Eigenvalues associated with Borel sets (Q2497006)

Let \(K\) be a closed Borel subset of an interval \([c, d]\) such that \([c, d] \backslash K\) consists of infinitely many disjoint open intervals \(I_k\) with \(\nu(I_1) \geq \nu (I_2) \geq \cdots\) (\(\nu\) denotes Lebesgue measure). It is known (see [\textit{F. V. Atkinson}, Discrete and continuous boundary problems. Mathematics in Science and Engineering. 8. New York-London: Academic Press (1964; Zbl 0117.05806)]) that the eigenvalue problem \[ u' = (1 - \chi_K (x)) v,\quad v' = -\lambda \chi_K(x) u, \text{ a.e. } x \in [c, d], \] \[ u(c, \lambda) = \sin \alpha,\quad v(c, \lambda) = \cos \alpha, \] can be transformed, via Prüfer angle \(\theta (x) = \theta (x, \lambda) = \arg (v(x, \lambda)+i u(x, \lambda))\), into the first-order differential equation \[ \theta' = (1 - \chi_K (x)) \cos^2 \theta + \lambda \chi_K(x) \sin^2 \theta,\quad \theta (c) = \alpha. \] Let \(\delta_k = \sum_{m=k}^\infty \nu (I_m)\). Firstly, an estimate \(\theta (d, \lambda) - \theta (c, \lambda) \leq 2 \lambda^{1/2} \delta_k^{1/2} \nu(K)^{1/2} + k \pi\) is got for all positive integer \(k\) and \(\lambda > 0\). Additionally, using a result from Atkinson, say \(\theta (d, \lambda_n) = \beta + n \pi\) (where \(\beta\) satisfies \(\cos \beta u(d) = sin \beta v(d)\)) and a clever control of the right-hand side of the inequality, a lower bound for the growth rate of \(\lambda_n\) is obtained and is realized by a Cantor-like symmetric perfect set \(K\) in \([c, d]\). This lower bound can be improved if the position of \(I_m\)s is known. As an application, some information on eigenvalues of a vibrating string with singular mass distribution is obtained, too.