Chapter 2 Canonical Sequences of Singular Cauchy Problems

Chapter 2 Canonical Sequences of Singular Cauchy Problems 2.1

The rank one situation.

In t h i s chapter we will

develop the group theoretic version of the EPD equations studied i n Chapter 1.

This leads t o many new classes of equations and

results parallel t o those of Chapter 1 as well as t o some i n teresting new situations.

Moreover i t exhibits the main results

of Chapter 1 i n t h e i r natural group theoretic context.

The

ideas of spherical symmetry, radial mean val'ues and Laplacians, etc. inherent i n EPD theory have natural counterparts i n terms of geodesic coordinates and one can obtain recursion relations,

Sonine formulas, etc. group theoretically.

The results are

based on Carroll [21; 221 in the semisimple case and were anticipated i n p a r t by e a r l i e r work of Carroll [18; 191, CarrollSilver [l5; 16; 17) and Silver [ l ] f o r some semisimple and Euclidean cases.

The group theory i s "routine" a t the present

time and r e l i e s heavily on Helgason's work (cf. Helgason [ l ; 2 ; 3; 4 ; 5; 6; 7; 8; 9; 101) b u t one must of course refer t o basic

material of Harish-Chandra (cf. Warner [ l ;

21

for a sumnary) as

well as lecture notes by Varadarajan, Ranga Rao, e t c . ) ; other specific references t o Bargmann [ l ] , Bargmann-Wigner [2], Bhanu-Munti [l], Carroll [27; 281, Coifman-Weiss [l], EhrenpreisMautner [ l ] , Flensted-Jensen [1], Furstenberg [ l ] , Gangolli [ l ] , Gelbart [ l ] , Gelfand e t a l , [ l ; 2 ; 3; 41, Godement [ l ] , Hermann 89

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

[ l ] , Jacquet-Langlands [l], Jehle-Parke [l], Kamber-Tondeur [l], Karpelevic" [ l ] , Knapp-Stein [l], Kostant [l], Kunze-Stein [ l ; 23, Lyubarskij [ l ] , Maurin [l], McKerrell [l], Miller [ l ; 23, Naimark [ l ] , Pukansky [ l ] , RGhl [l], Sally [ l ; 23, S i m s [ l ] , Smoke [ l ] , Stein

113,

Takahashi [l], Talman [ l ] , Tinkham [ l ] ,

Vilenkin [ l ] , Wallach [ l ] , Wigner [ l ] , etc., as well as the fundamental work of E. Cartan and H. Weyl, are t o be taken f o r granted, even i f n o t mentioned explicitly.

The basic Lie theory

i s developed somewhat concisely i n t h i s section; for a rather more leisurely treatment we refer t o Bourbaki [5], Carroll [23], Hausner-Schwartz [l], Helgason [ l ; 2; 31, Hochschild [ l ] , Jacobson [1], Loos [ l ] , Serre [ l ;

21,

V

Tondeur [ l ] , Zelobenko

Ell, etc. W e will s t a r t o u t w i t h the f u l l machinery f o r the rank one semisimple case, following Carroll [21; 221, and l a t e r will give extremely detailed examples f o r special cases.

T h i s avoids

some repetition and presents a "clean" theory immediately; the reader unfamiliar w i t h Lie theory m i g h t look a t the examples f i r s t where many d e t a i l s and definitions are covered.

We delib-

erately omit the treatment of invariant differential operators acting in sheaves o r i n sections of vector bundles even t h o u g h t h i s i s one of the more important subjects i n modern work (some references are mentioned above).

The preliminary material will

be expository and specific theorems will n o t be proved here. The Eucl idean group cases have been covered in Carroll -Si 1 ver [15; 16; 171 and especially Silver [ l ] s o that we will only give

90

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

a few remarks l a t e r about t h i s a t t h e end o f t h e chapter; t h e b a s i c r e s u l t s a r e i n any event i n c l u d e d i n Chapter 1.

Thus

l e t G be a r e a l connected noncompact semisimple L i e group w i t h f i n i t e c e n t e r and K a maximal compact subgroup so t h a t V = G/K -

.

-

d

L e t g = k + p be a

i s a symmetric space o f noncompact type.

Cartan decomposition, a c p a maximal a b e l i a n subspace, and we

-

-

w i l l suppose u n t i l f u r t h e r n o t i c e t h a t dim a = rank V = 1. One s e t s A = exp a, K = exp k, and

IsA for

N

-

= exp n where n =

A > 0 where t h e gA a r e t h e standard r o o t spaces corres-

ponding t o p o s i t i v e r o o t s a and p o s s i b l y 2a i n t h e rank one case.

1

One s e t s p = - c m A f o r 2 x element Ho

E

X > 0 where mA

= dim gA and we p i c k an

a such t h a t a(Ho) = 1. Thus p =

1 (H ma

+ m2a)a and

we can i d e n t i f y a Weyl chamber as a connected component a + C I

a c a w i t h (o,m) to

u

E

*

a

i n w r i t i n g a ( t H o ) = t where

by u ( t H o ) = U t .

u

E

R corresponds

The Iwasawa decomposition o f G i s G =

KAN which we w r i t e i n t h e form g = k ( g ) exp H(g) n ( g ) where t h e n o t a t i o n at = exp t H o i s used.

I

L e t M (resp. M ) denote t h e

c e n t r a l i z e r (resp. n o r m a l i z e r ) o f A i n K so t h a t t h e Weyl group I

i s W = M /M and t h e maximal boundary o f V i s B = K/M ( t h u s M = {k

E

H

K; AdkH = H f o r

E

a) and M

I

=

{k

E

K; Adka c a )

-

see

t h e examples f o r s p e c i f i c d e t a i l s ) . There are n a t u r a l p o l a r c o o r d i n a t e s i n a dense submanifold o f V a r i s i n g from t h e decomposition G = (kM,a)

-+

G+K (A+

=

exp a+) p r o v i d e d by t h e diffeomorphism

kaK : B x A+-+ V (one c o u l d a l s o work w i t h t h e decom-

p o s i t i o n G = KAK).

Thus t h e p o l a r coordinates o f ~ ( g )=

91

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

ak2)

V a r e (klM,a)

E

g i v e n v = gK -1

E

where

:G

IT

V and b = kM

+ .

V i s t h e n a t u r a l map.

B one w r i t e s A(v,b)

E

k ) and t h e F o u r i e r t r a n s f o r m o f f

=

2 L ( V ) i s d e f i n e d by

E

Y

F f = f where ( u s i n g Warner's n o t a t i o n ) Y

f(u,b)

(1.1) for

u

E

a

*

and b

= E

r 1, f(v) B.

exp (iu+p)A(v,b)dv

--

A l l measures a r e s u i t a b l y normalized i n

t h i s treatment. T h i s s e t s up an i s o m e t r i c isomorphism f f 2 * between L ( V ) and t h e space L2(a: x B) ( c f . here below f o r a+) w i t h i n v e r s i o n formula

Here t h e 1 / 2 comes from t h e o r d e r o f t h e Weyl group (which i s two) and c ( u ) i s t h e standard Harish-Chandra f u n c t i o n .

For a

general expression o f C(M) we can w r i t e ( c f . a l s o (3.14)) c ( u ) = I(ip)/I(p)

where ( c f . G i n d i k i n - K a r p e l e v i c [l],Helgason

[3; 81, Warner [l;21)

Here B(x,y)

= I'(x)r(y)/I'(x+y)

i s t h e Beta f u n c t i o n and one

d e f i n e s (v,a) i n t h i s c o n t e x t i n terms o f t h e K i l l i n g form B(-,*)

as B(H ,Ha)

A(h) = B ( j , H )

V

where f o r X

for H

E

l i n e a r maps o f a i n t o

E

*

aC, HA

E

*

aC i s determined by

a ( n o t e here t h a t aC i s t h e space of R w h i l e aC i s t h e c o m p l e x i f i c a t i o n o f a

which i s f o r m a l l y t h e s e t o f a l l sums A + i u f o r

92

A,u

E

a).

Then

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

*

a+ i s t h e preimage o f a+ under t h e map A chamber i n a later.

* . *

HA and i s a Weyl

S p e c i f i c examples o f c ( p ) w i l l be w r i t t e n down

Now a / W

- a+* (.recall W

= (1,s)

o r e q u i v a l e n t l y sat = a-t = a t ’ )

where, f o r $

+.

where sH = -H f o r

H

E

a

and one can w r i t e

L2 (B),

E

The q u a s i r e g u l a r r e p r e s e n t a t i o n L o f G on L2 (V), d e f i n e d by L ( g ) f ( v ) = f(g-’v)

(1.5)

L =

decomposes i n t h e form

r Ja*,W

‘plC(w)

dp

where L a c t s i n H by t h e same r u l e as L. L i s i n f a c t i r P Fc 1-1 r e d u c i b l e and u n i t a r y and i s e q u i v a l e n t t o t h e s o - c a l l e d c l a s s one p r i n c i p a l s e r i e s r e p r e s e n t a t i o n induced from t h e p a r a b o l i c subgroup MAN by means o f t h e c h a r a c t e r man

-f

a”

= exp i p l o g a.

We r e c a l l here a l s o t h e d e f i n i t i o n o f t h e mean value o f a f u n c t i o n @ over t h e o r b i t o f g r ( h ) = gu under t h e i s o t r o p y = gKg-’ subgroup Iv

Thus w r i t i n g MhC$ = Mu$ we have

a t v = .rr(g).

( r e c a l l i n g t h a t r ( h ) = u)

93

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

( c f . Helgason [l]) when D

D(G/K).

E

The symbol D(G/K) denotes

t h e l e f t i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on G/K which a r e d e f i n e d as follows.

If $ : G

G i s a diffeomorphism and f i s

-+

a f u n c t i o n on G one sets f $ ( g ) = ( f

while i f D i s a -1 l i n e a r d i f f e r e n t i a l o p e r a t o r we w r i t e DQ : f + (Of$ )$. Thus

G o r on V = G/K.

$-')(g)

W r i t i n g akg = gk f o r g

G and k T

space of d i f f e r e n t i a l o p e r a t o r s on G such t h a t D D i s denoted by D(G/K);

E

K, t h e

= D and D"k =

t h i s i s i d e n t i c a l w i t h t h e space o f

l i n e a r d i f f e r e n t i a l o p e r a t o r s on G/K ( c a l l e d l e f t i n v a r i a n t ) such t h a t

T

D

= D (the condition

proof obvious).

Dak

D i s automatic on G/K

=

-

Now t h e zonal s p h e r i c a l f u n c t i o n s on G a r e

d e f i n e d by t h e formula

(u

E

*

a )

. ..,

and we can w r i t e by i n v a r i a n t i n t e g r a t i o n 41 (9) = $ (gK) (ac-

u

..,

u

..,

t u a l l y t h e $ (9) a r e K b i i n v a r i a n t i n t h e sense t h a t $ ( 9 ) =

u

* -

..,

$ ( k g k ) ) and K i s u n i m d u l a r

u

-

u

c f . Helgason [l], Maurin [l],

-

Nachbin [2], Wallach [l],o r Weil [l] and thus

.,K CI$

f(k-')dk

-u

f(ikk)dk = ..,

f(k)dk. I t i s known f u r t h e r t h a t $,(g) = K ( g - l ) ( c f . Harish-Chandra [2] o r Warner [l;21) w h i l e t h e

E

=

C"(G/K),

@u X ($ ) u ..,u of

1

I,

for D

and a r e c h a r a c t e r i z e d b y t h e i r eigenvalues AD = E

D(G/K), w i t h $u(n(e)) = 1, p l u s t h e b i i n v a r i a n c e

(I~.We now demonstrate a lemma which has some i n t e r e s t i n g

consequences.

