In nature, one observes that a K-theory of an object is defined in two steps. First a structured category is associated to the object. Second, a K-theory machine is applied to the latter category to produce an infinite loop space. We develop a genera
l framework that deals with the first step of this process. The K-theory of an object is defined via a category of locally trivial objects with respect to a pretopology. We study conditions ensuring an exact structure on such categories. We also consider morphisms in K-theory that such contexts naturally provide. We end by defining various K-theories of schemes and morphisms between them.
Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...). One then
applies to the category A_C a $K$-theory machine, which provides an infinite loop space that is the $K$-theory K(C) of the object C. We study the first step of this process. What are the kinds of objects to be studied via $K$-theory? Given these types of objects, what structured categories should one associate to an object to obtain $K$-theoretic information about it? And how should the morphisms of these objects interact with this correspondence? We propose a unified, conceptual framework for a number of important examples of objects studied in $K$-theory. The structured categories associated to an object C are typically categories of modules in a monoidal (op-)fibred category. The modules considered are locally trivial with respect to a given class of trivial modules and a given Grothendieck topology on the object Cs category.
The Schrodinger equation incorporating the long-range Coulomb potential takes the form of a Fredholm equation whose kernel is singular on its diagonal when represented by a basis bearing a continuum of states, such as in a Fourier-Bessel transform. S
everal methods have been devised to tackle this difficulty, from simply removing the infinite-range of the Coulomb potential with a screening or cut function to using discretizing schemes which take advantage of the integrable character of Coulomb kernel singularities. However, they have never been tested in the context of Berggren bases, which allow many-body nuclear wave functions to be expanded, with halo or resonant properties within a shell model framework. It is thus the object of this paper to test different discretization schemes of the Coulomb potential kernel in the framework of complex-energy nuclear physics. For that, the Berggren basis expansion of proton states pertaining to the sd-shell arising in the A ~ 20 region, being typically resonant, will be effected. Apart from standard frameworks involving a cut function or analytical integration of singularities, a new method will be presented, which replaces diagonal singularities by finite off-diagonal terms. It will be shown that this methodology surpasses in precision the two former techniques.
Is there a connection between the branch point singularity at the particle emission threshold and the appearance of cluster states which reveal the structure of a corresponding reaction channel? Which nuclear states are most impacted by the coupling
to the scattering continuum? What should be the most important steps in developing the theory that will truly unify nuclear structure and nuclear reactions? The common denominator of these questions is the continuum shell-model approach to bound and unbound nuclear states, nuclear decays, and reactions.
The isospin breaking effects due to the Coulomb interaction in weakly-bound nuclei are studied using the Gamow Shell Model, a complex-energy configuration interaction approach which simultaneously takes into account many-body correlations between val
ence nucleons and continuum effects. We investigate the near-threshold behavior of one-nucleon spectroscopic factors and the structure of wave functions along an isomultiplet. Illustrative calculations are carried out for the T=1 isobaric triplet. By using a shell-model Hamiltonian consisting of an isoscalar nuclear interaction and the Coulomb term, we demonstrate that for weakly bound or unbound systems the structure of isobaric analog states varies within the isotriplet and impacts the energy dependence of spectroscopic factors. We discuss the partial dynamical isospin symmetry present in isospin-stretched systems, in spite of the Coulomb interaction that gives rise to large mirror symmetry breaking effects.
Demonstrating the completeness of wave functions solutions of the radial Schrodinger equation is a very difficult task. Existing proofs, relying on operator theory, are often very abstract and far from intuitive comprehension. However, it is possible
to obtain rigorous proofs amenable to physical insight, if one restricts the considered class of Schrodinger potentials. One can mention in particular unbounded potentials yielding a purely discrete spectrum and short-range potentials. However, those possessing a Coulomb tail, very important for physical applications, have remained problematic due to their long-range character. The method proposed in this paper allows to treat them correctly, provided the non-Coulomb part of potentials vanishes after a finite radius. Non-locality of potentials can also be handled. The main idea in the proposed demonstration is that regular solutions behave like sine/cosine functions for large momenta, so that their expansions verify Fourier transform properties. The highly singular point at k = 0 of long-range potentials is dealt with properly using analytical properties of Coulomb wave functions. Lebesgue measure theory is avoided, rendering the demonstration clear from a physical point of view.
The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently
immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. In order to solve these problems, a generalization of R.C. Forreys transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when |a|,|b|,|c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.
