Infinitely many singular interactions on noncompact manifolds
Abstract
We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional noncompact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004).  Highlights: • Schrödingeroperator for infinitely many singular interactions on noncompact manifolds. • Proof of the finiteness of the groundstate energy.
 Authors:
 Publication Date:
 OSTI Identifier:
 22451174
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Annals of Physics; Journal Volume: 356; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; DELTA FUNCTION; GEOMETRY; GROUND STATES; KERNELS; MATHEMATICAL MANIFOLDS; RIEMANN SPACE; TWODIMENSIONAL CALCULATIONS
Citation Formats
Kaynak, Burak Tevfik, Email: burak.kaynak@boun.edu.tr, and Turgut, O. Teoman, Email: turgutte@boun.edu.tr. Infinitely many singular interactions on noncompact manifolds. United States: N. p., 2015.
Web. doi:10.1016/J.AOP.2015.03.016.
Kaynak, Burak Tevfik, Email: burak.kaynak@boun.edu.tr, & Turgut, O. Teoman, Email: turgutte@boun.edu.tr. Infinitely many singular interactions on noncompact manifolds. United States. doi:10.1016/J.AOP.2015.03.016.
Kaynak, Burak Tevfik, Email: burak.kaynak@boun.edu.tr, and Turgut, O. Teoman, Email: turgutte@boun.edu.tr. Fri .
"Infinitely many singular interactions on noncompact manifolds". United States.
doi:10.1016/J.AOP.2015.03.016.
@article{osti_22451174,
title = {Infinitely many singular interactions on noncompact manifolds},
author = {Kaynak, Burak Tevfik, Email: burak.kaynak@boun.edu.tr and Turgut, O. Teoman, Email: turgutte@boun.edu.tr},
abstractNote = {We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional noncompact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004).  Highlights: • Schrödingeroperator for infinitely many singular interactions on noncompact manifolds. • Proof of the finiteness of the groundstate energy.},
doi = {10.1016/J.AOP.2015.03.016},
journal = {Annals of Physics},
number = ,
volume = 356,
place = {United States},
year = {Fri May 15 00:00:00 EDT 2015},
month = {Fri May 15 00:00:00 EDT 2015}
}

This work is intended as an attempt to study the nonperturbative renormalization of bound state problem of finitely many Diracdelta interactions on Riemannian manifolds, S{sup 2}, H{sup 2}, and H{sup 3}. We formulate the problem in terms of a finite dimensional matrix, called the characteristic matrix {phi}. The bound state energies can be found from the characteristic equation {phi}({nu}{sup 2})A=0. The characteristic matrix can be found after a regularization and renormalization by using a sharp cutoff in the eigenvalue spectrum of the Laplacian, as it is done in the flat space, or using the heat kernel method. These two approachesmore »

Underlying structure of singular perturbations on manifolds
A perturbation problem is singular when the straightforward expansion in powers of a small parameter fails to describe the exact solution qualitatively in some region of interest. I formulate singular perturbations on manifolds in a coordinatefree way suitable for treating problems in general relativity and other field theories. I define uniformity as the ideal sought. Uniformity leads to internal criteria, such as matching, for validity. Subsequent papers will apply this formalism to (1) the problem of motion in general relativity (and other theories), (2) problems in which the causal structure or topology change qualitatively as the result of an apparentlymore » 
The structure of optimal synthesis in a neighbourhood of singular manifolds for problems that are affine in control
The question of the classification of the phase portraits of optimal synthesis in a neighbourhood of a singular universal manifold is discussed for systems of constant rank that are affine in control. Both phase state and control are assumed to be manydimensional. The classification is based on the order of the singular extremals and the property of involutiveness (or otherwise) of the velocity indicator. The synthesis of optimal trajectories is shown to be a space fibred over the base W consisting of singular optimal trajectories; its fibres are nonsingular optimal trajectories. If the control is manydimensional, then W is amore » 
Crichton ambiguities with infinitely many partial waves
We construct families of spinless twoparticle unitary cross sections that possess a nontrivial discrete phaseshift ambiguity, with in general an infinite number of nonvanishing partial waves. A numerical investigation reveals that some of the previously known finite Crichton ambiguities are merely special cases of the newly constructed examples.