Comment on "Quantum bound states with zero binding energy"

The purpose of this Comment is to show that the solutions to the zero energy Schr\"odinger equations for monomial central potentials discussed in a recently published Letter, may also be obtained from the corresponding free particle solutions in a straight forwardly way, using an algorithm previously devised by us. New solutions to the zero energy Schr\"odinger equation are also exhibited.Comment: Accepted for publication in PHISICS LETTERS

Closing a window for massive photons

Working with the assumption of non-zero photon mass and a trajectory that is described by the non geodesic world line of a spinning top we find, by deriving new astrophysical bounds, that this assumption is in contradiction with current experimental results. This yields the conclusion that such photons have to be exactly massless.Comment: 5 pages, 1 figur

Multi-Lagrangians, Hereditary Operators and Lax Pairs for the Korteweg-de Vries Positive and Negative Hierarchies

We present an approach to the construction of action principles for differential equations, and apply it to field theory in order to construct systematically, for integrable equations which are based on a Nijenhuis (or hereditary) operator, a ladder of action principles which is complementary to the well-known multi-Hamiltonian formulation. We work out results for the Korteweg-de Vries (KdV) equation, which is a member of the positive hierarchy related to a hereditary operator. Three negative hierarchies of (negative) evolution equations are defined naturally from the hereditary operator as well, in the context of field theory. The Euler-Lagrange equations arising from the action principles are equivalent to the original evolution equation + deformations, which are obtained in terms of the positive and negative evolution vectors. We recognize the Liouville, Sinh-Gordon, Hunter-Zheng and Camassa-Holm equations as negative equations. The ladder for KdV is directly mappable to a ladder for any of these negative equations and other positive equations (e.g., the Harry-Dym and a special case of the Krichever-Novikov equations): a new nonlocal action principle for the deformed system Sinh-Gordon + spatial translation vector is presented. Several nonequivalent, nonlocal time-reparametrization invariant action principles for KdV are constructed. Hamiltonian and Symplectic operators are obtained in factorized form. Alternative Lax pairs for all negative flows are constructed, using the flows and the hereditary operator as only input. From this result we prove that all positive and negative equations in the hierarchies share the same sets of local and nonlocal constants of the motion for KdV, which are explicitly obtained using the local and nonlocal action principles for KdV.Comment: Final version, accepted in JMP; RevTeX, 31 page

On the Measurement of Poverty Dynamics

This paper introduces a family of multi-period poverty measures derived from commonly used static poverty measures. Our measures trade-off poverty levels and changes (gains and losses) over time, and are consistent with loss aversion. We characterize the partial ranking over income dynamics induced by these measures and use it in two empirical applications with longitudinal household level data. Comparing two decades of income dynamics in the United States we find that the income dynamics of the 1990s -post Welfare reform- dominates the income dynamics of the 1980s -pre Welfare reform. Next, we compare the contemporary income dynamics of three industrialized countries and conclude that United Kingdom dominates Germany and United States, and Germany dominates the United States if poverty stocks are given more importance than poverty flows. The differences between our ranking and those obtained using other welfare criteria such as social mobility suggest that our measures capture critical information about the evolution of poverty.