Absolute and uniform convergence of expansions in the root vector functions of the Schrödinger operator with a matrix potential (Q378195)

The authors consider the Schrödinger operator \[ L\psi = - I\frac{{d^2 \psi }}{{dx^2 }} + U(x)\psi \] defined on the interval \(G = (0,1)\), where \[ U(x):= \left\{ {u_{ij}(x)} \right\}_{i,j = 1}^m ,\,u_{ij} \in L_1 \left( G \right), \] is complex-valued, and \(\psi \left( x \right) = \left( {\psi ^1 \left( x \right),\psi ^2 \left( x \right),\dots,\psi ^m \left( x \right)} \right)^T\). The paper analyzes the absolute and uniform convergence of the biorthogonal expansion of a vector function from the class \(W_{p,m}^1 \left( G \right)\), \(p \geq 1\), in terms of the root functions of the operator \(L\). Sufficient conditions for absolute and uniform convergence are established and the rate of the uniform convergence of these expansions on \(\overline G = [0,1]\) is estimated.