94

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

h The F o u r i e r t r a n s f o r m o f 1.1 = MU

Lemma 1.1

.rr(h)) i s g i v e n b y FMh = (M‘f)(v)

-

A

1-I

(h) and i f h = kak w i t h a

Proof:

A+ then

o f u = .rr(h).

h The a c t i o n o f M o r MU as a d i s t r i b u t i o n i n E ’ ( V )

determined by ( c f . (1.6)) b e i n g t h e i d e n t i t y i n G).


h

h

,$> = (14 $ ) ( w )

Thus i n (1.6)

A

is

where w = r ( e ) ( e

take Q = exp (ill+ p )

A

where b = kM.

Since A(ekhK,kM)

t h e u n i m o d u l a r i t y o f K ( c f . (1.8) (1 -9)

E

(u =

( M a f ) ( v ) so t h a t (Muf)(v) depends o n l y on t h e r a d i a l

=

component a i n t h e p o l a r decomposition (kM,a)

A(w,b)

E’(V)

E

-

(Mh$)(w) =

I

= -H((kh)-’;)

one has by

and comments t h e r e a f t e r )

exp [-(ip+p)H(h -1 k -1,- k)dk

K

j

=

exp [-(ip+p)H(h -1 kk)dk =

K

,. = $

-1-1

(h-’)

=

-

0

1-I

j

e-(i’+p)H(h-lk)dk

K h (h) = FM

h One notes here t h a t M = MU works on f u n c t i o n s i n V such as exp (i’+p)A(v,b) placed by 14‘

= $

h

= M

h and i s p r e c i s e l y (1.1) w i t h f r e -

and $ = exp (iu+p)A(w,b);

no i n t e g r a t i o n over

V i s i n v o l v e d s i n c e v = w i n $, o n l y a d i s t r i b u t i o n e v a l u a t i o n A

i s o f concern here.

-

F i n a l l y we n o t e t h a t w i t h h = kak as above

t h e r e f o l l o w s from (1.6) (1.10)

(Mhf)(v) =

J K f(gkiaK)dk

=

i,

f(gkaK)dk = (Maf)(v)

T h i s l e a d s t o a symmetric space v e r s i o n o f theorem o f Zalcman [l],g e n e r a l i z i n g an o l d formula o f P i z z e t t i ; here G/K

95

QED

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

i s of a r b i t r a r y rank and we refer t o Helgason [l ; 23 o r Warner [ l ; 23 f o r general information on h i g h e r rank s i t u a t i o n s . Theorem 1.2

Suppose the o p e r a t o r s

(Pizzetti-Zalcman).

A k w i t h eigenvalues X k g e n e r a t e D(G/K) as an a l g e b r a ; then l o -

cal l y (1.11)

(MUf)(V)

=

bqu,

Ak)f)(V)

00

1 Pn(u,X k ( 4 v ) ) i s expressed i n terms o f i t s n=l eigenvalues a s $ (u,Xk).

where $,(u)

= 1 +

P

Zalcman [ l ] , i n an Rn c o n t e x t , has g e n e r a l i z e d t h i s kind o f theorem considerably Proof:

.

h

One n o t e s from Helgason [ l ] t h a t (M @,)(v)

-

i,

=

= z X ( g ) $ X ( h ) and the l o c a l expres$X(gk.rr(h))dk = ;Jgkh)dk K O3 sion (Muf)(v) = ([1 + 1 Pn(u,Ak)]f)(v) holds. Consequently n=l we o b t a i n $ X ( h ) = applying t h i s l o c a l e x p r e s s i o n t o

-

m

1 P , ( U , A ~ ( $ ~ ) =) $,(u) and (1.11) follows. One should n=l remark a l s o t h a t the polynomials P a r e without cons t a n t terms. 1 +

n

2.2

Resolvants.

The o b j e c t s o f interest n a g e n e r a l i z e d

EPD theory a r e the r a d i a l components o f a b a s i s f o r the H

v spaces

of (1.4), m u l t i p l i e d by a s u i t a b l e weight funct on, the r e s u l t

o f w h i c h we w i l l denote by @ ( t , u ) ( c f . C a r r o l l [21; 221, C a r r o l l S i l v e r [15; 16; 171, S i l v e r [ l ] ) and we mention t h a t m can denote

96

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

a m u l t i i n d e x here.

F i r s t we remark t h a t V i s endowed w i t h t h e

Riemannian s t r u c t u r e induced by t h e K i l l i n g form B ( * , * )

(but

t h e Riemannian s t r u c t u r e does n o t p l a y an i m p o r t a n t r o l e here, e.g.,

$(*,*)

c o u l d serve

-

see Example 3.2) and f o r rank

V

=

1, D(G/K) i s g e n e r a t e d by a s i n g l e Laplacian A, determined by

-

t h e standard Casimir o p e r a t o r C i n t h e enveloping algebra o f g ( c f . Remark 3.15 about C).

We l o o k a t t h e r a d i a l component AR o f

passing t h i s from t h e coord n a t e t i n at E A t o r ( A ) i n an a obvious manner, and s e t t i n g Mt = M w i t h Ro(t,p) = FMt = $,,(at) A,

-

A

one o b t a i n s an eigenvalue e q u a t i o n ( c f . Helgason [5]) (2.1)

[of. +

(ma+mZa)cotht Dt + mZatanht D

141

t u

The s o l u t i o n o f (2.1), a n a l y t i c a t t = 0, i s e.g., A

(2.2)

^Ro(tyu) = (I (exp t H o ) = F(G,B,y,

u

"0

"0

where e v i d e n t l y R ( 0 , ~ =) 1 and Rt(o,p)

B

=

-

2

sh t )

0 ( 6 = (ma+2m2a+ 2 i p ) / 4 ,

= ( m +2mZa-2iu)/4,

and y =(ma+m2a+1)/2). The general EPD a "0 s i t u a t i o n i n v o l v e s embedding R i n a "canonical" sequence o f "m l l r e s o l v a n t s " R (t,p), f o r m > 0 a p o s i t i v e i n t e g e r o r a m u l t i index, such t h a t t h e r e s o l v a n t i n i t i a l c o n d i t i o n s (2.3)

"m R ( 0 , ~ =) 1;

are s a t i s f i e d .

"m Rt(oYp) = 0

There w i l l a l s o be "canonical" r e c u r s i o n r e l a t i o n s

97

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

“m between t h e R of i n d i c e s d i f f e r i n g by k1 o r k 2 which w i l l a r i s e group t h e o r e t i c a l l y f r o m c o n s i d e r i n g a f u l l s e t o f b a s i s elements i n the H

spaces.

1-I

2 Thus we must f i r s t determine a b a s i s f o r L ( B ) and t h i s i s w e l l known (thanks a r e due here t o R. Ranga Rao f o r some h e l p f u l information).

We l e t { I T ~ , V ~ Iw i t h dim VT = dT be a complete s e t

o f i n e q u i v a l e n t i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s o f K and l e t

M VT C V T be t h e s e t o f elements f i x e d by M.

One knows by a r e -

s u l t o f Kostant [l]t h a t dim V y = 1 o r 0 i n t h e rank one case ( c f . a l s o Helgason [5]) and f o r t h e s e t

T E

we l e t w i be a b a s i s v e c t o r f o r VTM w i t h

T where dim

wY(1

2

normal b a s i s f o r VT under a s c a l a r p r o d u c t < example t h e c o l l e c t i o n o f f u n c t i o n s

(T E

T)kM

= 1

i 5 dT) an o r t h o >T.

-+

!V

Then f o r

T

is

2 known t o be a b a s i s f o r L ( B ) and we d e f i n e ( c f . (1.4) w i t h v = atK and b = kM) n -

(2.4)

$;,(atK)

= El 9T(BT:-ip:a -1 )w T

t

where BT

E

j

HomM(VT9$) i s determined by t h e r u l e Bw‘:

= 61,s

(Kronecker symbol) so t h a t B T r T ( k )-1 wT. = T = J J T <~~(k)-’w;,w;>~ (note t h a t $(k)dk = $(km)dm)db and K B M

J

I (I

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

H(g-’km)

= H(g-’k)

w i t h nT(km) = nT(k)nT(m)).

above

The

are E i s e n s t e i n i n t e g r a l s as d e f i n e d i n Wallach [l]and t h e A .

( a K) a r e t h e r a d i a l components o f b a s i s elements i n H t FC ( c f . below). I t i s p o s s i b l e t o o b t a i n an e x p l i c i t e v a l u a t i o n o f M,T

these f u n c t i o n s , u s i n g r e s u l t s o f Helgason [5;

111 as f o l l o w s .

One d e f i n e s ( c f . Helgason 151)

Then i t i s proved i n Helgason [5;

f o r x = gK.

111 ( t o whom we

g r a t e f u l l y acknowledge some conversation .on t h e m a t t e r ) t h a t

where Y

,T

( x ) i s d e f i n e d by

j

e (-iA-p)H(g-’k) < wl,T a T ( k ) w i >T dk K Thus r e c a l l i n g t h e p o l a r decomposition (kM,a) + kaK we have yA,T(x) =

(2.7)

Theorem 2.1

;’

(2.8)

MrT

h

General b a s i s elements i n

(iatK) =

1

K

e (iu-p)H(at k

dT- l l 2 fT,J -’.(katK)

I n p a r t i c u l a r one has s i n c e ti1

H me FC

-1^-1

A

=

A

G’

k ) ., T

dk

A T = < w i , ~ ~ ( k ) w Y-FC,T(atK) ~>~

;’

-’

( a K) = ( a K) = Y (atK) t 1-I.T t ,T ( t h i s “ c o l l a p s e ” was observed i n VaT

= ,j J T 1 T s p e c i a l cases by S i l v e r [l]). A

Now l e t KO = {nT,VTl f o r T ET (i.e.,

dim VTM = 1 ) and f o l l o w -

i n g Helgason [5] we use a p a r a m e t r i z a t i o n due t o Johnson and

99

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Wallach (see Helgason [5] f o r references and c f . a l s o Kostant h

[l]).I f mZcl = 0, KO = {(p,q)> w i t h p ..

1, KO = {(p,q)> w i t h (p,q)

E

E

Z+ and q = 0; i f mZa =

Z+ x Z where p

f

q

E

i f mZa =

22,;

h

3 o r 7, KO = {(p,q)>

w i t h (p,q)

E

Z+ x Z+ where p

&

q

E

One

22.,

s e t s R = (ip+p)(HO) and then ( c f . Helgason [5; 111) Theorem 2.2

The r a d i a l components o f b a s i s elements i n

H

IJ

are given by

= cI.1'

STt h P t

ch-'tF(-,--p

R+

where t h = tanh, ch = cosh, and

+

R+p-q+l -m2a

+

ma+m

+1

,th2t)

F i s t h e standard hypergeometric

function. Note here t h a t o u r R i s t h e n e g a t i v e o f t h e R i n Helgason [5;

111 where R = ( i A - p ) ( H ) = ( - i p - p ) ( H

i n (1.1))

0

0

) ( c f . here o u r n o t a t i o n

and r e c a l l t h a t a(Ho) = 1 w i t h at = exp tHo.

now sets dZa = -4q(q+mZcl-1) and da = -p(p+m +mZa-l) i t i s shown i n Helgason [5] t h a t Y ( t ) = Y

the d i f f e r e n t i a l equation

100

-1-I ,T

I f one

+ q(q+mZcl-l)

(atK) a l s o s a t i s f i e s

2.

