One-dimensional Schrödinger operators with \(\delta'\)-interactions on a set of Lebesgue measure zero (Q2873573)

In recent years, the spectral theory of one-dimensional Schrödinger operators with \(\delta\) or \(\delta'\) potentials has become an intensely investigated field of research. The survey article by \textit{A. Kostenko} and \textit{M. Malamud} [``1-D Schrödinger operators with local point interactions: a review'', Proc. Symp. Pure Math. 87, 235--262 (2013; Zbl 1319.34003)] has a list of references with 112 items and mentions the arXiv version of the paper under review [\url{arXiv:1112.2545}] on pages 251 and 256.NEWLINENEWLINEIn the present paper, a compact set \(\Gamma\subset(a, b)\) with Lebesgue measure zero is equipped with a Radon measure \(\mu\). With the help of a function \(\beta:(a,b)\to \mathbb{R}\) (the intensity of the \(\delta'\)-interaction) which is absolutely integrable with respect to \(\mu\), a selfadjoint operator \(L_{\Gamma,\beta}\) is defined in \(L^2(\mathbb{R})\). For \(x\in\Gamma\) and functions \(\psi\) in an intermediate space between the Sobolev spaces \(W^2_2(\mathbb{R})\) and \(W^2_2(\mathbb{R}\setminus\Gamma)\), the operator \(L_{\Gamma,\beta}\) can be described by the ``boundary conditions'' NEWLINE\[NEWLINE{d\over d\mu} \psi'(x)= 0,\;{d\over d\mu} \psi(x)= \beta(x){1\over 2}[\psi'(x+ 0)- \psi'(x- 0)].NEWLINE\]NEWLINE It is shown that the essential and the absolutely continuous spectrum of \(L_{\Gamma,\beta}\) are the nonnegative half-line and that a neighbourhood to the left of zero belongs to its resolvent set. Under additional assumptions on \(\beta\), it is shown that the set of its negative eigenvalues is infinite and unbounded from below.