In this work, we study phenomena in weakly interacting dilute bosonic gases, which evade a mean-field treatment and require the inclusion of quantum fluctuations. An example of this is the recent observation of droplets in both dipolar quantum gases and Bose-Bose mixtures. For these systems, the interplay of attractive and repulsive interactions leads to a vanishing mean-field contribution. The quantum fluctuations become dominant and stabilize the gas against the collapse. Despite the inhomogeneous character of the droplets, the analysis has mostly been based on the local-density approximation. In dipolar Bose gases, the stabilization due to quantum fluctuations also allows for a supersolid phase in one-dimensional geometries. The confinement in combination with the dipolar interaction leads to a roton instability, resulting in the formation of arrays of droplets. These arrays feature an additional Goldstone mode due to the broken translational symmetry. The theoretical description of the supersolid relies on the extended Gross-Pitaevskii equation. While this allows for reliable results in finite size systems, accessing the thermodynamic limit is difficult. It is also well-known that quantum fluctuations strongly influence spontaneous symmetry breaking in one dimension, which puts into question whether the use of the extended Gross-Pitaevskii formalism is justified for one-dimensional supersolids. Here, we investigate quantum fluctuations in confined geometries and test the validity of the local-density approximation in a harmonic trap with a model system. For a one-dimensional geometry, we derive the excitation spectrum across the superfluid-to-supersolid phase transition in the thermodynamic limit and demonstrate its stability. We also investigate the influence of quantum fluctuations on the formation of the one-dimensional supersolid and show that for current experimental parameters, the use of the extended Gross-Pitaevskii equation is justified.weiterlesen