(2.11)

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

ytt

+ (ma+in2a)coth t Yt + m2a tanh t i t

+

[d sh-2t + d2ash-22t + p(Ho) 2 + v(Ho) 2 19' = 0 a

where sh = s i n h , and we have used t h e i d e n t i t y c o t h 2 t = h c o t h t + tanh t). R e c a l l i n g t h a t p ."(-in21 a+m 2a ) one sees t h a t the t r i v i a l representation 1 g i v e s r i s e t o (2.1),

T, corresponding t o p = q = 0,

E

so t h a t (since c

R+I-M,

.w

1) we should have +I 2a , a 2a , t h t). This =

m +m

(atK) = +(at) = ch-'t F(L/2, ,1 i s borne o u t b y t h e Kummer r e l a t i o n F(a,b,c,z)

y-lJ

w i t h z = -sh 2 t, a = 6

c,z/z-1)

( l - ~ ) -= ~ch-'t

and c

-

b = y

B

=

(l-z)-aF(a,c-b,

b , = 8 , and c = y, so t h a t

= R/2,

-

=

(ma+2+2iv)/4 = (R+1-m2a)/2;

. "

hence Y

-v,1

2.3 h

(atK) = +(a ) as i n d i c a t e d . t = 0.

Examples w i t h

Now t o c o n s t r u c t r e s o l v a n t s

A

( a K) one m u l t i p l i e s t h e t by a s u i t a b l e f a c t o r i n o r d e r t o o b t a i n t h e r e s o l v a n t i n i -

Rm(t,v) = RPsq(t,p) from t h e Y

y-v ,T

-!J,T

. "

t i a l c o n d i t i o n s (2.3)

and t o produce I$ (a,)

v

when p = q = 0.

These requirements are n o t alone s u f f i c i e n t t o produce t h e "cano n i c a l " r e s o l v a n t s s i n c e one needs t o i n c o r p o r a t e c e r t a i n group t h e o r e t i c r e c u r s i o n r e l a t i o n s i n t o t h e t h e o r y which serve t o " s p l i t " t h e second o r d e r s i n g u l a r d i f f e r e n t i a l equation f o r t h e A

Rp,q

a r i s i n g from (2.11)

i n t o a composition o f two f i r s t o r d e r

equations ( c f . here I n f e l d - H u l l [ l ] ) .

Even then we remark t h a t

t h e r e s o l v a n t s w i l l n o t be unique since one can always m u l t i p l y ^Rm(t,p) by a f u n c t i o n $, I+

0

( t ) E 1.

E

C2 such t h a t $,(o)

=

1, $ i ( o ) = 0,

T h i s w i l l simply g i v e a d i f f e r e n t second o r d e r

101

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

s i n g u l a r d i f f e r e n t i a l equation f o r t h e new r e s o l v a n t and d i f -

-

ferent s p l i t t i n g recursion r e l a t i o n s b u t the resolvant i n i t i a l c o n d i t i o n s w i l l remain v a l i d and t h e r e d u c t i o n t o 0 (a ) f o r m = U

0 i s unaltered.

t

Thus we w i l l choose the s i m p l e s t form o f r e s o l -

vant f u l f i l l i n g t h e s t i p u l a t i o n s imposed w h i l e r e f e r r i n g t o these r e s o l v a n t s and equations as canonical. Example 3.1 -p

2

, d2,

Take t h e case where ma = 1 , mZa = 0, da =

= 0, and p = 1/2.

T h i s corresponds t o G = SL(2,IR)

with

K = S O ( 2 ) and t h e r e s o l v a n t s a r e given i n C a r r o l l [18; 19; 223,

C a r r o l l - S i l v e r [15; 16; 171, S i l v e r [l]as

where 5 = c h t and P i m denotes t h e standard associated Legendre f u n c t i o n o f t h e f i r s t kind.

Now we r e c a l l a formula ( c f . Snow

111, P. 18)

2 S e t t i n g z = t h t i n (3.2) and n o t i n g t h a t 1 have fi = sech t and 1 as above.

-

-

2 2 t h t = sech t we

n / ( l + Z ) = 5-1/<+1 w i t h 5 = c h t

Now one observes t h a t

102

(t,U)

i n (3.1)

i s symmetric

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

t h i s i s e x h i b i t e d f o r example i n the form of

i.n !J (and t ) and

t h e r e s o l v a n t (3.1) when one uses t h e formula ( c f . Robin [l]) 1 1 2-mshmt m+-+ip m+--ip 2 2 (3.3) P-m (cht) = F( , ,m+l,-sh t ) ip-

'2

z

= F(b,a,c,z))

Hence (3.1) can be w r i t t e n ( r e c a l l t h a t F(a,b,c,z) (3.4)

1 1 "m m+T+ip m+q-ip R (t,u) =F( , , m + 1, -sh 2 t ) Now, r e t u r n i n g t o (3.2) and (2.9)

T

-

-

(2.10), we w r i t e f o r and R = i u

( p , ~ ) w i t h ct = i d 2 , B = 1/2(p + 1/2),

= c

R

-!J,T

t h P t ch- t (

1 t sech t ) -

1/2

2(ct+B)

1

+2 + p,

= c-!J ,Ts h P t ( y ) - ' - P F ( i p

+

ill +

1

7, p+l,

5- 1

5+1)

R e w r i t i n g t h e f i r s t equation i n (3.1) w i t h p replaced by -!J i n t h e r i g h t hand s i d e we o b t a i n (3.6)

i m ( t , p ) = ($!-)

-R-m F (1z + ip,z+ 1 ip+m,m+l,-) 5-1

5+1

Then i d e n t i f y i n g p w i t h m we can say from (3.5) h

(3.7)

RP(t,p) =

sh-Pt Y

-!J,T

103

( a K) t

-

(3.6)

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

For completeness we check t h e d i f f e r e n t i a l e q u a t i o n s a t i s f i e d by t h e Rp o f (3.7),

g i v e n t h a t (2.11) holds.

An elementary c a l c u l a -

t i o n y i e l d s then

which agrees w i t h p r e v i o u s c a l c u l a t i o n s ( c f . C a r r o l l [21; C a r r o l l - S i l v e r [15;

223,

16; 171, S i l v e r [l]).The canonical r e c u r s i o n A

r e l a t i o n s a s s o c i a t e d t o t h e Rp o f (3.1),

(3.4),

(3.6),

o r (3.7)

can o f course be r e a d o f f from known formulas f o r t h e associated Legendre f u n c t i o n s f o r example b u t t h e y can a l s o be o b t a i n e d group t h e o r e t i c a l l y ( c f . Example 3.5 o f b a s i s elements i n t h e

-

Theorem 3.8)

H spaces. P

by u s i n g a f u l l s e t

For now we s i m p l y l i s t them

i n t h e form

i!

+ 2p c o t h t

^Rp

= 2p csch t

iP-'

E v i d e n t l y t h e composition o f these two r e l a t i o n s y i e l d s (3.8) and t h i s i s t h e sense i n which we speak o f " s p l i t t i n g " t h e r e s o l v a n t equation (3.8). Example 3.2 Example 3.1.

We g i v e now e x p l i c i t m a t r i x d e t a i l s t o c l a r i f y

(The E u c l i d i a n case can a l s o be t r e a t e d group

t h e o r e t i c a l l y as i n C a r r o l l - S i l v e r [15;

16; 171 and e s p e c i a l l y

S i l v e r [1] and a s i m p l e example i s worked o u t a t t h e end o f t h i s chapter; t h e r e s u l t s o f course agree w i t h those o f Chapter 1.)

104

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

-

Thus l e t G = SL(2,lR)

-

and K = SO(2) w i t h L i e algebras g = sR(2,R)

We r e c a l l t h a t G i s connected and semisimple,

and k = so(2).

-

c o n s i s t i n g o f r e a l 2 x 2 matrices o f determinant one, w h i l e the

- -

matrices i n t h e compact subgroup K a r e orthogonal; g consists o f r e a l 2 x 2 m a t r i c e s o f t r a c e zero and k c g i s composed o f skew We w r i t e V = G/K and s e t

symmetric matrices.

X =

(3.10)

1 7

-

O

1 1 ., Y = 1z

1

0

so t h a t k = lR Z and we w r i t e p = { J R X +.lRY) f o r the subspace

-

--

( o f v e c t o r spaces) w i t h [k,k] Cartan i n v o l u t i o n 8 :

-

5 +n

b r a automorphism o f g.

- -

One has a Cartan decomposition g = k + p

o f g spanned by X and Y.

-t

-

- -

c k, [p,p] c k , [k,p] c p and the

6-

Q

-

(6 E

k,

n

E

p ) i s a L i e alge-

Recall here t h a t i f P = exp p then G =

PK i s t h e standard p o l a r decomposition o f g

E

G i n t o a product The K i l l i n g

o f a p o s i t i v e d e f i n i t e and an orthogonal m a t r i x . form B(5,n) = t r a c e ad5

adn

(=4 t r a c e

"

CQ)

-

i s negative d e f i n i t e

on k and p o s i t i v e d e f i n i t e on p ( w i t h B(k,p) = 0).

One checks

e a s i l y t h a t X and Y form an orthonormal basis i n p f o r t h e s c a l a r product

1 ((6,~))= p(5,n); we

repeat t h a t t h e 1/2 f a c t o r

i s o f no p a r t i c u l a r s i g n i f i c a n c e here i n c o n s t r u c t i n g resolvants, etc. and i s used mainly t o be c o n s i s t e n t w i t h some previous work and w i t h t h e e x p o s i t i o n i n Helgason [l]. Now s e t t i n g Xa = X

+

Z, X-a

= X

- Z, and Ha

standard ( o r "canonical") t r i p l e ( c f . Serre [ l ] )

105

= 2Y we have a

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

= RX,

where t h e r o o t subspaces g,

(5

i z e d by t h e r u l e gA = w h i l e t h e map a : t Y r o o t ) i n t h e dual a

*

and g

.-E

-a

RX

=

are character-,

g; ad H< = X(H)S f o r a l l H

a = RY),

E

t determines then an element ( c a l l e d a

-+

( n o t e t h a t ,(Ha)

-a i s a l s o a Cartan subalgebra of g.

= 2 and

R

N

a

*

=

Ra).

Here a i s a maximal a b e l i a n subspace o f p and i n t h i s case a = .--

{I,,g,

> 01,

.--

N = exp n, and A

g = k a n

(3.13)

e t t

-

< IT)

= exp B Z exp t Y exp

[b

=

The Iwasawa decomposi-

= exp a.

e

t i o n o f G can be w r i t t e n ( 0 2

-.

Set now n = g,

exa

which we express as b e f o r e i n t h e form g = k(g) exp H(g)n(g).

[-:

Next we s e t M = {+ 1

I4 = M {k

E

u {+

:]I

f o r t h e c e n t r a l i z e r o f A i n K and

w i l l be t h e n o r m a l i z e r o f A i n K ( t h u s M =

K; AdkH = H f o r H

- . - -

I

E

a and M = { k

E

K; Adk a c a) ).

Again

I

W = M /I4 has o r d e r two and B = K/M i s e s s e n t i a l l y K w i t h t h e angle v a r i a t i o n c u t i n h a l f . b = kM

E

B A(v,b)

= -H(g-lk)

We r e c a l l t h a t f o r v = gK and here p = 1/2

106

l m A A (A>O)

E

V and equals

2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

1

Thus p ( t Y ) = t / 2 and f o r 1-1

p.

R, 1-1

E

"* a

E

i s determined again

-*

by t h e r u l e u ( t Y ) = tu(Y) = 1-1t (such l~ o f course exhaust a ct

- 1-1

(1.1)

= 1).

The F o u r i e r t r a n s f o r m F f o f f

E

and

L2 ( V ) i s given by

and a+ i s d e f i n e d as i n S e c t i o n 1, as i s t h e s c a l a r prod-

u c t (X,v) = B(HA,HV).

