Renormalized contact potential in two dimensions

We obtain for the attractive Dirac δ-function potential in two-dimensional quantum mechanics a renormalized formulation that avoids reference to a cutoff and running coupling constant. Dimensional transmutation is carried out before attempting to solve the system, and leads to an interesting eigenvalue problem in N−2 degrees of freedom (in the center of momentum frame) when there are N particles. The effective Hamiltonian for N−2 particles has a nonlocal attractive interaction, and the Schrodinger equation becomes an eigenvalue problem for the logarithm of this Hamiltonian. The three-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given.