Point interactions in two- and three-dimensional Riemannian manifolds

Abstract

We present a non-perturbative renormalization of the bound state problem of n bosons interacting with finitely many Dirac delta interactions on two and three dimensional Riemannian manifolds using the heat kernel. We formulate the problem in terms of a new operator called the principal or characteristic operator Φ(E). In order to investigate the problem in more detail, we then restrict the problem to one particle sector. The lower bound of the ground state energy is found for general class of manifolds, e.g., for compact and Cartan-Hadamard manifolds. The estimate of the bound state energies in the tunneling regime is calculated by perturbation theory. Non-degeneracy and uniqueness of the ground state is proven by Perron-Frobenius theorem. Moreover, the pointwise bounds on the wave function is given and all these results are consistent with the one given in standard quantum mechanics. Renormalization procedure does not lead to any radical change in these cases. Finally, renormalization group equations are derived and the β function is exactly calculated. This work is a natural continuation of our previous work based on a novel approach to the renormalization of point interactions, developed by S. G. Rajeev.

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

“…The many-body version of the same problem, where the particles interact via two-body delta potentials, has also been studied [25,26,49]. Recently, we have derived the generalization of the RG equations of the above one-body model with N delta centers into two-and three-dimensional Riemannian manifolds [31]. Here, we will show that the interacting version of the problem can also be studied explicitly, as we will see.…”

“…This orthofermion wavefunction corresponding to the pairing could be quite regular; yet its multiplication with the heat kernel integrated over the time variable produces a function singular as the two variables of the heat kernel approach one another. This singularity is the same as the singularity of the bound state wavefunction of a particle interacting with a delta source [31], hence it is square integrable.…”

A Perturbative Approach to the Tunneling Phenomena

“…where n = 0, 1 and Γ is the gamma function, it is easy to see that the integral that we consider is finite around η i = 0 (s i = s i ). For non self-intersecting curves, the integrals in the diagonal and off-diagonal terms in (55) are finite whenever s i = s i due to the upper bounds of the Bessel functions [14] K 0 (x)…”

“…Furthermore, point like Dirac delta interactions have also been extended to various more general cases. For our approach, to illustrate the main ideas, we are mainly concerned with the delta potentials supported by points on flat and hyperbolic manifolds [13,14,15], and delta potentials supported by curves in flat spaces, and its various relativistic extensions in flat spaces [16,17,18,19].…”

Geometric scattering of a scalar particle moving on a curved surface in the presence of point defects

“…The case of a single delta-function potential, which corresponds to taking N = 1 in (5), has been thoroughly investigated in [14,15,16,17,18,19,20,21]. See also [22] and references therein. The mathematical reason for the emergence of divergences in the treatment of the Hamiltonian operator (5) is that it fails to be a genuine self-adjoint operator [23].…”