In this paper we present our partial results of numerical investigation of the quantum Coulomb field |u\rangle . In particular we investigate the matrix of the C₁=-1/2 M_{\mu \nu} M^{\mu \nu} operator in Jacobi base, obtained from orthonormalization of C₁ⁿ|u\rangle vectors and photon distribution in the bound state of the C₁ operator, in search for some critical values of the coupling constant e² of the theory. So far our results are negative, that is, all characteristics we studied are smooth functions of 0²<\pi .

PACS numbers: 03.70.+k, 12.20.Ds