MAXWELL - DIRAC EQUATIONS

21

As | V|

p _ 1 / 2

0 | V|

p _ 1 / 2

0

I:EP

—

E1/2

is an isomorphism and as the representation

(A

- 1

0 A

- 1

0 / )

U1

is a unitary representation on

E1/2,

it follows that V is unitary in

Ep.

This proves the theorem.

To be able to use the implicit functions theorem in the Frechet space E^, we shall

establish the existence of smoothing operators (cf. [25]). As this can be done in a general

context of unitary representations, in the following lemma and theorem, V will be an

arbitrary unitary representation of a Lie group Q on a Hilbert space H and S will be the

corresponding representation on H^ (the space of C°°-vectors) of the Lie algebra g of Q

by differentiation and II is a general basis of g. The only fact which is really used is that

the Laplace operator for the representation V, A = ]Cxen(^x)

2 m

H with domain H^

is essentially self-adjoint (see [34] Theorem 4.4.4.3). Let A denote the closure of A.

Lemma 2.3. The domain of (1 — A)

n

/

2

is Hn (the space of

Cn-vectors

ofV), and the

norms || • ||H and ||(1 — A)

n

/

2

• \\H are equivalent, i.e.

Cn'll/lk ||(1

- S )

B / 2

/ I I H

Cn\\f\\Hn, Cn0JeHn,n 0.

Proof. Let II = {Xi,..., Xr}, r — dimg. The statement is true for n — 0. Let n 1 and

let f eH^. Then

WfWk ii/iik-! + £ I I ^

•••**»/II2 (2-2)

l i i , . . . , i

n

r

= ii/iik_i+ E (-infsXtn...xtlxil...Xinf),

l i i , . . . , i

n

^

as Sx, X G 0, is skew-symmetric on H^. By successive commutations we obtain

(-1)" E Xin-.-XilXil---Xin (2.3)

lii,...,inr

= (-l)» E ( 4 )

2

- ( ^ )

2

+ 4 n - l + - + ^l

l i i , . . . , i

n

r

= (-*)"+ A2n_1 + --- + Al,

where A\ is a (noncommutative) polynomial of degree I in Xj; € II, 1 j r. As Sx, is a

skew-symmetric operator on ifoo, we have

\(f,SAlf)\C\\f\\HJ\f\\Hi_q, 0ql, (2.4)

where C depends on A\.

Equality (2.3) and inequality (2.4) give

E

\\Sxtl...xtJ\\2 (f,(-±)nf)

+ Cn\\f\\HJf\\Hni

l i i , . . . , i

n

r

{f,(i-Arf)+cn\\f\\Hjf\\H