64 research outputs found
Sub-Gaussian estimates of heat kernels on infinite graphs
We prove that a two sided sub-Gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the Green kernel admits a uniform polynomial decay
On isoperimetric profiles of product spaces
Let p ∈ [1,+∞]. Given the Lp-isoperimetric profile of two non-compact Riemannian manifolds M and N, we compute the Lp-isoperimetric profile of the product M×N
Relativistic Lee Model on Riemannian Manifolds
We study the relativistic Lee model on static Riemannian manifolds. The model
is constructed nonperturbatively through its resolvent, which is based on the
so-called principal operator and the heat kernel techniques. It is shown that
making the principal operator well-defined dictates how to renormalize the
parameters of the model. The renormalization of the parameters are the same in
the light front coordinates as in the instant form. Moreover, the
renormalization of the model on Riemannian manifolds agrees with the flat case.
The asymptotic behavior of the renormalized principal operator in the large
number of bosons limit implies that the ground state energy is positive. In 2+1
dimensions, the model requires only a mass renormalization. We obtain rigorous
bounds on the ground state energy for the n-particle sector of 2+1 dimensional
model.Comment: 23 pages, added a new section, corrected typos and slightly different
titl
Point Interaction in two and three dimensional Riemannian Manifolds
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. 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 beta-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.Comment: 43 page
Existence of Hamiltonians for Some Singular Interactions on Manifolds
The existence of the Hamiltonians of the renormalized point interactions in
two and three dimensional Riemannian manifolds and that of a relativistic
extension of this model in two dimensions are proven. Although it is much more
difficult, the proof of existence of the Hamiltonian for the renormalized
resolvent for the non-relativistic Lee model can still be given. To accomplish
these results directly from the resolvent formula, we employ some basic tools
from the semigroup theory.Comment: 33 pages, no figure
Heat kernel upper bounds on a complete non-compact manifold
Let M be a smooth connected non-compact geodesically complete Riemannian manifold, ? denote the Laplace operator associated with the Riemannian metric, n = 2 be the dimension of M. Consider the heat equation on the manifold ut - ?u = 0, where u = u(x,t), x Î M, t > 0. The heat kernel p(x,y,t) is by definition the smallest positive fundamental solution to the heat equation which exists on any manifold (see [Ch], [D]). The purpose of the present work is to obtain uniform upper bounds of p(x,y,t) which would clarify the behaviour of the heat kernel as t ? +8 and r = dist(x,y) ? +8
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