On an eigenvalue problem with a reciprocal-linear term

Abstract

We discuss the nonlinear eigenvalue problem and fully resolve it in two different settings. Namely, we first pose the problem in the space of the classical Dirichlet series. We find a complete set of eigenfunctions in this space, and their corresponding eigenvalues. Secondly, we pose the problem in the space of functions on the unit circle which admit a holomorphic extension to an open neighborhood of the unit disk. We obtain a complete set of solutions in this case also. In both cases the eigenfunctions are scale-similar. Both types of solutions are obtained via a reduction of the underlying problem to a set of discrete conditions on series coefficients. In fact, the discrete problem is identical for both settings. We discuss the intriguing correspondence between the two sets of solutions in detail. The methods developed here appear applicable to a broader range of differential eigenvalue problems. Furthermore, the results are applicable to circuit analysis as well as signal analysis, particularly in the context of nanocircuits with memristance.