Asymptotics of eigenvalues in a problem of high even order with discrete self-similar weight (Q2849054)

The authors study the eigenvalue problem NEWLINE\[NEWLINE(-1)^{n}y^{(2n)} - \lambda \rho y = 0NEWLINE\]NEWLINE subject to the boundary conditions NEWLINE\[NEWLINEy^{(k)}(0) = y^{(k)}(1) = 0\text{ for }0 \leq k < n.NEWLINE\]NEWLINE Here, the weight \(\rho\) is the generalized derivative (in a distribution sense) of a so-called self-similar function \(P \in L_2[0,1]\) of zero spectral order. Roughly speaking, this means that outside of a subinterval of \([0,1]\) \(P\) is a step function and inside this subinterval it is ``similar'' to the whole function \(P\) up to a factor and another ``step''. The main results of the paper are statements on (positive and negative) eigenvalue asymptotics of exponential order. These statements extend known results for \(n=1\) obtained by the same authors in 2010. Additionally, for some particular cases the first eigenvalues are calculated numerically.