When 1-1

E

-* a

= 2 i t follows t h a t H

s i n c e B(Y,Y)

i s c h a r a c t e r i z e d as above then = p1

*

Y

and o b v i o u s l y a+ can 1-1 "*" thus be i d e n t i f i e d w i t h ( 0 , ~ ) . E v i d e n t l y f o r 1-1, v E a one has

Example 3.3

A geometrical r e a l i z a t i o n o f t h e present s i t u a -

t i o n can be described i n terms o f G a c t i n g on t h e upper h a l f a b p l a n e I m z > 0 by t h e r u l e g z = (az+b)/(cz+d) f o r g = ( c d) G.

E

Then K i s t h e i s o t r o p y subgroup o f G a t t h e p o i n t z = i

(i.e.,

K

i = i ) and t h e upper h a l f plane can be i d e n t i f i e d

w i t h G/K under t h e map gK d e s i c p o l a r coordinates

-+

g

(T,e)

i ( c f . Helgason [ l ] ) .

I n geo-

a t i t h e Riemannian s t r u c t u r e i s

2 2 2 2 described by ds =d-c + s i n h .cde where we use t h e m e t r i c tensor determined by gv(<,q)

for

= $(C,q)

5,

q

E

p.

I f w = r ( e ) (which

corresponds t o i under o u r i d e n t i f i c a t i o n s i n c e r ( e ) = K) then, denoting l e f t t r a n s l a t i o n i n G by

t h e Riemannian s t r u c t u r e i s g' determined b y g V ( ( T )*<,(T~)*T-I) = gw(S,q) ( f o r t h e n o t a t i o n ( T )* 9 9 see e.g. Kobayashi-Nomizu [ l ; 21). The geodesics on V through

w =.rr(e) a r e o f t h e form y : s

t h e square .rr(exp

T'

-+

T

(exp sC)w and i f

5

= aX

+ BY

E

o f t h e geodesic d i s t a n c e from w t o (exp <)w =

5 ) i s equal t o

a2

+ B2 ( s i n c e X and Y form an orthonormal

b a s i s f o r p under t h e s c a l a r product ((<,q)) = 1/2 B(5,n)) 107

and

p

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

the polar angle i s determined by tan 8 = B/a.

Many calculations

can easily be made using standard formulas and one can describe the action g

i for example i n general (relevant material for

the computations appears i n Gangolli [l], Helgason [ l ; 2 ; 31, and

21).

Kobayashi-Nomizu [ l ;

We will spare the reader the results

of o u r computations since they are essentially routine.

Let us

however remark that the volume element dv i n (1.1) can be written i n this example as dv = shT d.r d8 ( u p t o a normalization factor). F i r s t we refer back t o the formula for c(v)

Example 3.4

given a f t e r ( 1 . 2 ) and mention t h a t i n general (cf. Helgason [2; 31, Warner [Z])

where

n=

Y

On = g-ahere,

c(-ip) = 1.

for 5

exp

n, and

dii i s normalized by

Here 8 i s the Cartan involution 8 : 5 + rl

Y

E

K=

k and rl

E

+

E

- rl

p and one observes t h a t i t is an involutive

Y

automorphism of g such that

-.-.

(c,rl)

-+

-B(€,,Orl)

i s s t r i c t l y positive

definite on g

x

Since

1 / 2 , ( i p , a ) = i p , and ( p , a ) = 1/4 there results

(cl,a) =

g.

In terms of Beta functions we obtain

(cf. Bhanu Murti [l])

108

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

-

Thus we can r e c o v e r f ( v ) f r o m f ( u , b ) Example 3.5

u s i n g (3.16)

and (1.2).

We now want t o show how t h e r e s o l v a n t s (3.1)

can be o b t a i n e d by a cumbersome and " c l a s s i c a l " method i n o r d e r t o f u r t h e r c o n f i r m (.3.7) as a l e g i t i m a t e r e s o l v a n t ( t h e equival e n c e o f (3.1)

and (3.7)

has a l r e a d y been e s t a b l i s h e d ) .

The

t e c h n i q u e has a c e r t a i n i n t e r e s t and moreover we w i l l o b t a i n a

H

w h i l e showing how t h e r e c u r s i o n r e l a t i o n s (3.9) 1-1 can be o b t a i n e d group t h e o r e t i c a l l y . Some o f t h e more t e d i o u s f u l l basis f o r

c a l c u l a t i o n s w i l l be o m i t t e d .

We remark t h a t Theorem 2.1

does

n o t seem c o n v e n i e n t h e r e t o o b t a i n a f u l l s e t o f b a s i s elements

in

Hu s i n c e t h e u n i t a r y i r r e d u c i b l e r e p r e s e n t a t i o n s o f K

(see V i l e n k i n [l]).Thus

a r e one dimensional o f t h e form ein4 h

h

= SO(2)

h

h

f i r s t we can remark t h a t atk0 = k a n where a = as w i t h (3.17)

es = ( c h t t sh t cos 0 )

(cf..Helgason

[l]and Warner [l;23).

Thus pH(atke) = p(sY) =

s/2 and f o r ~ ( s V ) = us one has (iu-p)H(atkg) (3.18)

e

(i1-1-

2I) s

= e

=

(ch t

-+ sh

t

COS

0)

ill-

I

Now, g i v e n b a s i s v e c t o r s $m(b) = exp im(0-n) f o r example i n

LL(B) (recall t h a t B t i o n o f (3.13), in

-

K w i t h t h e a n g l e v a r i a t i o n , i n o u r nota-

c u t f r o m ( 0 , 4n) t o (0, IT)), we g e t b a s i s v e c t o r s

H by means o f (1.4) ( t h e f a c t o r o f exp ( - i m ) i s i n s e r t e d f o r

u

convenience i n c a l c u l a t i o n ) . o b j e c t s " when v = atK i n (1.4)

I n p a r t i c u l a r we o b t a i n " r a d i a l and s i n c e A(gK,kOM) = -H(g-'kO)

109

we

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where s ' i s determined as i n

consider H(ailkg) = s'Y i n (1.4). (3.17) b u t w i t h t replaced by -t. b a s i s vactors i n H

(3.19)

9

u =

u

w i l l be I

( i u - p ) s Yeimede

e

-imr

=

Thus the r a d i a l p a r t o f the

21T

1

TT

(ch t

-lT

-

I iv- 7 e irned0 sh t cos 0 )

'

iv- 1 - m' .o (ch t ) (see V i l e n k i n [ l ] ) .

%l

I n order now t o produce r e s o l v a n t s R ( t a u )

as i n Section 2.2 from Pm

a0

-

1/2(ch t ) o f (3.19)

consider t h e growth o f these f u n c t i o n s as t

+ .

one must f i r s t

0 and i n t r o d u c e

s u i t a b l e weight f a c t o r s i n o r d e r t o s a t i s f y t h e r e s o l v a n t i n i t i a l c o n d i t i o n s (2.3)

(note t h a t

a / a t = ( ~ ~ - 1 ) 3/22). l/~

L e t us r e c a l l

i n t h i s d i r e c t i o n t h a t ( c f . V i l e n k i n [l]and Robin [l]) (3.20)

where P!

P:am(~) =

r ( k t m t l ) r ('R-mtl r2(R+i)

)

R

pm,o(z)

i s the standard associated Legendre f u n c t i o n o f the

f i r s t kind.

Then we can e x h i b i t t h e r e s o l v a n t s i n t h e form

110

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

z + l R-m

z-1 F( -R ,m-R ,m+l -,)z+l

= (T)

where z = c h t, R = -1/2 + i p , rn

2 0 i s an i n t e g e r , and F denotes

a standard hypergeometric f u n c t i o n ( c f . C a r r o l l - S i l v e r [15; 171 and C a r r o l l [18;

191).

16;

R e c a l l i n g t h a t r(R+m+l)Pim(z) =

r ( a - m + l ) P ~ ( z ) f o r m 2 0 an i n t e g e r , we remark i n passing t h a t , for m

E

C n o t a n e g a t i v e i n t e g e r , r e s o l v a n t s can be determined

b y t h e formula

(which reduces t o (3.21)

when m i s an i n t e g e r ) and t h i s should be *

taken as t h e generic expression f o r Rm i n t h i s case ( c f . (3.1)). Evidently the

^Rm

o f (3.21)

s a t i s f y (2.3).

We r e c a l l t h e n e x t w e l l known formulas

(3.23)

,

dP! mz ( ~ ~ - 1 )" ~ + /2 dz (22-1)

P!

=

(a+m) (R-m+l )P!-'

( c f . V i l e n k i n [l]and Robin [l])and these, w i t h (3.22), (3.9),

where p i s r e p l a c e d by m ( n o t e t h a t R(R+1) =

lead t o

- ( 1B + u 2 ) ) ;

consequently (3.8) w i l l f o l l o w as before. The most i n t e r e s t i n g f a c t however about (3.9)

i s t h a t these

r e c u r s i o n r e l a t i o n s have a group t h e o r e t i c s i g n i f i c a n c e (cf. 111

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

-

C a r r o l l - S i l v e r [15; 16; 171) and t h e v a r i a b l e t i n (3.8)

(3.9)

may be i d e n t i f i e d w i t h t h e geodesic d i s t a n c e T o f Example 3.3. For purposes o f c a l c u l a t i o n i t i s convenient t o c o n s i d e r X+ = -X + i Y ;

(3.25)

X

and t o n o t e t h a t (H, X+,

-

-

= -X

iY;

-

H = -2iZ

X-) form a "canonical" t r i p l e r e l a t i v e -.

t o t h e r o o t space decomposition gt = kt

Y

+ t X + + CX - , kt

= (CH, i n

t h e sense t h a t (3.26)

= 2X+;

[H,X+]

[H,X-]

= -2X

[X+,X-]

;

(one notes t h e d i f f e r e n c e here between (3.12) -.

9Q; = Q;Ha

+ t$ +

(H,X+,X-)

tX,

= H

c o m p l e x i f i e d as

and t h e decomposition above r e l a t i v e t o

based on (3.26)

-

such decompositions occur r e l a t i v e

t o any Cartan subalgebra such as (CH o r ICY as i n d i c a t e d i n Serre

[l]f o r example).

It w i l l be u s e f u l t o e x p l o i t t h e isomorphism

Q between S L ( 2 , R ) and t h e group SU(1,l)

o f unimodular q u a s i u n i 1 i t a r y m a t r i c e s g i v e n by Q(g) = g = m-lg m where m = (i l). Thus A

g =

(y E)

+

6

=

a b (6:)

w i t h a = 1/2[a+6+i(B-y)]

and b =

A

1 / 2 [ ~ + y + i ( a - 6 ) ] w h i l e d e t g = d e t g = 1 ( c f . V i l e n k i n [l]). There i s a n a t u r a l p a r a m e t r i z a t i o n o f SU(1,l) a l i z e d E u l e r angles

h

(I$,T,$)

so t h a t any g

w r i t t e n i n t h e form

112

E

i n terms o f gener-

SU(1,l)

can be

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

where 0 I cp < ZIT, 0 5

<

T

h

pressed as g = (cp,~,$).

I: $ < ZIT;

and -2n

m,

t h i s w i l l be ex-

I n p a r t i c u l a r , s e t t i n g wl(s)

=

exp sX,

wz(s) = exp sY, and w3(s) = exp sZ, we have Qwl(s) = (o,s,o), = (IT/Z, s, -v/Z),

Qw,(s) that

to,.,$)-’

and Qw ( s ) =(s,o,o) 3

= (o,o,s).

