PhD thesis in English

Appendix A Numerical solution of the GP equationFor the numerical solution of the GP equation, we use algorithms describedin Ref. [87]. Original codes are written using the Fortran programming language,while we implemented the numerical procedure in the C programming language. Forcompleteness, in this Appendix we outline the main steps of the applied numericalmethod for the simplest, spherically-symmetric case.In general, there are two different type of questions that we want to answer bysolving the GP equation: either we are interested in the equilibrium configurations,i.e. stationary solutions, or we simulate real-time dynamics of the system. It turnsout that both situations can be treated on an equal footing by using propagation inreal and imaginary time, t and τ respectively, which are connected by the expression:i t = τ .(A.1)The imaginary-time propagation is very useful and efficient technique for obtainingstationary states of both linear and nonlinear systems. Essentially, it is equivalentto the minimization of the energy functional, and here we explain its basics for thecase of a linear system. The main underlying identity is given by|ψ 0 〉 = limτ→∞e −τĤ|ψ initial 〉 ,(A.2)where |ψ initial 〉 is an arbitrary initial state, which has a nonzero overlap with theground-state |ψ 0 〉 of a system. Eq. (A.2) states that after long enough propagationin the imaginary time, we will obtain the ground state of the system. This canbe easily understood by decomposing the initial state into the eigenvectors of theHamiltonian Ĥ, e −τĤ|ψ ∞∑initial 〉 = 〈ψ k |ψ initial 〉e −τE k|ψ k 〉 , (A.3)k=0by noting that the coefficients in front of all eigenstates decay exponentially inthe imaginary time τ, and that the slowest decaying coefficient is the one in front137