Spectral asymptotics for solutions of 2× 2 system of ordinary differential equations of the first order

2022/12/12 by Alexey Pavlovich Kosarev, Kosarev, A. P., Андрей Андреевич Шкаликов +1
Mathematics · #Spectral Theory in Mathematical Physics #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods

paper · pdf · doi:10.48550/arxiv.2212.06227

Abstract

The aim of the paper is to find representation for solutions of 2× 2 system of ordinary differential equations y^′ - B(x)y = λA(x)y, x ∈ [0, 1], where A(x) = diag\a₁(x), a₂(x)\, B(x) = (b_ij(x)), a₁(x) > 0, a₂(x) < 0 and all the functions a_i, b_ij belong to the Sobolev spaces Wⁿ₁[0,1] for given integer n\geqslant 0. We prove that there exists a fundamental matrix of solutions for the above system, which have representation Y(x, λ) = M(x)(I + \fracR¹(x)λ + … + (Rⁿ(x))/(λⁿ) + o(1)λ^-n)E(x, λ), where o(1) → 0 uniformly for x∈ [0,1] as the spectral parameter λ→ ∞ in the half plane \Re λ>-κ or \Re λ<κ, where κ is any fixed real number. The main novelty is that we give explicit formulae for all matrices M,E and R^m in this representation.

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