On the relationship between the Carleman condition and the equivalence of norms of root functions on various compact sets (Q2461896)

Let \(G\) be an arbitrary finite interval of the real axis and let \(L\) be a formal differential operator of order \(2m\) with real valued coefficients. The root functions of \(L\) are treated in the sense of \textit{V. A. Il'in} [Differ. Equations. 16, 479--496 (1980; Zbl 0473.42018), Differ. Equations 16, 613-633 (1980; Zbl 0527.42012)]. Namely, they are regular solutions \(u_n\) of the equation \[ Lu_n+(-1)^m\mu_n^{2m}u_n=\theta_n\mu_n^{n-1}u_{n-1}, \] where each of the numbers \(\theta_n\) is either zero (in this case, \(u_n\) is referred to as an eigenfunction) or unity (then it is required that \(\mu_n=\mu_{n-1}\) and \(u_n\) is referred to as an associated function of order \(l\), where \(\theta_n=\theta_{n-1}=\dots=\theta_{n-l+1}=1\) and \(\theta_{n-l}=0\)); moreover, \(\arg\mu_n\in(-\pi/(2m),\pi/(2m)]\). An important role in studying properties of systems of root functions is played by the following two possible assumptions on the spectrum: (a)~the Carleman condition \(\sup| \Im\mu_n| <+\infty\), and (b)~the equivalence of norms of root functions \(\| u_n\| _{L_2(K')}\leq C(K,K')\| u_n\| _{L_2(K)}\), where \(K\subset K'\) are arbitrary compacts contained in~\(\overline{G}\). The authors prove that condition (b) implies condition (a) in the following two cases: (1) \(Lu=u''+q(x)u\), \(q\in L_2(G)\), and (2) \(Lu=u^{(4)}\).