We describe non-flat standard Friedmann cosmology of canonical scalar field with barotropic fluid in form of non-linear Schr{o}dinger-type (NLS) formulation in which all cosmological dynamical quantities are expressed in term of Schr{o}dinger quant
ities as similar to those in time-independent quantum mechanics. We assume the expansion to be superfast, i.e. phantom expansion. We report all Schr{o}dinger-analogous quantities to scalar field cosmology. Effective equation of state coefficient is analyzed and illustrated. We show that in a non-flat universe, there is no fixed $w_{rm eff}$ value for the phantom divide. In a non-flat universe, even $w_{rm eff} > -1$, the expansion can be phantom. Moreover, in open universe, phantom expansion can happen even with $w_{rm eff} > 0$. We also report scalar field exact solutions within frameworks of the Friedmann formulation and the NLS formulation in non-flat universe cases.
Aspects of non-linear Schr{o}dinger-type (NLS) formulation of scalar (phantom) field cosmology on slow-roll, acceleration, WKB approximation and Big Rip singularity are presented. Slow-roll parameters for the curvature and barotropic density terms ar
e introduced. We reexpress all slow-roll parameters, slow-roll conditions and acceleration condition in NLS form. WKB approximation in the NLS formulation is also discussed when simplifying to linear case. Most of the Schr{o}dinger potentials in NLS formulation are very slowly-varying, hence WKB approximation is valid in the ranges. In the NLS form of Big Rip singularity, two quantities are infinity in stead of three. We also found that approaching the Big Rip, $w_{rm eff}to -1 + {2}/{3q}$, $(q<0)$ which is the same as effective phantom equation of state in the flat case.