We n o t e

and ( ~ ~ , o , o ) ( o , T , o )(o,o,$)

= (IT-$,T,-IT-@) A

A

(+,T,$) w h i l e ifg1 = (0, T ~ , O ) and g2 =

(cp2,~2,~)

= A

h

t h e n gigz =

with

(cp,T,$)

t a n cp = ch

(3.28)

= chTl

ch T

’

tan

T~

sh -rz s i n cpz sh -r2 cos cp + sh T 1 ch 2

+ sh

ch T~

T~

sh

T~

Tz ’

cos cpZ ;

sh T, s i n cpz = sh -r1 ch T~ cos

cpZ + ch

T~

sh

‘c2

A

Now g o i n g t o t h e n o t a t i o n o f Example 3.3 w i t h p = exp (ax = a2 +

T‘

$,

and t a n

e

= @ / aan

+

BY),

easy c a l c u l a t i o n shows t h a t

A

Q ( p ) = (O,-r,-O). A

Consequently t h e geodesic p o l a r c o o r d i n a t e s o f A

h

~ ( p )= pK can be r e a d o f f d i r e c t l y f r o m t h e E u l e r angles o f Q(p). Now t h e r e p r e s e n t a t i o n L

!J

H

o f G on

u

a r e p r e s e n t a t i o n , which we a g a i n c a l l L

W

u’

u

E

( c f . (1.5))

induces

o f i o n dense subspaces

H o f d i f f e r e n t i a b l e f u n c t i o n s by t h e r u l e

u

R e c a l l h e r e t h a t L,,(h)f(n(g)) a p p l y (3.29) f(n(gi(s)))

= f(h-lv(g))

=

f(v(h-lg))

and t o

f o r 5 = X, Y , o r Z one wants t h e n t o d i f f e r e n t i a t e w i t h r e s p e c t t o s where gi(s) 113

= wi(-s)g

=

wil(s)g

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(and v

=

W e work i n the geodesic polar coordinates

T(g)).

( ~ ( s ) , e(s)) o f .rr(g(s))

A

=

A

T ( p ( s ) k ( s ) ) = ~ ( p ( s ) )and consider

therefore Q ( g ( s ) ) = Q ( ; ( s ) ) Q ( k ( s ) ) = (e(s), ~ ( s ) , -e(s))

(o,o,ds)) =

(W, ~ ( s ) ,+(s)-e(s))

=

( e ( s ), T ( s ) ,$ts)).

In par-

t i c u l a r Q ( g l ( s ) ) = Q(exp(-sX))Q(g)=(o,-s,o)(8,T,o)(o,o,u-8) where Q ( g ) = Q ( p ) Q ( k ) = (O,T,-e)(o,o,u).

Then one uses (3.28)

t o compute (o,-s,o)(O,T,o) w i t h

T =

T~ = T,

and 0,

e

=

S imi 1arly Q( g2 ( s ) )

$ = e(s),

~ ( s ) , T~ = -s,

(evidently $ ( s ) i s of no i n t e r e s t here). ( ~ / ,-s 2 ,- ~ / 2 )(8, T ,-8) ( o ,o,

=

U)

= ( ~ / ,o 2 ,o)

(o,-s,o)(~-T/~,T,o)(o,o,u-~) and (3.28) can be applied t o the middle two terms.

Finally Q ( g 3 ( s ) ) = (-s,o,o)(B,T,o)(o,o,u-8)

e write H+ = which lends i t s e l f immediately t o calculation. W 1 Lv(X+), H- = L ( X ), and H3 = 4 ( H ) and using the relation u 2 v I d/ds f(T(s),e(s))ls=O = f T T ( 0 ) + f e 0 ( 0 ) a routine calcula-

tion yields (cf. Carroll-Silver [16]) In geodesic polar coordinates (.r,8) on V

Proposition 3.6 one has (3.30)

H+ = e-ie [-i coth

T

a/ae

+

a/aT]

(3.31)

H- = e i e

T

a/ae

+

a1a-c-j

(3.32)

H3

=

i

From genera

[i coth

a/ ae known facts about irreducib e unitary principal

series representations o f G , o r SU(1,1), and t h e i r complexifications (see e.g.

Miller [ l ;

21 o r

Vilenkin [l])

114

-

other

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

representations contribute nothing new i n t h i s context

-

it is

natural now t o look f o r "canonical" dense d i f f e r e n t i a b l e basis vectors in H of the form

u

(3.33)

H+f:

satisfying

fk(T,e)

= (m-R)f:+

H3fi = mfi

where R = -1/2 + i p ( u > 0 ) and m = 0 , k l , k2,

....

( I t seems

excessive here t o go i n t o the general representation theory of SL(2, R ), o r SU(1,l); the references c i t e d here -and previously a t the b e g i n n i n g of t h e chapter

-

a r e adequate and accessible.)

I t i s then easy t o v e r i f y , using (3.30)

-

(3.32), t h a t the f o l -

lowing proposition holds. Proposition 3.7

Canonical b a s i s vectors i n H can be taken

u

i n the form

(3.34)

f i ( T , e ) = ( - l ) m exp(-im8) P

where R = -1/2

t

R 0 ,m

(ch

ip(u > 0) and m = 0, 21,

T)

k2,

....

In view of (3.20) and the d e f i n i t i o n (3.21) ( o r (3.22)) of

8 we

can write f o r m 2 0, z = ch

Using (3.30)

-

T,

and R = -1/2 + iu

(3.31) one can then e a s i l y prove ( c f . Carroll-

S i l v e r [16])

115

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 3.8

The " c a n o n i c a l " r e l a t i o n s (3.33)

a r e e q u i v a l e n t t o t h e r e c u r s i o n r e l a t i o n s (3.9)

and H-

f o r H,

"m for R

.

I n par-

t i c u l a r t h e v a r i a b l e t i n Example 3.1 can be i d e n t i f i e d w i t h t h e geodesic d i s t a n c e Proof:

T,

from w = K t o k a K. UJt

The r e c u r s i o n r e l a t i o n s (3.9)

low immediately from (3.30)

-

(3.31)

(and hence (3.8))

and (3.34)

suggests o f course t h e i d e n t i f i c a t i o n o f

T

-

(3.35).

A

n

This

and t b u t one can g i v e

We w r i t e k , a k 2 = pk

a computational p r o o f also.

fol-

n

so t h a t kla =

3

pk4 and f o r p = exp (aX+BY) we o b t a i n from Example 3.3 t h e d i s -

tance r 2 = a2 + B

2

n

between w = T ( e ) = K and pK t o g e t h e r w i t h

angle measurements cos

e

=

a/T

and s i n 8 =

B/T.

Set now kl =

a = aty and k4 = kg, u s i n g t h e n o t a t i o n o f (3.13); kJ, r e s u l t a number o f equations connecting t h e v a r i a b l e s

there

( T y t , $ y $ y a y f 3 ) y t h e s o l u t i o n o f which i n v o l v e s t h e i d e n t i f i c a t i o n o f t w i t h T.

Thus t h e r a d i a l v a r i a b l e t i n t h e n a t u r a l p o l a r

expression ( k M y a ) f o r u = T(h) = kUJatK i s i n f a c t t h e geoJ , t d e s i c r a d i u s T o f T ( h ) = pK. The a c t u a l c a l c u l a t i o n s a r e someA

what t e d i o u s b u t completely r o u t i n e so we w i l l o m i t them. We n o t e now t h a t t h e second e q u a t i o n i n (3.9) i n t h e form (shZmt @ ) ' = 2mshzm-lt.

QED

can be w r i t t e n

8-l which y i e l d s ,

i n view

of (2.3), (3.36)

shZmt i m ( t y u )= 2m

1:

sh2m-1

One r e w r i t e s t h e i n t e g r a n d i n (3.36),

116

rl

In-1 R (rlyil)drl

i n terms o f (3.36)m-1

and

2.

CANONICAL SEQUENCES OF SINGUALR CAUCHY PROBLEMS

i n t e g r a t e s o u t t h e rl v a r i a b l e ; i t e r a t i n g t h i s procedure we o b t a i n I f in 2 k 2 1 a r e i n t e g e r s t h e "Sonine" formula

Theorem 3.9 (3.37)

shZmt km(t,p) =

r

m-k+l

r

i s a d i r e c t consequence o f (3.9). have

jt(cht-chrl)k-'

I n p a r t i c u l a r f o r m = k we

t

(3.38)

shZmt @(t,p)

=

2%

0

(cht-chn)m-lsho

s p h e r i c a l f u n c t i o n s (1.8)

and u s i n g t h e n o t a t i o n o f (3.17)

we see t h a t ( c f . (3.19))

,.,

(3.39)

+v(at)

where P

=

&

I

27T

( c h t + s h t cose)

ip-

0

'

-

1

dB = P o

,(cht)

1v-7

1 ip- '(cht) l ( c h t ) = Po

'p-7

We r e c a l l

io(q,u)do

R e f e r r i n g back t o t h e formula f o r t h e zonal

Remark 3.10

(3.18)

I

k

i s t h e standard Legendre f u n c t i o n . ,.. a l s o by Lemma 1.1 t h a t FMat = FMt = ( a ), which SO

+U

t

equals i o ( t , p ) . Using (3.38)

we can d e f i n e Rm(t,-)

integrating i n E'(V)

=

F

-l*m

R (t,v)

E I ( V ) and

E

( c f . C a r r o l l 1141) we have f o r m

2

1 an

integer, (3.40)

shZmt Rm(t,-) = Zmm

(cht-cho)m-lshn

We r e c a l l t h a t i n t h e p r e s e n t s i t u a t i o n D(G/K)

M drl rl

i s generated by

t h e Laplace-Beltrami o p e r a t o r A which a r i s e s from t h e Casimir

117

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

.

o p e r a t o r C i n t h e enveloping a l g e b r a E(g) (here A corresponds t o 2C due t o o u r choice o f Riemiannian s t r u c t u r e determined by 2 2 2 and C = 1/2(X +Y -Z )

1/2B(-,*)

-

see Remark 3.15 below on Casi-

-3

m i r operators).

I n E(g ) we have then a l s o 2C = ($1)'

B

+ X-X,)

$X+X-

+

and i n geodesic p o l a r coordinates ( c f . Proposi-

t i o n 3.6) (3.41)

A = H:

+ $H+H-

+ H-H,)

a2

=

+ coth

T

a/a-c

+ csch2T a2/ a 0 2 We n o t e here t h a t an a l t e r n a t e expression f o r A i s given by A = H+H

.

+ H'3 -

-

H3 (cf.

Theorem 3.11

C a r r o l l - S i l v e r [15;

16; 171 and S i l v e r

Ill).

For v f i x e d (Muf)(v) depends o n l y on t h e geo-

d e s i c r a d i u s T = t o f u = n ( h ) = k a K and can be i d e n t i f i e d

9 t

w i t h t h e geodesic mean value M(v,t,f). Proof:

T h i s can be proved b y t e d i o u s c a l c u l a t i o n based on

t h e coordinates i n t r o d u c e d i n p r e v i o u s examples b u t i t i s simply a consequence o f Lemma 1.1 and Theorem 3.8. A

Indeed, s e t t i n g

AA

A

h

g = k ak w i t h h = k a k we have v = n ( g ) = k aK and u = n ( h ) = 1 2 1 t 2 1 klatK w i t h (3.42)

a ( M u f ) ( v ) = (M t f ) ( v ) = ( M t f ) ( v )

A

A

A

A

We n o t e t h a t t h e d i s t a n c e from v = klaK t o k aka K i s t h e same as 1 t

118

2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS A

t h e d i s t a n c e from w = n ( e ) = K t o katK = (exp E ) w = pw, which as

n d i c a t e d i n Theorem 3.8, ~n

V '

Hence as k

d e s c r i b e a geodesic " c i r c l e " around

v a r es t h e p o i n t s klak.rr(at) n

w i l l s i m p l y be T = t.

h

k aw o f r a d i u s T = t; (3.42) i s , upon s u i t a b l e normal z a t i o n

1 a l r e a d y provided, t h e d e f i n i t i o n o f M(v,t,f). Remark 3.12

QED

The v a l i d i t y o f t h e Darboux equation ( .7) f o r and D = A i n spaces o f constant

geodesic mean values M(v,t,f)

n e g a t i v e curvature, harmonic spaces, etc. has been discussed, u s i n g p u r e l y geometrical arguments by G h t h e r [l],O l e v s k i j [l], and Willmore [l]( c f . a l s o Fusaro [l],G'u'nther [2;

31, Weinstein

and Remark 5.6).

