The Hilbert and Bessel properties of systems of eigenfunctions of second-order operators (Q5926550)

Let \(q\in L_2[0,1]\) and \(Lu=u''+qu\). For a sequence \((\mu_n)_{n=1}^\infty\) of complex numbers, let \((u_n)_{n=1}^\infty \) be a sequence of nontrivial solutions to \(Lu+\mu^2u=0\). The related functions \[ \widetilde u_n(x)=\begin{cases} u_n(0)+xu_n'(0)& \text{for } \mu_n=0,\cr u_n(0)\cos(\mu_n x)+\mu_n^{-1}u_n'(0)\sin(\mu_n x)& \text{for } \mu_n\neq 0,\end{cases} \] are solutions to \(u''+\mu_n^2u=0\) satisfying \(\widetilde u_n(0)=u_n(0)\) and \(\widetilde u_n'(0)=u_n'(0)\). A system \((e_n)_{n=1}^\infty \) of elements in a Hilbert space is called a Hilbert (Bessel) system if there is a constant \(\alpha >0\) (\(\beta >0\)) such that for all \(f\in H\) \[ \sum_{n=1}^\infty|(f,e_n)|^2\geq \alpha \|f\|^2\;(\leq \beta \|f\|^2). \] Conditions on \((\mu_n)_{n=1}^\infty\) and \(q\) are given such that, if \((\|\widetilde u_n\|^{-1}\widetilde u_n)_{n=1}^\infty\) is a Hilbert (Bessel) system, then \((\|u_n\|^{-1}u_n)_{n=1}^\infty\) is a Hilbert (Bessel) system. Conditions for the reversed conclusions are also given.