Connection between closeness of classical orbits and the factorization of radial Schr\"{o}dinger equation

It was shown that the Runge-Lenz vector for a hydrogen atom is equivalent to the raising and lowering operators derived from the factorization of radial Schr\"{o}dinger equation. Similar situation exists for an isotropic harmonic oscillator. It seems that there may exist intimate relation between the closeness of classical orbits and the factorization of radial Schr\"{o}dinger equation. Some discussion was made about the factorization of a 1D Schr\"{o}dinger equation.Comment: 14 pages, no figure

The roles of quantum correlations in quantum cloning

In this paper, we study the entanglement and quantum discord of the output modes in the unified $1\rightarrow 2$ state-dependent cloning and probabilistic quantum cloning. The tripartite entanglement among the output modes and the quantum cloning machine is also considered. We find that the roles of the quantum correlations including the bipartite and tripartite entanglement and quantum discord strongly depend on the quantum cloning machines as well as the cloned state. In particular, it is found that this quantum cloning scheme can be realizable even without any quantum correlation.Comment: Accepted by Eur. Phys. J.

Explosive Behavior in the 1990s Nasdaq: When Did Exuberance Escalate Asset Values?

A recursive test procedure is suggested that provides a mechanism for testing explosive behavior, date-stamping the origination and collapse of economic exuberance, and providing valid con?dence intervals for explosive growth rates. The method involves the recursive im- plementation of a right-side unit root test and a sup test, both of which are easy to use in practical applications, and some new limit theory for mildly explosive processes. The test procedure is shown to have discriminatory power in detecting periodically collapsing bubbles, thereby overcoming a weakness in earlier applications of unit root tests for economic bubbles. An empirical application to Nasdaq stock price index in the 1990s provides con?rmation of ex- plosiveness and date-stamps the origination of ?nancial exuberance to mid -1995, prior to the famous remark in December 1996 by Alan Greenspan about irrational exuberance in ?nancial market, thereby giving the remark empirical content.Explosive root, irrational exuberance, mildly explosive process, Nasdaq bubble, periodically collapsing bubble, sup test, unit root test

Explosive Behavior in the 1990s Nasdaq : When Did Exuberance Escalate Asset Values?

A recursive test procedure is suggested that provides a mechanism for testing explosive behavior, date-stamping the origination and collapse of economic exuberance, and providing valid conOdence intervals for explosive growth rates. The method involves the recursive im- plementation of a right-side unit root test and a sup test, both of which are easy to use in practical applications, and some new limit theory for mildly explosive processes. The test procedure is shown to have discriminatory power in detecting periodically collapsing bubbles, thereby overcoming a weakness in earlier applications of unit root tests for economic bubbles. An empirical application to Nasdaq stock price index in the 1990s provides conOrmation of ex- plosiveness and date-stamps the origination of Onancial exuberance to mid -1995, prior to the famous remark in December 1996 by Alan Greenspan about irrational exuberance in Onancial market, thereby giving the remark empirical content.Explosive root, irrational exuberance, Mildly explosive process, Nasdaq bubble, periodically collapsing bubble, sup test, unit root test