An Indefinite Inverse Spectral Problem of Stieltjes Type

Abstract

We consider a regular indefinite Krein–Feller differential expression of Stieltjes type \(-D_mD_x\). This can be regarded as an indefinite generalization of a vibrating Stieltjes string wearing only concentrated (now positive or negative) “masses” which accumulate at a finite right endpoint. From the general theory of indefinite Krein–Feller operators we conclude a number of spectral properties. In particular, we obtain a spectral function \(\sigma \) which is non-increasing on \((-\infty ,0)\) and non-decreasing on \((0,\infty )\) and which allows the existence of all moments \(\int \lambda ^n \; d\sigma \) for \(n \in {\mathbb N}\). The main result of the present paper is an inverse statement: Starting from a function \(\tau \) with properties like \(\sigma \), the (unique) “masses” of an indefinite Krein–Feller operator of Stieltjes type are reconstructed such that \(\tau \) belongs to the same so-called spectral class like the associated spectral function. All elements of this class are identified as the spectral functions of similar operators with a generally “heavy” right endpoint (and one additional function).