[12],

We can now define, f o r m c o n v o l u t i o n ) o f Rm(t,.) means o f (3.40).

E

2 1 an i n t e g e r , a composition ( o r

E'(V) w i t h a f u n c t i o n f ( - ) on V by

Thus f o r v = n ( g ) we w r i t e

(3.43) where <

,>

denotes a d i s t r i b u t i o n p a i r i n g as i n Lemma

( c f . He gason [2] f o r a s i m i l a r n o t a t i o n ) . (Rm(t,=

1.1

Then, s e t t i n g

# f ( * ) ) ( v ) = um(t,v), we have from (3.40) a Sonine

formula (3.44)

sh2mt urn(t,v)

Since F

= R(R+l)FT w i h

Ehrenpre s-Mautner [1])

R

=

-1/2 + i u when T

I

E

t h e r e f o l l o w s from (3.8),

119

E (V) (cf. (3.9),

(3.43)

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

and t h e d e f i n i t i o n o f u" = u"(t,v)

F o r m 2 1 an i n t e g e r and f

Theorem 3.13

= um(t,v)

t i o n um(t,v,f)

= um(t,v,f)

=

(Rm(t,-)

E

2 C ( V ) t h e func-

# f(*))(v) satisfies

qq)sh [A-m(m+l)]~"+~

(3.45)

u;

(3.46)

uy + 2m c o th t um = 2m csch t urn-'

(.3.47)

utt + (2m+l) c o th t uy + m(m+l)um = Aum

(3.48)

um(o,v) = f ( v ) ;

=

m

where (3.45),

and (3.48)

=

0

h o l d f o r m 2 0.

The n o t a t i o n here d i r e c t s t h a t (AMt # f ) ( v ) =

Proof: Av(Mtf)(v)

(3.47),

uT(o,v)

= (MtAf)(v)

(see e.g.

(1.7)

and r e c a l l a l s o t h a t A =

6d + d6 i n t h e n o t a t i o n o f de Rham [1] i s f o r m a l l y symmetric) w h i l e (3.48) example.

i s immediate fro m t h e d e f i n i t i o n s and (3.45)

for

One can a l s o prove Theorem 3.13 d i r e c t l y from t h e de-

f i n i t i o n (3.44)

and t h e Darboux e q u a ti o n (1.7) w i t h o u t u s i n g t h e

F o u r i e r t r a n s f o r m a t i o n ( c f . C a r r o l l - S i l v e r [15; (3.46)

f o l l o w s immediately from (3.44)

(3.45)

and (3.47)

16; 173).

by d i f f e r e n t i a t i o n w h i l e

r e s u l t then by i n d u c t i o n u s i n g (1.7)

r a d i a l Laplacian DU = A = m = 0 t oget her w i t h (3.46)

a 2/ a t 2 +

Indeed

coth t

and t h e

a / a t ( c f . (3.41)) f o r

(see C a r r o l l - S i l v e r [16] f o r d e t a i l s ) . QED

D e f i n i t i o n 3.14

The sequence o f s i n g u l a r Cauchy problems

120

2.

(3.47)

-

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

sequence. (3.8)

f o r m 2 0 an i n t e g e r w i l l be c a l l e d a canonical

(3.48)

The canonical r e s o l v a n t sequence corresponding t o

under i n v e r s e F o u r i e r t r a n s f o r m a t i o n i s ( c f . a l s o (2.3))

+ ( 2 1 ~ 1 c) o t h

(3.49)

RTt

(3.50)

Rm(o,*) = 6 ;

t RY + rn(m+l)Rm = ARm m

= 0

Rt(0,*)

where 6 i s a D i r a c measure a t w = .rr(e). t i o n s (3.9)

S i m i l a r l y from t h e equa-

we have

2(m+l) sh [A-m(m+l)]Rm+’

(3.51)

Rt =

(3.52)

R Y + 2m c o t h t R” = 2m csch t Rm-’ Remark 3.15

We r e c a l l a t t h i s p o i n t a few f a c t s about .-.

Casimir o p e r a t o r s f o r a semisimple L i e algebra g ( c f . Warner [l]).

. . ., Xn i s any b a s i s f o r gE l e t gij .-.

I f X1,

= B(Xi,X.)

where J be t h e elements o f

B(=,*) denotes t h e K i l l i n g form and l e t gij the matrix (g’j)

i n v e r s e t o (g..). 1J

The Casimir o p e r a t o r i s then

d e f i n e d as

T h i s o p e r a t o r C i s independent o f t h e b a s i s {Xi} .-.

c e n t e r Z o f t h e enveloping algebra E(gt).

Another f o r m u l a t i o n

. “ -

can be based on a Cartan decomposition g = k

Cartan subalgebra j o f g

.-.

.

such j always e x i s t .

(i.e.,

. - . . - . - . . - .

j = j f) k +

L e t a b a s i s Hi

121

and l i e s i n t h e

+ p and a e s t a b l e

jfl p

= j- + jp);

k ( 1 <, i 5 m) f o r j-. and a k

SINGULAR AND DEGENERATE CAUCHY PROBLEMS = -6ij for b a s i s H.(m+l
t

Then

When G has f i n i t e c e n t e r w i t h K a maximal compact subgroup then the operator i n D ( G / K )

determined by C i s the Laplace-Beltrami

operator f o r the Riemannian s t r u c t u r e induced by t h e K i l l i n g form B(*,-)

(see Helgason [l]and cf.

Example 3.16

(3.41)).

We consider now the case where G = SOO(3,1) =

SH(4) i s the connected component o f the i d e n t i t y i n t h e Lorentz group L = SO(3,l)

and K = SO(3) x SO(1) Z SO(3).

The r e s u l t i n g

Lobazevskij space V = G/K has dimension 3 and rank 1.

I n R4 G

c o n s i s t s o f so c a l l e d proper Lorentz transformations which do n o t

2 2 reverse the time d i r e c t i o n . Thus s e t [x,x] = -r (x,x) = -xo + 2 2 2 + x2 + x3 and t h i n k o f xo as time. Then G corresponds t o 4 x 4 x1 matrices o f determinant one l e a v i n g [x,x] t h e s i g n o f xo.

I n particular i f

2 p o s i t i v e l i g h t cone r (x,x)

i n v a r i a n t and p r e s e r v i n g

n4 denotes

the i n t e r i o r o f the

> 0, xo > 0, then G : R4

+

R4.

The

LobaEevskij space V can be i d e n t i f i e d w i t h t h e p o i n t s o f t h e 2 "pseudosphere" r (x,x) = 1, xo > 0 (which l i e s w i t h i n R4). coordinates

122

If

2.

(3.55)

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

x1 = r sh

e3

s i n €I2 s i n

-

e 1’

x3 = r sh €I3 cos €I

x2 = r sh €I3 s i n €I2 cos €I1; with 0

5

€I1 <

ZIT, 0 5 O 2 <

IT, 0

s c r i b e d i n R4 then coordinates

5

ei(i

x0 = r ch €I3

e3

<

m,

and 0 5 r <

= 1,2,3)

m

are pre-

can be used on V and

t h e i n v a r i a n t measure dv on V i s g i v e n by dv = sh

(i= 1,2,3).

2;

2

e3

s i n O2lIdei

We w i l l now f o l l o w Takahashi [l]i n n o t a t i o n because

o f h i s more “canonical” f o r m u l a t i o n b u t r e f e r a l s o t o V i l e n k i n

[l]f o r many i n t e r e s t i n g geometrical i n s i g h t s . ,.. Thus as a b a s i s o f t h e L i e algebra g o f G we t a k e

Io (3.56)

Y1 =

Y3 =

‘1 3

1

0

O 0

0’ 0 Y

Y2 =

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0)

0

0

0

1’

0

0

0’

0

0

0

0

0

1

0

-1

0

0

3

‘12

=

0

0

0

0

1

0

0

0,

0

0

0,

0

0

0

0’

0. 0

0’

0

0

0

1

0

0

0

0

0

1

0

-1

0,

0

0

0

0

0

-1

0

0,

Y

‘23

Y

3

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Note t h a t i f E

denotes the matlrix w i t h 1 i n the (p,q) Pq (pth row and qth column) w i t h zeros elsewhere then Y P E ( p = 1,2,3) and X E (p
-

position

+

EoP Then

Y

R Y 3 we

~

Y

.

.

,

+ p and K

have a Cartan decomposition g = k

Y

= exp k.

Take now

a = R Y 1 as a maximal a b e l i a n subspace o f p and s e t X

P

+ X

= Yp

1P

(p = 2,3) so t h a t

(3.57)

x2

=

’0

0

1

0’

0

0

1

0

1 - 1

0

0

0

0,

,o

0

;

x3 =

0

0

0

1

0

0

0

0

1

-1

0

0,

-

Then w r i t i n g n = RX, + R X an Iwasawa decomposition o f g i s 3 ,.L given b y g = k + a t n and w r i t i n g A = exp a w i t h N = exp n one

- -

has G =

KAN.

We w i l l use the n o t a t i o n at = exp t Y 1 so t h a t

-

sh t

0

0

sh t

ch t

0

0

0

0

1

0

( 0

0

0

1

‘ch

(3.58)

at =

One can take Y1 = Ho i n the general rank one p i c t u r e w i t h a(Ho) = 1 since [Yl,X2]

= X2 and [Y, ,X3]

= X3.

Thus a, which i s the

1 o n l y p o s i t i v e r o o t , has m u l t i p l i c i t y two and p = -in a = a 2 a p(tY1) = t.

We r e c a l l t h a t M = (kek; AdkH = H f o r H

t h a t Mi = (kEk; Adka C a).

E

with

a1 and

Since exp AdkH = k exp Hk-’ we see

124

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

t h a t AdkH = H f o r H a for H

E

a i m p l i e s k exp Hk-l = exp H w h i l e AdkH

a i m p l i e s k exp Hk”

E

mediate t h a t M = {exp

m =

(3.59)

0

= exp Hi f o r H I

E

a.

E

It i s im-

c o n s i s t s o f matrices

1

0

A

m

;

E

SO(2)

A

whereas M

I

exp I T X ~ ~ ) where , exp ITX

= M U {exp vXl2,

( p = 2,3) 1P Thus W = M /M has the p r o p e r t i e s i n d i c a t e d beI

sends at t o a,t.

f o r e i n t h e general rank one s i t u a t i o n and B = K/M can be i d e n t i -

2

f i e d w i t h the two sphere S c R

3

.

This i d e n t i f i c a t i o n can be

s p e l l e d o u t i n a t l e a s t two ways and we w i l l adopt the one leadi n g t o the s i m p l e s t i n t e g r a l expressions l a t e r .

Thus, f o l l o w i n g

Takahashi [l], we w r i t e u = exp 0X12 and v = exp I$X23 f o r 0 5 0 I$ 0 5 IT and -IT < I$ 5 TI ( i n p a r t i c u l a r v E M). Then any k E K can

0

be w r i t t e n i n t h e form k = m u v f o r m E M so t h a t i f m = v 0 0 JI then kll = cos 0 , k12 = s i n 8 c o s b k13 = s i n 0 s i n I$, kZ1 = -sin 8

COSI),

(kll,k21,k31)

and k31 = s i n 8 s i n +

:k

+

One can pass the map k

R3 t o q u o t i e n t s t o o b t a i n a map K/M

+

+

S

x =

2

v u M and x + ( J I , 0 ) ; here -IT < JI < TI and 0 < 0 < IT $ 0 2 on the domain o f uniqueness. The i n v a r i a n t measure on S corres-

where k

+

l ponding t o t h i s i d e n t i f i c a t i o n i s given by ds = ( 4 1 ~ ) - sined0dJI and the f u n c t i o n s

125

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

with c = [(2n+l)(n-1n)!/(n+m)!]"~, -n 5 m 5 n, and n = 0,1,2, m ,n 2 2 form a complete orthanormal system i n L ( S ) ( c f . Talman

. . .,

[l]and V i l e n k i n [ l ] ) . We can now g i v e e x p l i c i t formulas f o r a canonical s e t o f radial objects i n

H, ( c f . (1.4)).

