Consider the following NLS:

$$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$

where $F(u):=(u + \bar{u} + |u|^2)u.$

In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. Nakanishi, and T.-P. Tsai used a change of variables to get the free evolution as a unitary group:

$$u \mapsto v:=V^{-1}u:= U^{-1} \operatorname{Re} u + i \operatorname{Im}u,$$

where $U:=\sqrt{- \Delta (2-\Delta)^{-1}}$.

Then $v$ satisfies the equation

$$i u_t v - \sqrt{- \Delta (2-\Delta)} v = - i V^{-1} i F(V v).$$

I have been trying to understand the motivation of this change of variable but no result! The main obstacles for me is the squire root, I can not see where the square root came from? Any help is appreciated.

Updates

I tried to write the equation as a dynamical system of two PDEs, real and imaginary parts, and tried to diagonalized the operator but I couldn't get the same result.