Complex powers of degenerating differential operators connected with the Klein–Gordon–Fock operator

We develop a theory of complex powers of the generalized Klein–Gordon–Fock operator
$$ m^2-\square-i\lambda\frac{\partial^2}{\partial x^2_1},\qquad\lambda>0. $$
The negative powers of this operator are realized as potential-type integrals with nonstandard metrics, while positive powers inverse to negative ones are realized as approximative inverse operators.