The Lippmann–Schwinger Formula and One Dimensional Models with Dirac Delta Interactions

We show how a proper use of the Lippmann–Schwinger equation simplifies the calculations to obtain scattering states for one dimensional systems perturbed by N Dirac delta equations. Here, we consider two situations. In the former, attractive Dirac deltas perturbed the free one dimensional Schrödinger Hamiltonian. We obtain explicit expressions for scattering and Gamow states. For completeness, we show that the method to obtain bound states use comparable formulas, although not based on the Lippmann–Schwinger equation. Then, the attractive N deltas perturbed the one dimensional Salpeter equation. We also obtain explicit expressions for the scattering wave functions. Here, we need regularisation techniques that we implement via heat kernel regularisation

Efek Variasi Waktu Ball Milling terhadap Karakteristik Elektrokimia Sel Superkapasitor Berbasis Karbon

Supercapacitor electrodes from rubber wood saw dust (RWSD) have been fabricated using experiment method to study the ball milling variation time on performance of the supercapacitor cells. The carbon electrodes were prepared with time variation of 20, 40, and 80 hours and thickness of 0.2 mm. Carbon electrodes were carbonized at 600oC and followed by physical activation method in CO2 gas atmosphere on the constant temperature of 900o C, and chemical activation was performed by KOH as an activating agent. Densities of the electrodes were 0.849 g/cm3 , 0.892 g/cm3 , 0.982 g/cm3 respectively. XRD measurement showed the peaks of carbon electrodes at 2θ of 24.091o and 44.473o which represented the presence of carbon materials with their crystal orientation of (002) and (100). SEM micrograph on magnification of 1000X showed that the pore distribution of the carbon electrodes dominant on macropores. This study found that the effects of increasing of ball milling time influenced the electrochemical properties of supercapacitor electrodes fro m RWSD. The optimum supercapacitor performance was found on 20 hour milling time electrode and had a specific capacitance of 55.414 F/g

Stock Pricing Analysis of PT. LEN as Alternative Sources of Fund Through the Initial Public Offering (IPO)

PT. LEN is one of state owned companies (BUMN) in Indonesia which is engaged in technology industries. As one of state owned companies (BUMN) which enroll in strategic industries, PT. LEN has a very important role in supporting the growth of Indonesia\u27s development, PT. LEN\u27s business performance has successfully led PT. LEN as a trillion company. In 2011, the life cycle of this company has been projected from phase stability to phase growth. Nevertheless, one issue has arisen when the growth of the company is hampered by funding issue, especially the lack of capital to run the projects. The growth of revenues become unbalanced with working capital, especially liquid working capital (cash). The objective of the company is to Go Public with huge amount of capital which can support the business activities and also to strengthen the capital structure and the liquidity of assets. The sales revenue from some of its stocks is used as an alternative source of fund apart from bank loans. The fund is used to expand the company\u27s activities, to build the facility of developing system, and also to build fabrication workshop. The total amount of fund needed is 590 billion. During the IPO process, there are several things to be concerned, internally and externally. From the company\u27s internal perspective, several things to be be concerned are company\u27s feasibility studies in doing IPO. From the external perspective will affect investors\u27 interest to the company. This thesis uses discounted cash flow as a method to calculate the valuation of the company. This method is used to value stock\u27s price and the amount of stocks issued by PT. LEN for IP process so the price will be interesting to the investors. Other than to fulfill the needs of source funding, IPO process is also one of the program to privatize the state owned companies in Indonesia which is included in BUMN\u27s Masterplan 2010-2014

A Many-body Problem with Point Interactions on Two Dimensional Manifolds

A non-perturbative renormalization of a many-body problem, where non-relativistic bosons living on a two dimensional Riemannian manifold interact with each other via the two-body Dirac delta potential, is given by the help of the heat kernel defined on the manifold. After this renormalization procedure, the resolvent becomes a well-defined operator expressed in terms of an operator (called principal operator) which includes all the information about the spectrum. Then, the ground state energy is found in the mean field approximation and we prove that it grows exponentially with the number of bosons. The renormalization group equation (or Callan-Symanzik equation) for the principal operator of the model is derived and the $\beta$ function is exactly calculated for the general case, which includes all particle numbers.Comment: 28 pages; typos are corrected, three figures are adde

Finitely Many Dirac-Delta Interactions on Riemannian Manifolds

This work is intended as an attempt to study the non-perturbative renormalization of bound state problem of finitely many Dirac-delta interactions on Riemannian manifolds, S^2, H^2 and H^3. We formulate the problem in terms of a finite dimensional matrix, called the characteristic matrix. The bound state energies can be found from the characteristic equation. The characteristic matrix can be found after a regularization and renormalization by using a sharp cut-off in the eigenvalue spectrum of the Laplacian, as it is done in the flat space, or using the heat kernel method. These two approaches are equivalent in the case of compact manifolds. The heat kernel method has a general advantage to find lower bounds on the spectrum even for compact manifolds as shown in the case of S^2. The heat kernels for H^2 and H^3 are known explicitly, thus we can calculate the characteristic matrix. Using the result, we give lower bound estimates of the discrete spectrum.Comment: To be published in JM

Tube Models for Rubber-Elastic Systems

In the first part of the paper we show that the constraining potentials introduced to mimic entanglement effects in Edwards' tube model and Flory's constrained junction model are diagonal in the generalized Rouse modes of the corresponding phantom network. As a consequence, both models can formally be solved exactly for arbitrary connectivity using the recently introduced constrained mode model. In the second part, we solve a double tube model for the confinement of long paths in polymer networks which is partially due to crosslinking and partially due to entanglements. Our model describes a non-trivial crossover between the Warner-Edwards and the Heinrich-Straube tube models. We present results for the macroscopic elastic properties as well as for the microscopic deformations including structure factors.Comment: 15 pages, 8 figures, Macromolecules in pres

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