Singular potentials and annihilation

We discuss the regularization of attractive singular potentials $-\alpha _{s}/r^{s}$, $s\geq 2$ by infinitesimal imaginary addition to interaction constant $\alpha_{s}=\alpha_{s}\pm i0$. Such a procedure enables unique definition of scattering observables and is equal to an absorption (creation) of particles in the origin. It is shown, that suggested regularization is an analytical continuation of the scattering amplitudes of repulsive singular potential in interaction constant $\alpha_{s}$. The nearthreshold properties of regularized in a mentioned way singular potential are examined. We obtain expressions for the scattering lengths, which turn to be complex even for infinitesimal imaginary part of interaction constant. The problem of perturbation of nearthreshold states of regular potential by a singular one is treated, the expressions for level shifts and widths are obtained. We show, that the physical sense of suggested regularization is that the scattering observables are insensitive to any details of the short range modification of singular potential, if there exists sufficiently strong inelastic short range interaction. In this case the scattering observables are determined by solutions of Schrodinger equation with regularized potential $-(\alpha_{s}\pm i0)/r^{s}$. We point out that the developed formalism can be applied for the description of systems with short range annihilation, in particular low energy nucleon-antinucleon scattering.Comment: 10 page

Regularization of the Singular Inverse Square Potential in Quantum Mechanics with a Minimal length

We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heun's functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.Comment: 30 pages, 3 figure

Novel Quantum States of the Rational Calogero Models Without the Confining Interaction

We show that the N-particle A_{N-1} and B_N rational Calogero models without the harmonic interaction admit a new class of bound and scattering states. These states owe their existence to the self-adjoint extensions of the corresponding Hamiltonians, labelled by a real parameter z. It is shown that the new states appear for all values of N and for specific ranges of the coupling constants. Moreover, they are shown to exist even in the excited sectors of the Calogero models. The self-adjoint extension generically breaks the classical scaling symmetry, leading to quantum mechanical scaling anomaly. The scaling symmetry can however be restored for certain values of the parameter z. We also generalize these results for many particle systems with classically scale invariant long range interactions in arbitrary dimensions.Comment: Latex file, 21 pages; minor changes in text, some references added, to appear in Nucl. Phys.

Effective theories of scattering with an attractive inverse-square potential and the three-body problem

A distorted-wave version of the renormalisation group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wave function satisfies a Schroedinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalisation of the three-body interactions, with the renormalisation-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. The perturbations around the cycle start with a marginal term whose effect is simply to change the phase of the short-distance oscillations, or the self-adjoint extension of the singular Hamiltonian. The full power counting in terms of the energy and two-body scattering length is constructed for short-range three-body forces.Comment: 19 pages (RevTeX), 2 figure

Black-hole concept of a point-like nucleus with supercritical charge

The Dirac equation for an electron in the central Coulomb field of a point-like nucleus with the charge greater than 137 is considered. This singular problem, to which the fall-down onto the centre is inherent, is addressed using a new approach, based on a black-hole concept of the singular centre and capable of producing cut-off-free results. To this end the Dirac equation is presented as a generalized eigenvalue boundary problem of a self-adjoint operator. The eigenfunctions make complete sets, orthogonal with a singular measure, and describe particles, asymptotically free and delta-function-normalizable both at infinity and near the singular centre $r=0$. The barrier transmission coefficient for these particles responsible for the effects of electron absorption and spontaneous electron-positron pair production is found analytically as a function of electron energy and charge of the nucleus. The singular threshold behaviour of the corresponding amplitudes substitutes for the resonance behaviour, typical of the conventional theory, which appeals to a finite-size nucleus.Comment: 22 pages, 5 figures, LATEX requires IOPAR

Франчайзинг как способ эффективного трансфера инноваций (на примере ООО «Сибирская сбытовая компания»)

Франчайзинг в РФ на сегодняшний день стремительно развивается, и это позволяет развиваться малому и среднему бизнесу более активно. Цель работы – исследование франчайзинга как одной из возможных форм развития предпринимательства в том числе засчет эффективного трансфера инноваций компании. В результате исследования сформированы рекомендации для ООО "Сибирская Сбытовая Компания" по дальнейшему развитию. Проведенный анализ и разработанные рекомендации могут быть использованы как ООО "Сибирская сбытовая Компания", так и другими компаниями, планирующими начать бизнес АЗС с помощью франшизы и двигаться далее в направлении инновационного развития.Franchising in the Russian Federation today is rapidly developing, and this allows small and medium-sized businesses to develop more actively. The purpose of the work is to study franchising as one of the possible forms of entrepreneurship development, including through the effective transfer of company innovations. As a result of the study, recommendations were made for Siberian Sales Company LLC for further development. The analysis and recommendations developed can be used both by Siberian Retail Company LLC and other companies planning to start a gas station business with the help of a franchise and move further in the direction of innovative development