A Complete Spectral Characterization of the Double Commutation Method

We provide a complete spectral characterization of the double commutation method which introduces eigenvalues into arbitrary spectral gaps of a given background Schrödinger operator in L2(R) respectively in L2((0, ∞)). Our main result proves unitary equivalence of the doubly commuted operator, restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues, with the original background operator. Moreover, we explicitly compute the (matrix) spectral function of the doubly commuted operator in terms of the background spectral function.