Disconjugacy characterization by means of spectral of (k,n-k) problems

This paper is devoted to the description of the interval of parameters for which the general linear $n^{\rm th}$-order equation \begin{equation} \label{e-Ln} T_n[M]\,u(t) \equiv u^{(n)}(t)+a_1(t)\, u^{(n-1)}(t)+\cdots +a_{n-1}(t)\, u'(t)+(a_{n}(t)+M)\,u(t)=0 \,,\quad t\in I\equiv[a,b], \end{equation} with $a_i\in C^{n-i}(I)$, is disconjugate on $ I $. Such interval is characterized by the closed to zero eigenvalues of this problem coupled with $(k,n-k)$ boundary conditions, given by \begin{equation} \label{e-k-n-k} u(a)=\cdots=u^{(k-1)}(a)=u(b)=\cdots=u^{(n-k-1)}(b)=0\,,\quad 1\leq k\leq n-1\,. \end{equation}