State space resonance energy operator

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 resonance of a nucleus may be affected by the proximity of other nuclei in the molecule whose spins are non-zero these need not be of the same atomic number as the nucleus under scrutiny. Suppose we have two nuclei A with a spin IA and B with a spin /B. The resonance of nucleus A will be split into (2/B + 1) peaks, equally spaced and of equal intensity this happens because the precise frequency at which A absorbs depends upon the magnetic state of B. Because the (2/B + 1) states are so close in energy, they are for most practical purposes, equally occupied hence the equal intensities of the peaks in the resonance of nucleus A. The spacing between the peaks is the coupling constant J between the nuclei. This is expressed in frequency units the coupling constant is independent of the operating frequency of the spectrometer, in contrast to the chemical shift. If the nuclei A and B are chemically remote, the coupling may be so small that it cannot be observed. This is usually the case if the nuclei are separated by more than about three bonds in the molecule. 

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