We propose a novel numerical method to solve the two-body Euler-Lagrange equation derived by Krotscheck, Qian, and Kohn in the paired-phonon analysis for an inhomogeneous Bose liquid at zero temperature. The new algorithm is applied to thin films of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ supported by an external potential. Numerical results are reported for density profiles, chemical potentials, binding energies, and corrective Jastrow correlation factors as a function of the particle number of the film and the strength of the external potential. The stability of this kind of film is discussed in detail. Some evidence for a long-wavelength instability of free thin films is provided. In addition, in order to unify results obtained from different derivations, it is proved that the expression for the Hartree potential reported by Krotscheck et al. is equal, within the framework of the hypernetted-chain theory, to a previously published one by Saarela, Pietiläinen, and Kallio.

• Received 19 October 1988

DOI:https://doi.org/10.1103/PhysRevB.40.4391