We present a new method to compute deficiency indices of operators that are homogeneous polynomials in one pair of creation and annihilation operators. To this end we prove a classification theorem for special cubic forms by means of $SU(1,1)$ transformations and derive new non-selfadjointness criteria for Jacobi-like matrices. The method presented illuminates and systemizes former results of Rabsztyn.

Deficiency indices, creation and annihilation operators, Bogolyubov transformations, Jacobi matrices

81S05 (81Q10)