Asymptotics of solutions of the Sturm-Liouville equation with respect to a parameter (Q2754829)

The authors deal with the differential equation NEWLINE\[NEWLINE(a(x)y'(x))'+[\mu^2 \rho (x) + \mu \rho_1(x) + \rho_2(x)] y(x) = 0,NEWLINE\]NEWLINE with \(a(x),\rho(x)\in L_{\infty} [0, l], \) \( \rho_j(x) \in L_1[0, l], \) \(j=1,2;\) the functions \(a(x),\rho(x)\) satisfy the inequalities \(a(x)\geq m_0>0,\rho(x)\geq m_1>0\) almost everywhere; the function \(a(x)\rho(x) \) is absolutely continuous and the parameter \( \mu \in \mathbb{C}. \) The main result of the article is a theorem on the asymptotics of a fundamental system of solutions to the problem. An estimate is obtained on the asymptotic expansions for solutions to the equation in terms of the integral modulus of continuity \(\omega_1 (\cdot, |\mu|^{-1})\) of the functions \((a(x)\rho(x))'/ (a(x)\rho^3(x))^{1/2}\) and \(\rho_1(x)/\rho(x)\) as \(|\mu|\to\infty.\) The idea of obtaining asymptotic formulas is to reduce the initial equation to a system of some integral equations.