Levinson's theorem for the Schr\"{o}dinger equation in two dimensions

Levinson's theorem for the Schr\"{o}dinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schr\"{o}dinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison with Levinson's theorem in non-critical case, the half bound state for $P$ wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of $P$ wave at zero energy to increase an additional $\pi$.Comment: Latex 11 pages, no figure and accepted by P.R.A (in August); Email: [email protected], [email protected]

Sensitivity of neutron to proton ratio toward the high density behavior of symmetry energy in heavy-ion collisions

The symmetry energy at sub and supra-saturation densities has a great importance in understanding the exact nature of asymmetric nuclear matter as well as neutron star, but, it is poor known, especially at supra-saturation densities. We will demonstrate here that the neutron to proton ratios from different kind of fragments is able to determine the supra-saturation behavior of symmetry energy or not. For this purpose, a series of Sn isotopes are simulated at different incident energies using the Isospin Quantum Molecular Dynamics (IQMD) model with either a soft or a stiff symmetry energy for the present study. It is found that the single neutron to proton ratio from free nucleons as well as LCP's is sensitive towards the symmetry energy, incident energy as well as isospin asymmetry of the system. However, with the double neutron to proton ratio, it is true only for the free nucleons. It is possible to study the high density behavior of symmetry energy by using the neutron to proton ratio from free nucleons.Comment: 11 Pages, 9 Figure

Melosh rotation: source of the proton's missing spin

It is shown that the observed small value of the integrated spin structure function for protons could be naturally understood within the naive quark model by considering the effect from Melosh rotation. The key to this problem lies in the fact that the deep inelastic process probes the light-cone quarks rather than the instant-form quarks, and that the spin of the proton is the sum of the Melosh rotated light-cone spin of the individual quarks rather than simply the sum of the light-cone spin of the quarks directly.Comment: 5 latex page

Hyperon polarization in e^-p --> e^-HK with polarized electron beams

We apply the picture proposed in a recent Letter for transverse hyperon polarization in unpolarized hadron-hadron collisions to the exclusive process e^-p --> e^-HK such as e^-p-->e^-\Lambda K^+, e^-p --> e^-\Sigma^+ K^0, or e^-p--> e^-\Sigma^0 K^+, or the similar process e^-p\to e^-n\pi^+ with longitudinally polarized electron beams. We present the predictions for the longitudinal polarizations of the hyperons or neutron in these reactions, which can be used as further tests of the picture.Comment: 15 pages, 2 figures. submitted to Phys. Rev.

The Relativistic Levinson Theorem in Two Dimensions

In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number $n_{j}$ of the bound states and the sum of the phase shifts $\eta_{j}(\pm M)$ of the scattering states with the angular momentum $j$: $$\eta_{j}(M)+\eta_{j}(-M)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$~~~=\left\{\begin{array}{ll} (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=M ~~{\rm and}~~ j=3/2~{\rm or}~-1/2\\ (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=-M~~{\rm and}~~ j=1/2~{\rm or}~-3/2\\ n_{j}\pi~&{\rm the~rest~cases} . \end{array} \right.$$ \noindent The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.Comment: Latex 14 pages, no figure, submitted to Phys.Rev.A; Email: [email protected], [email protected]