On the theory of antiprior estimates in the sense of V.A. Il'in
Consider the regular Schrödinger operator \(L\) given by the expression \[ Lu=-u''(x)+q(x)u(x) \] defined on a real-line interval \(G\) with a complex-valued potential \(q\in L_1(G)\), and by some boundary conditions, whose particular form is not important for further considerations. Let \(\{u_{k0}(x),u_{k1}(x),\dots,u_{kp}(x)\}\), \(p\leq p_0\), be an unconditional basis in \(L_2(G)\), formed by eigen- and associated functions of \(L\) in the sense of \textit{V. A. Il'in} [Differ. Equations. 16, 479--496 (1980; Zbl 0473.42018) and ibid. 16, 613--633 (1980; Zbl 0527.42012)]. The authors state that the system \(\{u_{k0}(x),u_{k1}(x)+C_1u_{k0}(x),\dots, u_{kp}(x)+C_pu_{k0}(x)\}\) is an unconditional basis in \(L_2(G)\) if and only if the uniform estimate \(\| u_{k0}(x)\| \leq\text{const}\| u_{kj}(x)\| \) holds for any \(k\) and any \(j\), where \(1\leq j\leq p\leq p_0\). Further, let \(u_{kj}(x)\), \(1\leq j\leq p\) be a regular solution to the equation \(Lu_{kj}=\lambda_k(u_{kj}(x)+u_{kj-1}(x))\), \(j=1,2,\dots,p\), where \(| \Im\sqrt{\lambda_k}| \leq\text{const}\). Then the estimate \(\| u_{kj-1}(x)\| \leq\text{const}| \sqrt{\lambda_k}| ^{-1}\| u_{kj}(x)\| \) holds in the norm of \(L_2(G)\). No proofs are given.
scientific article; zbMATH DE number 5363474