Path integrals for two‐ and three‐dimensional δ‐function perturbations

Abstract

Abstract. The incorporation of two-and three-dimensional δ-function perturbations into the path-integral formalism is discussed. In contrast to the onedimensional case, a regularization procedure is needed due to the divergence of the Green-function G (V ) (x, y; E), (x, y ∈ IR 2 , IR 3 ) for x = y, corresponding to a potential problem V (x). The known procedure to define proper self-adjoint extensions for Hamiltonians with deficiency indices can be used to regularize the path integral, giving a perturbative approach for δ-function perturbations in two and three dimensions in the context of path integrals. Several examples illustrate the formalism.

On the Green function of linear evolution equations for a region with a boundary

“…We only point out that for most interesting cases of two and three-dimensional quantum problems there are no singularities within our approach, opposite to the models described in [11]. Indeed, the short distance behaviour of the Green function in d-dimensional spaces is ( [9], f.6.2.1.2)…”

“…For higher dimension (or) and other operator L, some sort of regularization should be used as e.g. in [11].…”

“…where we used the orthonormality of the functions (11). After summing up the geometrical progression we obtain…”

Semiclassics in a system without classical limit: The few-body spectrum of two interacting bosons in one dimension

“…The latter case was already discussed by Balian and Bloch [19,20] and by Sieber [21], but in all three cases the derivations were done via the energy dependent Greens function whereas we use a formalism based on 1D propagators [22,23]. For this we need the propagator for a δ-potential in 1D, which is known exactly from a path integral calculation [24] (see also [25] for higher dimensions). Notably, quantum mechanical properties, like the appearance of a bound state for attractive interaction, are well hidden inside the results (see, e.g., [26]).…”

Local realizations of contact interactions in two- and three-body problems

“…In this example the convergence of the Hamiltonian family (15) in the limit ǫ → 0 can be easily shown using elementary calculations. Since the eigenvalue problem of the scaled Pöschl-Teller operator (15) can be written in the form (10) the eigenvalues of (15) read…”