Quasi-projection operators

A. Temkin, A.K. Bhatia, Projection and quasi-projection operators for electron impact resonances on many-electron atomic targets, in A. Temkin (Ed.), Autoionization, Plenum Press, New York, 1985. 

Note that the projection operator P = (1 a, vF)/2 projects out the particle state, wF, and the anti-particle state, ip- (or more preciselyip-), from the Dirac spinor field k. The quasi-quarks in a patch carries the residual momentum l1 and is given as... 

Since Pcl j,) = and there are no quasi-open operators present to excite on the right side of Eq. (55), the right side of the quasi-open projection of Eq. (55) is zero. We have ... 

We refer to our two papers [60,73] for more detailed discussions and for the demonstration of the avoidance of intruders. Here we just emphasize that, to avoid intruders, it would be necessary to choose the IMS appropriately. The special IMS such as QCMS, where one usually puts quasi-degenerate orbitals in one class, and the nondegenerate orbitals in a different class are very appropriate for separability. There are other desirable simplifications also if we work in the QCMS. For the QCMS, the quasiopen operators cannot lead to excitations into the QCMS itself. This automatically separates the contributions of the quasi-open and the closed operators by simply using the projectors exp(T )/ exp(—T ) and exp(T )P exp(—T ), respectively. This is quite convenient for the practical applications. For the QCMS, we thus need to project Eq. (55) with exp(T )l<)), ) <)), lexp(—T ) to get every quasi-open operator present in for each and project onto exp(T )P exp(—T ) to generate W. Eq. (60) are all trivially zero for a QCMS. 

All quasi-relativistic Hamiltonians in the following may be thought of as being surrounded by proper projection operators onto electronic bound or... 

The quasi-continuous approach carries, however, a potentially dangerous hazard the time-evolution y contains projections and, therefore, it has zero eigenvalues. This renders it, in principle, impossible to find an inverse or logarithm operator. Sometimes it will be, however, possible to exclude the null-space of y from the consideration, which enables one to define these operators. This will be the case in our example below. 

In Eq. (56) or Eq. (57), no quasi-open or closed operators can project on to Xh since they have no inactive excitations in them, being labeled by active lines only. Hence, they do not appear in Eq. (56) or Eq. (57) for the Q-projection. Eq. (57) defines all the open components of Top, for every pu. By an entirely similar reasoning, we project all the quasi-open components of both sides of Eq. (55) for each /a and equate them (as the sufficiency conditions for defining Tq-opY... 

P projects into the inner space of n unstable quasi-bound states which are either true observable resonances or wave packets in the continuum (decay channels) taking a significant part in the dynamics. These latter may be either strongly or weakly coupled to the resonances. We shall treat in the same way the resonances and the quasi-bound states of interest and hence we call equally "resonance" or "quasi-bound state" any state belonging to the inner space. Q = 1 — P projects onto the complementary outer space. The energy-dependent wave operator 2(z) establishes a one-to-one correspondence between the n states belonging to the inner space and the states belonging to the outer space. f2(z) extends the concept of wave operator previously defined for bound sfafes [4, 5]. Let us transform the expression on the left of Eq. (3) info fhe basic equafion... 

The Chemical Warfare Service, in addition to conducting research and development on various aspects of chemical warfare, co-operated with other branches of the Army, with the U.S. Public Health Service, and with the Navy on projects of a quasi-public-health nature. In 1920 the service was directed to co-operate with the Medical Department and the Quartermaster Corps on the extermination of rodents and vermin. Later the CWS worked on methods of exterminating the boll weevil and on improved methods for fumigating ships. ... 

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