PhD thesis in English

of the ground state. Imaginary-time propagation does not preserve the norm ofthe state, and we have to renormalize the state manually after each iteration step.Therefore, it is obvious that after certain time τ, the contribution of higher stateswill be negligible, and thus we will arrive at the result given by Eq. (A.2). Asimilar reasoning is valid also for nonlinear systems, hence in the case of the GPequation we apply the transformation (A.1) and use imaginary-time propagation tofind the stationary states. From the numerical side, imaginary-time and real-timepropagation can be implemented in a very similar manner, and we briefly presentonly the details of numerical implementation of the real-time propagation.To simplify the expression for the Laplacian in a spherically-symmetric case, wefirst apply a commonly used rescaling:φ(r, t) =ψ(r, t)r, (A.4)and transform the spherically-symmetric GP equation into its simplified form:∂φ(r, t)i∂t[= − 1 ∂ 22 ∂r + 1 2 2 r2 + g∣φ(r, t)r] ∣ φ(r, t) . (A.5)∣2Next, we split the propagation into two steps, which correspond to the two parts ofEq. (A.5):∂φ(r, t)i∂t∂φ(r, t)i∂t=[12 r2 + g∂ 2∣φ(r, t)r= − 1 2 ∂r2φ(r, t) .(A.7)] ∣ φ(r, t) , (A.6)∣2This is the split-step approximation, valid for the short propagation time. It ismotivated by a possibility to treat each of the two previous equations in a speciallysuited way: the first equation deals with the part of the Hamiltonian which isdiagonal in the coordinate space, while the second equation considers the kineticterm. To perform the time discretization, we introduce the index n, which countsthe time slices φ(r, t) ≡ φ n (r), where t = nε, and ε is a discrete time-step of thepropagation. According to Eqs. (A.6) and (A.7), there are two different steps to beperformed for one time-step, in the propagation from t to t + ε. Correspondingly,we introduce the notation φ n+1/2 (r), which is the value of the wavefunction after the138