Analytical computation of quantum corrections to non-topological soliton (bright soliton) within the saddle-point approximation

Schr\"{o}dinger field theory with an attractive self-interaction possess non-topological extended solutions with a finite energy in both finite and infinite-volume cases, namely, bright solitons. The analytical form of the solution itself is well-known, though analytical investigation of the quantum fluctuations in this background still requires more thorough investigation, for instance, analytical computation of quantum corrections to this background within the saddle-point approximation. In the present work this gap is filled. Both 2-point Green's function and quantum corrections to the background are analytically computed and properly renormalized by means of momentum cut-off procedure. It is deduced that quantum corrections are indeed small provided that particle number is large. Also, we see that perturbation modes of continuum spectrum at bright soliton background generate a gap in the energy spectrum. Moreover, it turns out that the whole spectrum is continuous modulo zero-modes, which is similar to Sine-Gordon solitons.