MOLECULAR PROPAGATION THROUGH LEVEL CROSSINGS

15

C crossings, h(X) commutes with the orthogonal projections Pi and P2 onto the mutually

orthogonal carrier subspaces Hi and H2 associated with U\ and L^, respectively.

For X in a neighborhood of the origin, one can write the spectral projection P{X) for

h(X) associated with the eigenvalues Ej[(X) and Eg(X) as an integral of the resolvent of

h(X). From this it follows that P(X) is a Ck, rank 3 operator valued function of X near

X = 0 that commutes with Pi and Pi- Since U\ and U2 are inequivalent, it follows that

PA(X) = PiP(X) and Pjs(X) = P2P{X) are Ck, rank one and (respectively) rank two

orthogonal projections that project onto mutually orthogonal subspaces.

We construct a Ck unit-vector valued function $A(') exactly as in the case of a Type

C crossing, so that K$j±{X) — ^j^(X). For a Type F crossing we choose 3g i(0) to be an

arbitrary unit vector in the range of P#(0). We then let

* ^ P

B

pQ$B,l(0)

y(*B,i(°).

PBW*B,I(O))

and

*B,2(X) = K*B,lW.

Because K is antiunitary and D{K?) — — 1, it follows that $&i(X) and ^g^G^O comprise

an orthonormal basis for the range of Pg(X).

For Type G crossings, we let PQ and P p denote the orthogonal projections onto the

carrier subspaces for the two representations C and D of the subgroup H that are involved.

These projections commute with P&{X) and project onto mutually orthogonal subspaces.

Furthermore, P(jPg(X) and Pj^P^{X) are rank one projections. We choose £# i(0) to be a

unit vector in the range of P D P # ( 0 ) . We then let

* ^ PB(X)*B,I(P)

yJ(*B,l(0), PB(X)*Btl(0))

and

*B,2(X) = £ * 3 , l W -

From the structure theory of type III corepresentations, it follows that $B,2(X) belongs to

the range of PQPB(X), and that $gi(X) and $ # 2 p 0 comprise a Ck orthonormal basis

for the range of Pg(X).

Let h^-(X) denote the restriction of h(X) to the subspace orthogonal to the range

of P(X). By using ^ ( X ) , $ g i(X) and $ # 2 p 0 as a basis for the range of P(X) and