Thus l e t v = atK and b = kM

w i t h k = v+ue and consider ( c f . (3.19)) IT

A

(3.61)

@;"(atK)

=

-IT

I

(iP-l)H(a;'k)

IT 0

e

'n ,m

( e , $ ) s i n 8 dOd$

It f o l l o w s from general formulas i n Takahashi [l]t h a t

-

exp H(a-'v u ) = ch t t $ 0

I

1

2

IT

(ch t

-

sh t cos e)ip-lPn(cos

f 0 and P:(cos

reduces t o R (t,w)

=

u

io(t,p)

."

0 ) = Pn(cos 0 ) .

FMt = @ (a,)

= 1 and Po(cos 0 ) = 1 whereas,

calling that @

e ) s i n e de

o

"0

t h a t (3.62)

(3.63)

IT

= 0 for m

eimd$

-IT

0 3 0

,

h~

= 2 '0 n

c

becomes

"m 0', n (atK) = @ 'n (atK) = Z n ( t ) 1-I

(3.62)

since

sh t cos 8 and t h e r e f o r e (3.61)

1-I

when n = 0 since

s e t t i n g k = v, u v

+ @ @

and r e -

i s an even f u n c t i o n o f t, t h e expression f o r

can be w r i t t e n i n t h e form ( c f . Takahashi [l])

"0

We n o t e

R (t,p) =

I

(iu-l)H(ai'k)

e K

126

dk

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

=

1 JrO( c h t - s h t

cose)iv-l sinede = Z o ( t )

v

-

2ipsh t [(cht + s h t ) ”

=

sinpt/vsht

-

(cht

-

~ht)~’]

To evaluate (3.62) i n general one has recourse t o various formulas for special functions and we are grateful here t o R. G. Langebartel for indicating a general formula for such integrals i n terms of Meijer G functions (cf. Luke [l]).

The details are some-

what complicated so we simply s t a t e the r e s u l t here.

We note that f o r n

=

Thus, since

0 t h i s formula yields (cf. Magnus-

Oberhettinger-Soni [l])

(3.65)

=q

Z;(t)

sh-’t

=

1

1

l(ch t ) i v-7L

P-’

(L) sh-’t[(ch 21 v 1 21U

= (-)

t + sh t ) i v

sh-’t(e i v t

i n accordance w i t h (3.63).

-

-

(ch t + sh t)-iv]

e-ivt)

Further l e t us remark that (cf.

Robin [l])

127

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(3.66)

P

--1

1/2 t F l +( i vv 7 l-ill 2, 3 -sh 2 t )

l(ch t)=&sh 2

ill7

and hence from (3.65)

one has Z E ( t ) =

F(Y),Y, $

-sh 2 t)

as r e q u i r e d by (2.2). "n To o b t a i n r e s o l v a n ts R (t,p) m u l t i p l y t he Z n ( t ) o f (3.64) ll Since one has f o r example

(3.67) (cf.

P

--

:(ch

t) =

ip-7

n+21

'sh(t3)

2

r(n +

$

F (n +r l,+Fi p, n ++l - -i p, - s 3h 2

-

i t i s n a t u r a l t o ta k e ( c f . (3.21) n+l+iu n+l-iu

h

Rn(t,u)

We now use (3.23) Some r o u t i n e in. Theorem 3.17

3

= F ( T F n + p h

-

(3.24)

2

t o obtain recursion relations f o r the

c a l c u l a t i o n y i e l d s then The re s o l v a n ts

in for the

case o f t h r e e dimen-

iu-1)

^R!

(3.22)

t)

s i o n a l Lobazevskij space a re d e fi n e d by (3.68)

(3.69)

2t )

Robin [l]), i n o r d e r t o s a t i s f y t h e r e s o l v a n t i n i t i a l condi-

t i o n s (2.3), (3.68)

by an a p p ro p r i a t e w e i g h t f a c t o r .

-n 1

-n- 1

. . . we now

f o r n = 1, 2,

+ (2n+l) c o t h t

in = (2 n + l ) 128

csch t

and s a t i s f y (v

=

2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

h

Rt:

(3.71)

+(2n+2) c o th t

s:

+ n(n+2)^Rn = v(v+2)^Rn

These equations a re i n agreement w i t h S i l v e r [l]where they are obt ained i n a d i f f e r e n t manner and t h e r e s o l v a n t s a r e exof B ,Y V i l e n k i n [l]. One can now proceed as b e f o r e t o determine r e s o l -

pressed somewhat d i f f e r e n t l y i n terms o f t h e f u n c t i o n s Pa

vants Rn and canonical sequences o f s i n g u l a r Cauchy problems. S i m i l a r l y one can expand t h e m a t r i x th e o r y t o deal w i t h h i g h e r dimensional Lobazevskij spaces ( c f . S i l v e r [l])b u t we w i l l n o t s p e l l t h i s o u t here ( c f . Se c ti o n 4). i n g t h e r e c u r s i o n r e l a t i o n s (3.69)

-

The q u e s t i o n o f deduc-

(3.70)

from group t h e o r e t i c

i n f o r m a t i o n about t h e i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s o f G i n t h e f-l

u

w i l l be d e f e r r e d f o r t h e moment.

Remark 3.18 2.1

-

Going back t o th e general format o f Sections

2.2, Example 3.16 corresponds t o t h e case ma = 2 , mZa = 0,

p = 1, da = -p(p+l),

dpa = 0, and n = p.

Then u s i n g t h e Kumner

r e l a t i o n i n d i c a t e d a t t h e end o f S e c ti o n 2.2 w i t h z = t h L t now,

- (p,o)

2 we have z/z-1 = -sh t and f o r T (3.72

Y

-u,T

(a K) t = c

-uYT

t h P t ch-iu-lt

129

F(+,

ip+l+

iu+2+p

3 2 ,p+Z,th t )

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Hence we can write from (3.68) (3.73)

A

R P ( t , u ) = c-l

-uYT

exactly as i n (3.7).

shePt Y

-u ,'c (atK)

P u t t i n g this i n (2.11) yields (3.71) and

the recurrence relations follow as before from (3.23)

2.4

Expressions f o r general resolvants.

m and m2a = 0 so t h a t d2a = 0, while

T

Theorem 4.1

p =

- (p,o).

m/2, R = i v +

T, da

-

(3.24).

Let f i r s t ma = = -p( p+m-1)

, and

Resolvants f o r the case ma = m and m2a = 0 are

given by A

(4.1)

R P ( t , u ) = c-l

-uYT

= ch-P-R

s h w P t Y -u

,f(atK)

R+p+l +m+l 2 ,p 7 t h t )

t F(+,R+

These s a t i s f y the resolvant i n i t i a l conditions as well as the differential equation and sDlittinq recursion relations below. (4.2)

A

A

(4.4)

A

RtPt + (2p+m) coth t R! + [p(p+m) + p

A

2

+ ($m 2 ]R" p - - 0

A

R! + (2p+m-1) coth t Rp = (2p+m-l)csch t RP-'

130

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

Proof: and (3.73)

Equation (4.1)

f o l l o w s from d e f i n i t i o n s ( c f .

f o r m o t i v a t i o n ) , (2.9)

(3.7)

t h e Kumner r e l a t i o n F(a,b,c,z)=

(l-z)-aF(a,c-b,c,z/z-l),

and an extended v e r s i o n o f (3.67);

r e c u r s i o n formulas (4.3)

-

( c f. Robin [l]f o r d e t a i l s ) . o r (4.3)

-

(4.4)

the

f o l l o w from t h e known r e l a t i o n s

F i n a l l y (4.2)

r e s u l t s from (2.11)

(4.4).

QE D

I n t h e s i t u a t i o n now when mZa = 1 t h e s i t u a t i o n becomes somewhat d i f f e r e n t .

We r e c a l l t h a t

T

-+

(p,q) w i t h (p,q)

E

Z+

x

Z and

2 We take ma = m so t h a t da = -p(p+m) + q and dpa = q E 22., 2 m For t h e r e s o l v a n t s we use (3.7) again ( c f . -4q w i t h p = 2 + 1.

p

2

(3.73)

a l s o ) w i t h (2.9)

t o obtain

= ch-P-2XF(x+y,

x +

y,

y, t h 2 t )

where x = -1( i p + 2 + 1 ) = R and y = p + 2 + 1. An elementary 2 2 2 computation now y i e l d s th e re s o l v a n t e q u a t i o n from (2.11) i n t h e form (4.8)

*R;iq

+ [(Pp+m+l)coth t + tht]c!7q

+ [p(p+m+2) + p 2 + (!)+1)2 131

+ q 2 ~ e c h 2 t ] " R 9 q= 0

SINGULAR AND DEGENERATE CAUCHY PROBEMS

Theorem 4.2 given’by (4.7)

For the case

= 1 with

%

= in resolvants a r e

s a t i s f y i n g the resolvant i n i t i a l conditions

a

(2.3) and (4.8).

There a r e various spl i t t i n g recursion r e l a t i o n s

which we l i s t below.

(4.9)

p

-

^Rp’q

2x1 t h t

sh2t t h t

(4.10)

A

RPtYq = 2(y-l)coth t sech2t

ip-2sq

2(x+-l)(y-x

+ [ 2 ( 1 - 6 ) ~ 0 t ht + 1

-y-1)

Y-2

(4.11)

(4.12)

A

Rp’q

t

= -q t h

t

SPsq -Z(y-l)coth tiPsq

+ Z(y-l)csch t ^Rp-lSq-’ (4.13)

(4.14)

A

Rp’q

t

= qth

t

iPsq -Z(y-l)coth

+ E(y-l)csch t

t

iPsq

ip-l’q+l 132

6p+zq

- qlth

t]

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2 coth t iPsq + -=x+y) 9+ 1

h

RPtSq = q t h t

(4.15)

h

R!’q

(4.16)

= -q t h t

(y-x Proof: mula

d

iPsq + 2 coth t ( X + Y ) q-1

-

The r e c u r s i o n r e l a t i o n s are derived using the f o r -

F(a,b,c,z)

= (ab/c)F+ = (ab/c)F(a+l ,b+l ,c+1 ,z)

and var-

i o u s c o n t i g u i t y r e l a t i o n s f o r hypergeometric f u n c t i o n s ( c f . Magnus-Oberhettinger-Soni

[l].Thus t o o b t a i n (4.9)

-

(4.10) one

uses the e a s i l y derived formulas (4.17)

( l - z ) F + = F + w ,cz c+l F(a+l,b+l,c+2,z)

(4.18)

abz(1-z)F+ = c(c-l)F(a-1 ,b-1 , c - ~ , z )

+ For (4.11)

-

c

[

J

-

( z

-

(c-bz-l)]F

(4.12) one uses t h e formulas

(4.19)

b(1-z)F+ = CF + (b-c)F(a+l ,b,c+l ,z)

(4.20)

abz(1-z)F+ = c ( c - l ) F ( a - 1 ,b,c-1,z)-c(c-bz-l)F

whereas f o r (4.13) (4.21)

-

(4.14)

we u t i l i z e

abz(1-z)F+ = c(c-l)F(a,b-1,c-1

133

,z)

-

c(c-aa-1)F

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(4.22)

-

a(1-z)F+ = CF

and f i n a l l y f o r (4.15)

-

(c-a)F(a,b+l ,c+1 ,z) (4.16) one has

(4.23)

bZ(l-z)F+ = c ( z (b-c))F + ~ - ~ +- ~

(4.24)

abz(1-z)F+

=

F(a+l ,b-1 ,c,z)

-bco F(aa-b-1

Let us a l s o remark b r i e f l y about the s p l i t t i n g phenomenon.

