Fermi pseudopotential in arbitrary dimensions

Abstract

The theory of zero-range potentials is investigated in an arbitrary number of dimensions. Except for the trivial one-dimensional case the zero-range potentials are described by nonlocal operators called Fermi pseudopotentials. It is shown that in odd dimensions the Fermi pseudopotentials involve a very simple regularization operator. In even dimensions with the help of dimensional regularization, explicit formulas for the Fermi pseudopotentials are derived. The Green's functions, the propagators, and the exact solutions of the Lippmann-Schwinger equations are derived in explicit forms. In odd dimensions d=1 and 3 the Fermi pseudopotentials can be applied to describe multiphoton processes of atoms and molecules with very short-range interactions. In even dimension d=2 the Fermi pseudopotential can be applied to describe tunneling from laser-driven quantum wells. Physical applications involving higher d are also possible.