Spectral properties of Sturm–Liouville operators with local interactions on a discrete set

This paper deals with the spectral properties of the Sturm–Liouville operators generated by the differential expressionL(y)=1w(−(p(x)y′)′+υ(x)y) with singular coefficients υ(x)υ(x) in the sense of distributions. In particular, we study the operators with δ  -type point interactions at the centers xkxk on the positive half line in terms of energy forms. Necessary and sufficient conditions and also simple sufficient conditions are given for the spectrum of the operators to be discrete. We also prove sufficient conditions on the stability of the continuous spectrum.