Thus

f o r example (4.19) y i e l d s (4.25)

F

I

=

c(1 a -z)-[cF

+ (b-c)F(a+

whereas (4.20), a f t e r an index change (4.26)

I

[cF

F (a+l,b,c+l ,z) = z ( l

gives

-

(c-bz)F(a+l ,b,c+l ,z)]

Now d i f f e r e n t i a t e (4.25), insert (4.26), and then use (4.25)

Multiplying by z(1-z) one

again t o eliminate F ( a + l , b , c + l , z ) .

obtains then the hypergeometric equation z(1-z)F [c

-

(a+b+l)z]F

I

-

abF

=

I 1

+

I t i s easy t o show t h a t i f t h e

0.

hypergeometric equation s p l i t s i n t h i s manner then so does (4.8) under the composition of (4.11)

-

(4.12), f o r example.

QED

For completeness we will write down some formulas f o r t h e = 3 o r 7 b u t will omit t h e recursion r e l a t i o n s s i n c e 2a the pattern i s exactly as above. T h u s f o r m2a = 3 we s e t ma = m

cases m

and r e c a l l t h a t (p,q)

E

Z+

x

Z+ w i t h p

134

?

q

E

22.,

One has

2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

p =

-m2 + 3 and R

q(q+2).

(2.9)

=

iu +

We use (3.7)

p w i t h d2a = -4q(q+2) and da = -p(p+m+2) + h

and (3.73)

again t o d e f i n e R p y q and from

this yields

1 where x = R/2 = $ i u

+%+ 3)

and y = p +

e q uat ion a r i s i n g from (2.11) *

R!tq

(4.28)

m -i+

2.

The d i f f e r e n t i a l

i s then

+ [(2p+m+3)coth t + 3 t h t]i!yq = [p(p+m+6)

+ p 2 + =(!)+3)

For mZa = 7 one has p = !+

2 + q(q+2)sech2t]RPyq = 0 A

7 and s e t t i n g ma = m i t f o l l o w s

2

t h a t da = -p(p+m+6) + q ( q t 6 ) w i t h dZa = -4q(q+6). and (2.9)

from (3.7)

again R+

h

(4.29)

+

RPYq(t,u) = ch-P-a t F ( 9 ,

R;tq

,p+%+4,

2 th t )

th e d i f f e r e n t i a l e q u a ti o n f o r R P y q i s

h

+ [(Ep+m+7)coth t + 7th 2

+ [p(p+m+l4) + p +

Theorem 4.3 by (4.27)

Rtp-q-6

*

and from (2.11) (4.30)

I n t h i s case

t]6!’q

m 2 (9+ 7) + q ( q + 6 ) s e ~ h ~ t ]= ~0~ ’ ~

For m2a = 3 (resp. 7) t h e r e s o l v a n t s a r e given

(resp. (4.29))

and s a t i s f y (4.28)

135

(resp. 4.30)).

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

These r e s u l t s completely s o l v e t h e rank 1 case and t h e conn e c t i o n o f t h e F o u r i e r t h e o r y t o t h e associated s i n g u l a r p a r t i a l d i f f e r e n t i a l equations has been i n d i c a t e d a l r e a d y ( c f . Theorem 3.13 and Chapter 1). 2.5

The Euclidean case p l u s g e n e r a l i z a t i o n s .

here C a r r o l l - S i l v e r [15;

We f o l l o w

16; 171 and w i l l i n d i c a t e r e s u l t s o n l y

f o r one simple Euclidean case; a complete e x p o s i t i o n appears i n S i l v e r [l]. Thus l e t G = R2xySO(2) and K = SO(2) where (;,a)

(;,8)

=

(&y(ct)?,ct+B)

'cos ct (5.2)

g =

s i n ct

, o

i s a semidirect product w i t h

- s i n ct cos

ct

0

and m u l t i p l i c a t i o n i s f a i t h f u l .

."

xl' x2

1, As generators o f t h e L i e alge-

b r a g o f G we t a k e

136

2.

1 l;i 1; ; b;; j

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

0

(5.3)

al =

0

0

0

0

0

a2

0

a3 =

0

0

-1

0'

1

0

0

0

0

0,

with multiplication table

."

Thus g i s s o l v a b l e ( b u t n o t n i l p o t e n t , n o r semisimple).

Now s e t

V = G/K and s i n c e (;,a)(;,@) = (z,a+B) t h e r e i s an obvious g l o b a l a n a l y t i c diffeomorphism V f(g-%)

g-';

R

2

.

One d e f i n e s as b e f o r e L ( g ) f ( x ) =

where g = (;,a) and g - l = (y(-a)(-;),-a).

= f(y(-a)(;-;))

One t h i n k s here o f

-+

;;as

= g - l r ( h ) = IT(g-'h)

r ( h ) = IT(;,@) = = r(y(-a)(-;)

so t h a t

IT((;,O)(O,@))

+ y(-a);,@-a)

= y(-a)(;-G). . "

If f i s d i f f e r e n t i a b l e then L induces a r e p r e s e n t a t i o n o f g as i n we have f i r s t , f o r $ = ( x1 ) = x2 +. ( f o r s i m p l i c i t y o f n o t a t i o n ) , exp(-tal)x =

W r i t i n g Ai

(3.29). -+

r ( x , @ ) = (xl,x2) (xl-t,x2),

= L(ai)

e x p ( - t a 2 ) z = (xl,x2-t),

Consequently we o b t a i n A1 = -a/axl, x2 a/axl-xl

a/ax2.

(3.12)

and (3.26))

(5.5)

[H,H+]

A2 = -a/ax2,

W r i t i n g H+ = A1 + i A 2 , H

2iA3 i t f o l l o w s from (5.4)

= 2H+;

+.

and exp (-ta3);

= y(-t)x.

and A3 =

- = A1 -

iA2, and H =

t h a t ( n o t e t h e c o n t r a s t here w i t h

[H,H-]

= -2H ;

[H+,H-]

= 0

The Riemannian s t r u c t u r e on V i s described i n (geodesic) p o l a r c o o r d i n a t e s by ds2 = d t 2 + t 2d8 2 ( c f . Helgason [ l ] ) geometry i s " t r i v i a l . "

and t h e

There f o l l o w s immediately ( c f . V i l e n k i n

137

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Lemma 5.1 (5.6)

I n p o l a r coordinates ( t , e ) it3

H+ = -e

[a/at +

H- = -e (5.7)

A = H H

+-

[a/at

-

i

H = -2i

a/ae;

a/ael

1 a/at + a 2/ a t 2 + t

=

As b e f o r e (cf.

-i0

Lt a/ael;

on V

1 t

a 2/ae 2

P r o p o s i t i o n 3.7) we work w i t h (dense) d i f f e r -

e n t i a b l e b a s i s " v e c t o r s " f i i n H i l b e r t spaces tl a u n i t a r y i r r e d u c i b l e representation L

u

u

where G provides

and these are c h a r a c t e r -

i z e d by t h e c o n d i t i o n s ( c f . M i l l e r [l;23, V i l e n k i n [I]) =

iuf;+l;

H-f;

= ipfi-l;

= mfi

H3f;

where H3 = 1/2 H and 1-1 i s r e a l (see C a r r o l l - S i l v e r [15;

16; 171

Here we can w r i t e L =

and S i l v e r [ l ] f o r f u r t h e r d e t a i l s ) .

(1.5) ) , b u t emphasize t h a t t h e semi simp1 e t h e o r y does n o t apply. Theroem 5.2

There r e s u l t s (cf.

Vilenkin [l])

Canonical b a s i s v e c t o r s i n H

u

i n t h e form (5.9)

can be w r i t t e n

f i = (-i)m exp (ime)Jm(pt)

Proof:

Take f i = eimOwL(t)

requirement i n (5.8).

which w i l l assure t h e t h i r d

Then t h e w i must s a t i s f y

138

2.

(cf. (5.7

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

-

The solutions J,(ut)

(5.8)).

are chosen for f i n i t e -

ness cond tions and the factor ( - i ) mmakes valid the recursion relations indicated i n (5.8) (cf. Vilenkin [l]); the resolvants and s a t i s f y age given by "R(1-I,t)= (i)m2mr(m+l)(pt)-mwi(t) 2 (1.3.11) - (1.3.12) with A(y) replaced by u (resp. y by 1-11 similarly (1.3.4) Remark 5.3

-

(1.3.5) hold f o r

^Rm

with

?=

1.

QED

I t i s easy t o show that the mean value as de-

fined by (1.6) coincides i n t h i s case with the mean value u X ( t ) (cf. formula (1.2.1)

-

and see Carroll-Silver [16] for d e t a i l s ) .

Thus the results of Chapter 1 may be carried over t o t h i s group theoretic situation , which i s equivalent. W e conclude t h i s chapter with some "generalizations" of the

growth and convexity theorems (1.4.12) and (1.4.16) in the special case V = SL(2,R) /SO(2) Silver [15; 16; 173).

(cf. Theorem 3.13 and Carroll-

F i r s t i t i s clear that i f f _> 0 then,

referring t o (3.43), ( M t # f ) ( v ) teger (here we assume f

E

C'(V)

=

(Mtf)(v) 2 0 for m 2 0 an in-

f o r convenience l a t e r ) .

Hence

u m ( t , v , f ) = u m ( t , v ) 2 0 when f 2 0 by (3.44) f o r m > 0 an integer. Now write Am = A a M t, Theorem 5.4

- m(m+l) so Let A,f

t h a t by

3.45) we have

recall Mt

=

L 0, m 2 0 an integer; then u m ( t , v ) i s

monotone nondecreasing i n t f o r t 2 0. 139

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

AM f = M ( A f ) by (1.7)

Proof:

rl

rl

and Lemma 1.1.

QED

Now l e t $ be any f u n c t i o n such t h a t d$/dt = csch2m+1t so t h a t d/d$ = sh2m+1t d / d t ( c f . Weinstein [12]).

Then (3.47) can

be w r i t t e n

a 2/a$ 2 um ( t , v , f )

(5.11)

= ~

h um(t,v,A,f) ~

~

~

t

Consequently t h e r e f o l lows

L

Iff,A,,

Theorem 5.5

0 then um(t,v,f)

i s a convex f u n c t i o n

o f $. Working i n a harmonic space Hm(cf. Ruse-Walker-

Remark 5.6 Willmore [ l ] ) ,

Fusaro [l]proves (M = M(v,t,f)

3.11) (5.12)

Mtt

+

(w

+ log' g(t)'l2)Mt

where g = d e t (9. .), gij 1J

as i n Theorem

= AM

denoting t h e m e t r i c tensor, and g depends

on the geodesic distance t alone (A denotes t h e Laplace-Beltrami

I f A f 2 0, M w i l l be nondecreasing i n t and a convex

operator).

f u n c t i o n of $ where $ ' ( t ) = t'-m/g(t)'/2.

Weinstein 1123 works

2 i n spaces o f constant negative c u r v a t u r e -a (which a r e harmonic) and proves s i m i l a r theorems f o r (5.13)

Mtt

+

a(m-1) c o t h (at)Mt = A M

We r e f e r t o Helgason [4] f o r such "Darboux" equations i n a group context; t h e extension t o "canonical sequences" i s due t o C a r r o l l

140

2.

CANONICAL SEQUENCES OF SINGUALR CAUCHY PROBLEMS

[21; 221, Carroll-Silver [15; 16; 171, and Silver

141

113.