[Paper Review] Asymptotic formulas for fundamental system of solutions of high order ordinary differential equations with coefficients -- distributions

А. М. Савчук, Андрей Андреевич Шкаликов|arXiv (Cornell University)|Apr 10, 2017

Differential Equations and Boundary Problems16 references3 citations

TL;DR

This paper establishes asymptotic formulas for fundamental systems of solutions to high-order ordinary differential equations with distributional coefficients, using a transformation to a first-order system with a parameter-dependent potential. The key contribution is the derivation of precise asymptotic behavior of solutions as the spectral parameter λ → ∞, under integrability and smoothness conditions on the coefficients, extending classical results to the case of distributional coefficients in the operator structure.

ABSTRACT

This paper deals with differential equations of the form $$ τ(y)- λ^{2m} \varrho(x) y = 0, \quad τ(y) =\sum_{k,\,s=0}^m(τ_{k,\,s}(x)y^{(m-k)}(x))^{(m-s)}, $$ where $n=2m\geqslant 2$, $λ$ is the large complex parameter, the positive functions\ $\varrho$\ and\ $τ_{0,0}$ \ belong to $W^{1,1}[0,1]$ and the complex valued coefficients $τ_{k,s}$ are such that the anti-derivatives $τ_{k,s}^{(-l)}$ belong to $L_2[0,1]$, provided that $l=\min\{k,s\}$. Here the anti-derivatives are understood in the sense of distributions. The above equation can be reduced to the $n$-th order system of differential equations of the form $$ \mathbf y'=λρ(x)\mathrm B\mathbf y+\mathrm A(x)\mathbf y+\mathrm C(x,λ)\mathbf y $$ with constant matrix $\mathrm B$ and summable matrices $\mathrm A(x)$ and $\mathrm C(x,λ)$. The first objective of the paper is obtain new results on asymptotic representation for the matrix of fundamental solutions of the last equation with respect to $λ o\infty$ in certain sectors of the complex plane. The second objective is to apply the obtained results for analyzing the asymptotic representation of fundamental solutions of the first scalar equation with distribution coefficients.

Motivation & Objective

• To derive asymptotic formulas for fundamental solutions of high-order ordinary differential equations with coefficients in the space of distributions.
• To extend classical asymptotic analysis of differential equations to the case where coefficients are distributions, particularly in the context of Sturm-Liouville and quasi-differential operators.
• To analyze the behavior of solutions as the spectral parameter λ → ∞, under minimal regularity assumptions on the coefficients.
• To establish conditions under which the fundamental system of solutions admits uniform asymptotic expansions in angular sectors of the complex λ-plane.
• To generalize results from the regular coefficient case to the singular (distributional) coefficient case, preserving key spectral and asymptotic properties.

Proposed method

• Transform the 2m-th order equation into a first-order system of the form y′ = λV(x)y + A(x)y + C(x,λ)y on [0,1], with V(x) = ρ(x)B, where B is diagonal and ρ ∈ L₁[0,1].
• Use a change of variables u(x) = W(x)y(x) to reduce the system to a form with a diagonal leading term, enabling asymptotic analysis via variation-of-constants and integral representations.
• Apply techniques from spectral theory and oscillatory integrals, particularly estimating ∫₀ˣ e^{i r t} a(t) dt as r → ∞, to control the behavior of solutions.
• Employ L₁ and L₂ estimates on coefficients and their derivatives, including conditions such as τ₀⁻¹/² ∈ L₂ and τₖ,ₛ⁽⁻ˡ⁾ ∈ L₂ with l = min{k,s}, to ensure the existence and asymptotic control of solutions.
• Use the variation-of-constants formula to derive integral representations of the fundamental solution matrix Y(x,λ), and analyze its determinant and invertibility via trace and logarithmic derivative arguments.
• Establish asymptotic behavior in angular sectors arg(λ) ∈ (α, β) by analyzing the exponential growth/decay of solutions associated with eigenvalues of the diagonal matrix B.

Experimental results

Research questions

• RQ1 How do fundamental solutions of high-order ODEs behave asymptotically as the spectral parameter λ → ∞ when the coefficients are distributions?
• RQ2 What conditions on the coefficients ensure the existence and uniform asymptotic expansion of the fundamental solution matrix in angular sectors of the complex plane?
• RQ3 To what extent can classical asymptotic formulas for regular coefficients be extended to the case of distributional coefficients in quasi-differential operators?
• RQ4 How does the structure of the potential matrix V(x) = ρ(x)B influence the asymptotic behavior of solutions, particularly in terms of exponential factors exp(λ b_j p(x))?
• RQ5 What role do the L₁ and L₂ integrability conditions on the coefficients and their derivatives play in ensuring the validity of the asymptotic formulas?

Key findings

• The fundamental solution matrix Y(x,λ) exists and is invertible for |λ| > λ₀ and x ∈ [0,1], with det Y(x,λ) ≠ 0, ensuring a well-defined fundamental system.
• As λ → ∞, the solution matrix Y(x,λ) admits an asymptotic expansion of the form exp(λB p(x)) times a lower-order correction, where p(x) = ∫₀ˣ ρ(t) dt and B is diagonal.
• The error term in the asymptotic expansion is controlled by o(1) in L₁ norm as |λ| → ∞, and under stronger conditions, it is O(|λ|⁻¹), improving the rate of convergence.
• The asymptotic behavior is uniform in x ∈ [0,1] and in angular sectors arg(λ) ∈ (α + ε, β − ε) for any ε > 0, extending previous results to the distributional coefficient case.
• The conditions τ₀⁻¹/² ∈ L₂ and τₖ,ₛ⁽⁻ˡ⁾ ∈ L₂ with l = min{k,s} are sufficient to ensure the existence and asymptotic control of the fundamental system.
• The results generalize classical Birkhoff and Tamarkin-type asymptotics to equations with distributional coefficients, including cases such as −y′′ + qy = λ²ϱy with q ∈ W⁻¹₂[0,1], and extend to quasi-differential operators with singular potentials.