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Asymptotics in the Spectral Parameter for Solutions of $2 \times 2$ Systems of Ordinary Differential Equations
A. P. Kosarevab,
A. A. Shkalikovab a Lomonosov Moscow State University
b Moscow Center for Fundamental and Applied Mathematics
Abstract:
We consider a
$2 \times 2$ system of ordinary differential equations
$$
y'-By=\lambda Ay,
\qquad y=y(x),
\quad x \in [0, 1],
$$
where
$A=\operatorname{diag}\{a_1(x), a_2(x)\}$,
$B=\{b_{kj}(x)\}_{k, j=1}$, and all functions occurring in the matrices are complex-valued and integrable. In the case
$$
a_1,a_2, b_{21},b_{12} \in W^n_1[0,1],
\qquad
b_{11}, b_{22} \in W^{n-1}_1[0,1],
$$
we obtain
$n+1$ terms of the asymptotic expansion in powers of
$\lambda^{-1}$,
$\lambda \to \infty$, of the fundamental matrix of solutions of this equation. These asymptotic expansions are valid in the half-planes $\Pi_{\kappa}=\{\lambda \in \mathbb{C} \mid \operatorname{Re}{\lambda} \ge -\kappa \}$,
$\kappa \in \mathbb{R}$, and
$-\Pi_{\kappa}$ if
$a_1(x)-a_2(x) >0$. They hold in the sectors $S=\{\lambda \in \mathbb{C} \mid \lvert\operatorname{arg}\lambda\rvert \le \pi/2-\phi-\varepsilon\}$,
$\varepsilon > 0$, and
$-S$ under the condition that $\lvert\operatorname{arg}\{a_1(x)-a_2(x)\}\rvert<\phi<\pi /2$. The main novelty of the work is that we assume minimal conditions for the smoothness of the functions and in addition we obtain explicit formulae for matrices involved in asymptotic expansions. The results are also new for the Dirac system.
Keywords:
spectral asymptotics for solutions of ordinary differential equations and systems, regular and nonregular boundary value problems, spectral problems.
UDC:
517 Received: 27.06.2023
DOI:
10.4213/mzm14119
Fulltext:
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