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]

Naturally Small Seesaw Neutrino Mass with No New Physics Beyond the TeV Scale

If there is no new physics beyond the TeV energy scale, such as in a theory of large extra dimensions, the smallness of the seesaw neutrino mass, i.e. $m_\nu = m_D^2/m_N$, cannot be explained by a very large $m_N$. In contrast to previous attempts to find an alternative mechanism for a small $m_\nu$, I show how a solution may be obtained in a simple extension of the Standard Model, without using any ingredient supplied by the large extra dimensions. It is also experimentally testable at future accelerators.Comment: 9 pages, in final form for PR

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]

Left-Right Symmetry and Supersymmetric Unification

The existence of an SU(3) X SU(2)_L X SU(2)_R X U(1) gauge symmetry with g_L = g_R at the TeV energy scale is shown to be consistent with supersymmetric SO(10) grand unification at around 1O^{16} GeV if certain new particles are assumed. The additional imposition of a discrete Z_2 symmetry leads to a generalized definition of R parity as well as highly suppressed Majorana neutrino masses. Another model based on SO(10) X SO(10) is also discussed.Comment: 11 pages, 2 figures not included, UCRHEP-T124, Apr 199

Eigenmodes of Decay and Discrete Fragmentation Processes

Linear rate equations are used to describe the cascading decay of an initial heavy cluster into fragments. This representation is based upon a triangular matrix of transition rates. We expand the state vector of mass multiplicities, which describes the process, into the biorthonormal basis of eigenmodes provided by the triangular matrix. When the transition rates have a scaling property in terms of mass ratios at binary fragmentation vertices, we obtain solvable models with explicit mathematical properties for the eigenmodes. A suitable continuous limit provides an interpolation between the solvable models. It gives a general relationship between the decay products and the elementary transition rates.Comment: 6 pages, plain TEX, 2 figures available from the author

USF binding sequences from the HS4 insulator element impose early replication timing on a vertebrate replicator

The nuclear genomes of vertebrates show a highly organized program of DNA replication where GC-rich isochores are replicated early in S-phase, while AT-rich isochores are late replicating. GC-rich regions are gene dense and are enriched for active transcription, suggesting a connection between gene regulation and replication timing. Insulator elements can organize independent domains of gene transcription and are suitable candidates for being key regulators of replication timing. We have tested the impact of inserting a strong replication origin flanked by the β-globin HS4 insulator on the replication timing of naturally late replicating regions in two different avian cell types, DT40 (lymphoid) and 6C2 (erythroid). We find that the HS4 insulator has the capacity to impose a shift to earlier replication. This shift requires the presence of HS4 on both sides of the replication origin and results in an advance of replication timing of the target locus from the second half of S-phase to the first half when a transcribed gene is positioned nearby. Moreover, we find that the USF transcription factor binding site is the key cis-element inside the HS4 insulator that controls replication timing. Taken together, our data identify a combination of cis-elements that might constitute the basic unit of multi-replicon megabase-sized early domains of DNA replication