On conditions under which the sum of self-adjoint operators with given integer spectra is a scalar operator

Abstract

We describe the set ∑_{M1 ,...,Mn} и of parameters γ for which there exists a decomposition of the operator γI_H into a sum of n self-adjoint operators with the spectra belonging to the sets M₁ ,...,M_n .
The description of this set is performed for M_i = {0,1, ...,k_i } in the case of n ≥ 4 and in some cases for n = 3.