By E. McDaniel (Eds.)

ISBN-10: 0444108823

ISBN-13: 9780444108821

ISBN-10: 0720403316

ISBN-13: 9780720403312

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**Example text**

And, further because o f the form o f the relation, complete correspondence identities will always be possible for these systems through the spectral operator kernel (and sometimes, as well, through the propa gator and Green functions). T o illustrate the points we have just made we consider now those systems, other than ones with quadratic Lagrangians, for which complete correspondence iden tities are known to exist. Little more needs to be said about the bound states o f the Coulomb potential except that the existence o f the complete correspondence identity can now be regarded as a consequence o f 0 ( 4 ) symmetry.

What is important about this result is that it may be used to show that the spec tral operator kernel o f the/-dimensional hydrogen atom (w hen / is od d) can be ex pressed in terms o f the 3-dimensional kernel. It thus follows therefore from what we have shown earlier that the spectral operator kernel o f the /-dimensional hydro gen atom must also be expressible in terms o f the spectral operator o f a particle on a ring (w h e n / is odd). Further, bacause a complete correspondence identity exists for the particle on a ring via the spectral operator, then a complete correspondence identity must also occur for the/-dimensional bound-state Coulomb problem through the spectral operator.

Nowhere, however, have we really explained why we should expect a complete correspondence identity to exist for the Coulomb potential. It is true that Gutzwiller’ s work [50] did lead us to ex pect a complete correspondence identity for the spectral operator, but w e should note that Gutzwiller had no a p riori reason for suspecting the phase-integral approx imation for the momentum-space Green functions to be exact at the bound-state energy levels; neither does he offer any explanation o f this fact.

### Case Studies in Atomic Physics by E. McDaniel (Eds.)

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