The variational theorem for the scattering length in low dimensions and its applications to universal systems

Abstract

Abstract The variational theorem for the scattering length (Cherny and Shanenko 2000 Phys. Rev. E 62 1646–1659) is extended to one and two dimensions. It is shown that the arising singularities can be treated in terms of generalized functions. The variational theorem is applied to a universal many-body system of spinless bosons. The extended Tan adiabatic sweep theorem is obtained for interacting potentials of arbitrary shape with the variation of the one-particle dispersion. The pair distribution function is calculated at short distances by means of the variation of the potential. The suggested scheme is based on simple quantum mechanics; it is physically transparent and free from any divergence.