Uniform asymptotic integration of a family of linear differential systems (Q2732530)

The authors present an extension of Levinson's theorem on the asymptotic integration. Consider a family of equations, indexed by a parameter \(z\) (on a set \(Q\subset\mathbb{C})\), of the form NEWLINE\[NEWLINEy'(x,z)= \bigl(\Lambda (x,z)+R (x,z) \bigr)y(x,z),\;\left('={\partial \over\partial x}\right) \tag{1}NEWLINE\]NEWLINE where \(\Lambda\) is a diagonal matrix, \(\Lambda\) and \(R\) are continuous in \(z\), \(|\Lambda(x,z) |\leq a(x)\), \(|R(x,z)|\leq\rho(x)\) with \(a\in L_{1, \text{loc}} ([c,\infty))\), \(\rho\in L_1([c,\infty))\). The authors give conditions which guarantee that (1) has a solution of the form NEWLINE\[NEWLINEy_k(x,z)= \bigl(e_k+r_k (x,z)\bigr) \exp\mu_k (x,z),\;k=1,\dots,n,NEWLINE\]NEWLINE where \(r_k\) is continuous on \([c,\infty)\times Q\), \(\lim_{x\to\infty} \sup\{|r_k(x,z) |: z\in Q\}=0\), and \(e_k\) denotes the \(k\)th unit vector. An application of the general result is given to the one-parameter Schrödinger equation \(-y''(x)+V(x) y(x)=zy(x)\) with \(V\in L_1([0,\infty))\) and where \(Q=\mathbb{C} \setminus \{0\}\).