  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.
