Renormalized contact potential in two dimensions

Abstract

We obtain for the attractive Dirac δ-function potential in twodimensional 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 3-body case is examined in detail, and in this case a variational estimate of the ground-state energy is given.

Transfer matrix formulation of scattering theory in two and three dimensions

“…The perturbation series solution of the Lippmann-Schwinger equation [18] for the delta-function potential in two dimensions is plagued with the emergence of infinities. The same problem arises in the study of the corresponding spectral problem and has led to the development of intricate renormalization schemes that amount to the introduction of a length scale for the problem [29][30][31][32][33][34][35][36][37][38]. The problem of divergences also arises in the standard treatment of the delta-function potential in 3D.…”

“…Here ∂ p := (∂ px , ∂ py ), (p x , p y ) are cartesian coordinates of p, and v(i ∂ p , z) := v(i∂ px , i∂ py , z). For details of the derivation of (32), see Appendix E. According to (32), H(z, p) vanishes for the values of z for which v(x, y, z) = 0. This together with Eq.…”

Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line

“…According to this equation, whenever z 0 takes a positive imaginary value, i.e., Re(z 0 ) = 0 and Im(z 0 ) > 0, and |z 0 | < α, the potential has a spectral singularity at some wavenumber k ∈ [ |z 0 | 2 , α 2 ) and incident angle θ 0 = arccos(z 0 /2ik). 3 Another interesting consequence of (64) is that the potentials of the form (61) with z 0 = 0 are omnidirectionally invisible for any left-incident plane wave with wavenumber k < α/2. In particular, if z 0 = 0, the potential (61) is invisible for any left-incident plane wave whose wavenumber k and incident angle θ 0 satisfy k < α ≤ 2k and |θ 0 | < arcsin(α/k − 1).…”

A many-body problem with point interactions on two-dimensional manifolds

“…One of the earliest studies on the many-body or few-body version of this model in two or three dimensions dates back to the work of Flamand [12] and the unpublished thesis of Hoppe [13], and the ones in the Soviet Union, see the references given in [2]. More recently, a perturbative renormalization to the above n-body problem has been worked out in [14] and also the three-body problem in two dimensions is discussed in [15]. It has been proved that the perturbative treatment of the three-body problem shows new divergences in three dimensions after the renormalization of the two-body sector of the problem and these divergences appear for each added new particle [14].…”