Linear, second-order problems with Sturm-Liouville-type multi-point boundary conditions

We consider the linear eigenvalue problem \tag{1} -u" = \lambda u, \quad \text{on $(-1,1)$}, where $\lambda \in \mathbb{R}$, together with the general multi-point boundary conditions \tag{2} \alpha_0^\pm u(\pm 1) + \beta_0^\pm u'(\pm 1) = \sum^{m^\pm}_{i=1} \alpha^\pm_i u(\eta^\pm_i) + \sum_{i=1}^{m^\pm} \beta^\pm_i u'(\eta^\pm_i). We also suppose that: \alpha_0^\pm \ge 0, \quad \alpha_0^\pm + |\beta_0^\pm| > 0, \tag{3} \pm \beta_0^\pm \ge 0, \tag{4} (\frac{\sum_{i=1}^{m^\pm} |\alpha_i^\pm|}{\alpha_0^\pm})^2 + (\frac{\sum_{i=1}^{m^\pm} |\beta_i^\pm|}{\beta_0^\pm})^2 < 1, \tag{5} with the convention that if any denominator in (5) is zero then the corresponding numerator must also be zero, and the corresponding fraction is omitted from (5) (by (3), at least one denominator is nonzero in each condition). In this paper we show that the basic spectral properties of this problem are similar to those of the standard Sturm-Liouville problem with separated boundary conditions. Similar multi-point problems have been considered before under more restrictive hypotheses. For instance, the cases where $\beta_i^\pm = 0$, or $\alpha_i^\pm = 0$, $i = 0,..., m^\pm$ (such conditions have been termed Dirichlet-type or Neumann-type respectively), or the case of a single-point condition at one end point and a Dirichlet-type or Neumann-type multi-point condition at the other end. Different oscillation counting methods have been used in each of these cases, and the results here unify and extend all these previous results to the above general Sturm-Liouville-type boundary conditions.