Asymptotic formulas for the eigenvalues of linear differential boundary problems with indefinite weight function (Q1896657)

The problem (1) \(l(y) = \rho^n ry\), \(U_1 (y) = \cdots = U_n (y) = 0\) is considered, where \(l(y) = y^{(n)} + f_1 y^{(n - 1)} + \cdots + f_ny\), \(f_1 \in AC^0_* [0,1]\), \(f_2, \dots, f_n \in L_2 [0,1]\), \(r \in AC^k_*[0,1]\), \(\overline {r([0,1])} \subset \mathbb{R} \backslash \{0\}\), and \(f \in AC^k_* [0,1]\) means that there is a partition \(0 = a_0 < a_1 < \cdots < a_{m + 1} = 1\) such that the restriction of \(f\) to each interval \([a_\nu, a_{\nu + 1}]\), \(\nu = 0, \dots, m\), coincides almost everywhere with a function which has absolutely continuous derivatives of order up to \(k\). The conditions \(U_j (y) = 0\) are two-point boundary conditions. An eigenvalue of problem (1) is a complex number \(\lambda = \rho^n\) for which (1) has a nontrivial solution. The author derives asymptotic estimates for the characteristic determinant and arrives at the following asymptotic formulas for the eigenvalues of (1): If the boundary conditions are separated with \(\widetilde n\) conditions at 0 and \(n - \widetilde n\) conditions at 1, then there are two sequences of eigenvalues: \[ \lambda^\pm_k = \pm (-1)^{n + \widetilde n} \left( {k \pi \over \sin ({\widetilde n \over n} \pi) R^\pm} \right)^n \bigl( 1 + O(k^{-1}) \bigr),\;k \in \mathbb{N}, \] where \(R^\pm = \int^1_0 {\root n \of \max \{\pm r(t), 0\}} dt\). If the boundary conditions are regular, then similar asymptotic formulas hold, where the cases \(n = 2 \mu\), \(n = 2 \mu - 1\) and \(R^+ \neq R^-\), \(n = 2 \mu - 1\) and \(R^+ = R^-\) have to be distinguished.