2017/04/10 by А. М. Савчук, Savchuk, Artem, Андрей Андреевич Шкаликов +1
Mathematics · #34L40 #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Mathematical functions and polynomials #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1704.02736
openalex publication_date 2017/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper deals with differential equations of the form
tau(y)-
lambda\n2m
varrho(x) y = 0,
quad
tau(y)\n=
sumk,
,s=0m(
tauk,
,s(x)y(m-k)(x))(m-s), where n=2m geqslant\n2, \λ is the large complex parameter, the positive functions \n varrho and \τ0,0 belong to W1,1[0,1] and the complex valued\ncoefficients \τk,s are such that the anti-derivatives\n\τk,s(-l) belong to L2[0,1], provided that l=\min k,s . Here\nthe anti-derivatives are understood in the sense of distributions. The above\nequation can be reduced to the n-th order system of differential equations of\nthe form
mathbf y'=
lambda
rho(x)
mathrm B
mathbf y+
mathrm A(x)
mathbf\ny+
mathrm C(x,
lambda)
mathbf y with constant matrix mathrm B and\nsummable matrices mathrm A(x) and mathrm C(x,\λ). The first\nobjective of the paper is obtain new results on asymptotic representation for\nthe matrix of fundamental solutions of the last equation with respect to\n\λ\→\∞ in certain sectors of the complex plane. The second\nobjective is to apply the obtained results for analyzing the asymptotic\nrepresentation of fundamental solutions of the first scalar equation with\ndistribution coefficients.\n