This paper deals with singular Schrödinger operators of the form−d2dx2+∑k∈Zγkδ(⋅−zk),γk∈R, in L2(ℓ−,ℓ+), where δ(⋅−zk) is Dirac delta-function supported at zk∈(ℓ−,ℓ+) and (ℓ−,ℓ+) a bounded interval. It will be shown that interaction strengths γk points zk can chosen such way essential spectrum part discrete this self-adjoint operator coincide prescribed sets on real line.

We provide a complete spectral characterization of the double commutation method for general Sturm-Liouville operators which inserts any finite number of prescribed eigenvalues into spectral gaps of a given background operator. Moreover, we explicitly determine the transformation operator which links the background operator to its doubly commuted version (resulting in extensions and considerabl...