24
GEORGE A. HAGEDORN
of P(0). We define
MX) = ,
P ( x )
^
( 0 )
.
y/(M0), P(x)Mo))
We then define ip2(X) K^lPO- We let P\^{X) denote the orthogonal projection onto
the span of ipi(X) and ^2p0 an^ define
. ,
=
(i-Pi,2(x))P(x)v.3(o)
1 V ( ^ 3 ( 0 ) , ( l - ^ , 2 W ) P ( X ) ^ ( 0 ) "
We then define V^PO = Kip^(X). For each X in a neighborhood of the origin, these four
vectors form an orthonormal basis for the range of P(X).
In this basis, the restriction of
hx(X) = h{X) - \{EA{X) + EB{X))
to the range of P{X) is represented by a self-adjoint traceless 4 x 4 matrix valued function
M\{X) whose entries are
Ck
functions that all vanish when X = 0. Because h(X) commutes
with the action of /C, M\(X) commutes with
(Conjugation).
It follows that M\(X) must have the form
/ P(X) 0 y(X) + i6(X) c(X) + iZ(X)\
0 f3(X) -c(X) + tC(X)
7
( * ) - * * ( * )
1{X)-i8{X) -€(X)-iC(X) -0(X) 0
\e(X)-i((X) 7(X) + i6(X) 0 -0{X) )
where /?, 7, 5 , e, and ( are C real valued functions. The difference between EA(X) and
Eg(X) is the same as the difference between the eigenvalues of M\(X). By direct computa-
tion, the eigenvalues of M\(X) are
0
1
0
0
- 1
0
0
0
0
0
0
1
0
\
0
j
- 1
oy
±y//3(X)2 + i{Xf + 6{X)2 +e{Xf +
C(X)2.
Thus, the eigenvalues E^(X) and Eg(X) cross precisely at those points X where 0(X)
j(X) = 6(X) = e(X) = C(X) 0. Generically this defines a codimension 5 submanifold
T. Furthermore, it is clear that the eigenvalues Ej^{X) and Eg(X) are continuous, but
generically not differentiable near I\
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