1,944 research outputs found
Quite a Character: The Spectrum of Yang-Mills on S^3
We introduce a simple method to extract the representation content of the
spectrum of a system with SU(2) symmetry from its partition function. The
method is easily generalized to systems with SO(2,4) symmetry, such as
conformal field theories in four dimensions. As a specific application we
obtain an explicit generating function for the representation content of free
planar Yang-Mills theory on S^3. The extension to N = 4 super Yang-Mills is
also discussed.Comment: Based on a Brown University undergraduate thesis, 12 page
Pade approximation of the S-matrix as a way of locating quantum resonances and bound states
It is shown that the spectral points (bound states and resonances) generated
by a central potential of a single-channel problem, can be found using rational
parametrization of the S-matrix. To achieve this, one only needs values of the
S-matrix along the real positive energy axis. No calculations of the S-matrix
at complex energies or a complex rotation are necessary. The proposed method is
therefore universal in that it is applicable to any potential (local,
non-local, discontinuous, etc.) provided that there is a way of obtaining the
S-matrix (or scattering phase-shifts) at real collision energies. Besides this,
combined with any method that extracts the phase-shifts from the scattering
data, the proposed rational parametrization technique would be able to do the
spectral analysis using the experimental data.Comment: 20 pages, 6 figure
Antibound States and Halo Formation in the Gamow Shell Model
The open quantum system formulation of the nuclear shell model, the so-called
Gamow Shell Model (GSM), is a multi-configurational SM that employs a
single-particle basis given by the Berggren ensemble consisting of Gamow states
and the non-resonant continuum of scattering states. The GSM is of particular
importance for weakly bound/unbound nuclear states where both many-body
correlations and the coupling to decay channels are essential. In this context,
we investigate the role of l=0 antibound (virtual) neutron single-particle
states in the shell model description of loosely bound wave functions, such as
the ground state wave function of a halo nucleus 11Li
Multichannel coupling with supersymmetric quantum mechanics and exactly-solvable model for Feshbach resonance
A new type of supersymmetric transformations of the coupled-channel radial
Schroedinger equation is introduced, which do not conserve the vanishing
behavior of solutions at the origin. Contrary to usual transformations, these
``non-conservative'' transformations allow, in the presence of thresholds, the
construction of potentials with coupled scattering matrices from uncoupled
potentials. As an example, an exactly-solvable potential matrix is obtained
which provides a very simple model of Feshbach-resonance phenomenon.Comment: 10 pages, 2 figure
Supersymmetric transformations for coupled channels with threshold differences
The asymptotic behaviour of the superpotential of general SUSY
transformations for a coupled-channel Hamiltonian with different thresholds is
analyzed. It is shown that asymptotically the superpotential can tend to a
diagonal matrix with an arbitrary number of positive and negative entries
depending on the choice of the factorization solution. The transformation of
the Jost matrix is generalized to "non-conservative" SUSY transformations
introduced in Sparenberg et al (2006 J. Phys. A: Math. Gen. 39 L639). Applied
to the zero initial potential the method permits to construct superpartners
with a nontrivially coupled Jost-matrix. Illustrations are given for two- and
three-channel cases.Comment: 17 pages, 3 explicit examples and figures adde
Rigged Hilbert Space Approach to the Schrodinger Equation
It is shown that the natural framework for the solutions of any Schrodinger
equation whose spectrum has a continuous part is the Rigged Hilbert Space
rather than just the Hilbert space. The difficulties of using only the Hilbert
space to handle unbounded Schrodinger Hamiltonians whose spectrum has a
continuous part are disclosed. Those difficulties are overcome by using an
appropriate Rigged Hilbert Space (RHS). The RHS is able to associate an
eigenket to each energy in the spectrum of the Hamiltonian, regardless of
whether the energy belongs to the discrete or to the continuous part of the
spectrum. The collection of eigenkets corresponding to both discrete and
continuous spectra forms a basis system that can be used to expand any physical
wave function. Thus the RHS treats discrete energies (discrete spectrum) and
scattering energies (continuous spectrum) on the same footing.Comment: 27 RevTex page
Coulomb singularities in scattering wave functions of spin-orbit-coupled states
We report on our analysis of the Coulomb singularity problem in the frame of
the coupled channel scattering theory including spin-orbit interaction. We
assume that the coupling between the partial wave components involves orbital
angular momenta such that . In these conditions, the two
radial functions, components of a partial wave associated to two values of the
angular momentum , satisfy a system of two second-order ordinary
differential equations. We examine the difficulties arising in the analysis of
the behavior of the regular solutions near the origin because of this coupling.
First, we demonstrate that for a singularity of the first kind in the
potential, one of the solutions is not amenable to a power series expansion.
The use of the Lippmann-Schwinger equations confirms this fact: a logarithmic
divergence arises at the second iteration. To overcome this difficulty, we
introduce two auxilliary functions which, together with the two radial
functions, satisfy a system of four first-order differential equations. The
reduction of the order of the differential system enables us to use a
matrix-based approach, which generalizes the standard Frobenius method. We
illustrate our analysis with numerical calculations of coupled scattering wave
functions in a solid-state system
Jost Function for Coupled Partial Waves
An exact method for direct calculation of the Jost functions and Jost
solutions for non-central potentials which couple partial waves of different
angular momenta is presented. A combination of the variable-constant method
with the complex coordinate rotation is used to replace the matrix
Schr\"odinger equation by an equivalent system of linear first--order
differential equations. Solving these equations numerically, the Jost functions
can be obtained to any desired accuracy for all complex momenta of physical
interest, including the spectral points corresponding to bound and resonant
states. The effectiveness of the method is demonstrated using the Reid
soft-core and Moscow nucleon-nucleon potentials which involve tensor forces.Comment: 32 pages, RevTex, only latex pseudo-figure
On the inconsistency of the Bohm-Gadella theory with quantum mechanics
The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum
Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom
asserts that the solutions of the Lippmann-Schwinger equation are functionals
over spaces of Hardy functions. The preparation-registration arrow of time
provides the physical justification for the Hardy axiom. In this paper, it is
shown that the Hardy axiom is incorrect, because the solutions of the
Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also
shown that the derivation of the preparation-registration arrow of time is
flawed. Thus, Hardy functions neither appear when we solve the
Lippmann-Schwinger equation nor they should appear. It is also shown that the
Bohm-Gadella theory does not rest on the same physical principles as quantum
mechanics, and that it does not solve any problem that quantum mechanics cannot
solve. The Bohm-Gadella theory must therefore be abandoned.Comment: 16 page
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