Abstract
The variational theorem for the scattering length [Cherny and Shanenko, Phys. Rev. E 62, 1646 (2000)] 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.
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