\lambda,\mu)=
-\left(\frac{p_{2n}(b)}{q_{2m}(b)}\right)^{m/2k}
\frac{v_0^{(m)}(b,\lambda)}{h(b)\delta_1}
\left|
\begin{array}{cccc}
u_{2m+1}(x,\lambda,\mu)&\cdots&u_{2m+k}(x,\lambda,\mu)\
\varepsilon_1^{m+1}&\cdots&\varepsilon_k^{m+1}\
\cdots&\cdots&\cdots\
\varepsilon_1^{n-1}&\cdots&\varepsilon_k^{n-1}
\end{array}
\right|;
]
[
W_2(x,\lambda,\mu)=
-\left(\frac{p_{2n}(a)}{q_{2m}(a)}\right)^{m/2k}
\frac{v_0^{(m)}(a,\lambda)}{h(a)\delta_2}
\left|
\begin{array}{cccc}
u_{2m+k+1}(x,\lambda,\mu)&\cdots&u_{2n}(x,\lambda,\mu)\
\varepsilon_{k+1}^{m+1}&\cdots&\varepsilon_{2k}^{m+1}\
\cdots&\cdots&\cdots\
\varepsilon_{k+1}^{n-1}&\cdots&\varepsilon_{2k}^{n-1}
\end{array}
\right|;
]
({u_i(x,\lambda,\mu)}) ((2m+1\leqslant i\leqslant 2n)) are solutions of equation (1), representable in the form (4),
[
\delta_1=
\left|
\begin{array}{ccc}
\varepsilon_1^m&\cdots&\varepsilon_k^m\
\cdots&\cdots&\cdots\
\varepsilon_1^{n-1}&\cdots&\varepsilon_k^{n-1}
\end{array}
\right|,
\qquad
\delta_2=
\left|
\begin{array}{ccc}
\varepsilon_{k+1}^m&\cdots&\varepsilon_{2k}^m\
\cdots&\cdots&\cdots\
\varepsilon_{k+1}^{n-1}&\cdots&\varepsilon_{2k}^{n-1}
\end{array}
\right|;
]
[
c(\lambda,\mu)=1-\mu^{1/2k}\int_a^b \rho(x)v_0(x,\lambda)V_0(x,\lambda)\,dx.
]
- With the aid of Lemma 2 one can prove the following theorems.
Theorem 1. Suppose that in some open bounded domain (G) equation (5) has only simple roots. Then, for sufficiently small values of (\mu) ((0<\mu\le \mu_0(G))), problems I and II have in this domain one and the same number of eigenvalues, and
[
\lambda_j(\mu)=\lambda_j+\mu^{1/2k}\lambda_{j1}(\mu),
\tag{7}
]
where (\lambda_{j1}(\mu)) are functions analytic and bounded for sufficiently small values of (\mu) ((0<\mu\le \mu_{j0})).
Remark. Theorem 1 establishes a one-to-one correspondence between the eigenvalues of problems I and II. In paper ((1)) such a correspondence was established only in one direction—from the eigenvalues of problem II to the eigenvalues of problem I.
Theorem 2. If the operator (l[u]) is self-adjoint, then under the assumptions of Theorem 1 the relations (7) take the form
[
\lambda_j(\mu)=\lambda_j+\mu^{1/2k}c_k
\left[
\sqrt[2k]{p_{2n}(a)q_{2m}^{2k-1}(a)\bigl(v_j^{(m)}(a)\bigr)^2}
+
\sqrt[2k]{p_{2n}(b)q_{2m}^{2k-1}(b)\bigl(v_j^{(m)}(b)\bigr)^2}
\right]
+\mu^{1/k}\lambda_{j2}(\mu),
\tag{8}
]
where (v_j(x)) are the normalized eigenfunctions of problem II; (\lambda_{j2}(\mu)) are functions analytic and bounded for sufficiently small (\mu) ((0<\mu\le \mu_{j0}));
[
c_k=\sum_{\substack{\alpha_1+\cdots+\alpha_k=k-1\ \alpha_1,\ldots,\alpha_k}}
\varepsilon_1^{\alpha_1}\varepsilon_2^{\alpha_2}\cdots \varepsilon_k^{\alpha_k}.
]
In the case (n=2,\ m=1) this formula becomes the formula obtained in paper ((2)).
Remark. At each point (\lambda_j) which is a simple root of equation (5), at least one minor of order ((2n-1)) of determinant (5), for example the minor (\Delta(\lambda)), is nonzero. Thus, for the normalized eigenfunction of problem I (u_j(x,\mu)), belonging to the eigenvalue (\lambda_j(\mu)), we obtain
[
u_j(x,\mu)=u_0(x,\lambda_j(\mu),\mu),
\tag{9}
]
where (u_0(x,\lambda,\mu)) is the solution of problem Ia in a neighborhood of the point (\lambda_j). By virtue of formula (6), for the function (u_0(x,\lambda,\mu)) we shall have
[
u_j^{(s)}(x,\mu)=
\left(
1-\mu^{1/2k}\int_a^b \rho(x)v_0(x,\lambda_j)V_0(x,\lambda_j)\,dx
\right)
\left[v_0^{(s)}(x,\lambda_j(\mu))+\mu^{1/2k}V_0(x,\lambda_j)+O(\mu^{1/k})\right],
]
[
(0\le s\le 2n-1,\quad a+\delta\le x\le b-\delta,\ \delta>0),
\tag{10}
]
since the functions (W_1(x,\lambda,\mu)), (W_2(x,\lambda,\mu)) and their derivatives with respect to (x) decrease exponentially as (\mu\to 0) at any interior point of the interval ([a,b]). For the eigenfunction itself and its first ((m-2)) derivatives, formula (10) remains valid on the entire interval ([a,b]). The terms (W_1(x,\lambda,\mu)) and (W_2(x,\lambda,\mu)) turn out to be significant only when calculating the higher derivatives of the eigenfunction: the first in a neighborhood of the point (x=b), the second in a neighborhood of the point (x=a).
Moscow State University
named after M. V. Lomonosov
Received
15 I 1957
References
- J. Moser, Comm. Pure and Appl. Math., 8, 251 (1955).
- V. B. Glasko, DAN, 108, No. 5, 767 (1956).