Differential operators and adjoint systems derived from Green function and Hilbert space

Krall, A. M.

English

NASA Technical Reports Server

Allan M. YLralll” 
1.. in t roduct , ior , .  
p r e g r e s s  i n  Lhe a r e a  ot‘ lFnear  matr ix  d i f f e r e n t i a l  systems ucder  v a r  i i ) ~  .> 
u ~ ~ A l  jar; :or. auxF1 hry) condi t ions .  
Since t h e  t u r n  o €  t h e  centLry there has hcen s i y i ’ i c & r A  
Founit zky [ 3 1 Birkhoff and Langer [ 11, [ 21 considered syst e ~ s  w!ier~ 
:“e d i f f e r e n t i a l  cjperatzr i s  of the  form I,Y = Y‘ + ?Y ( P  i s  a continuous 
ri x n 
4.Yia) + BY(?;) = 0. 
mat r ix j  and t h e  boundary condi t ions  a r e  end poirit condi t ions  
Wilder [lo], Langer [7] and Cole [4] extended the r e s u l t s  t o  s j i ~ t e ~ s  
rn 
qaving L)omdary condi t ions  a t  i n t e r i o r  p o i n t s  1 
j =o 
A . k ’ ( a . )  = 6, W ~ ~ L ’ E  
J J  
a = a <: a ... < a  = 5. 
1 m 0 
Iv ’hybun  121 and mall  161 considered i c t e g r a l  boundary L.or~L?ic ior;e 
K(x)Y(x)dx = 0, P kY(a) + UY(b) + 
a 
I 
1. I!: h l l i s t e r  Building, The Pennsylvania State Uaive r s i ty ,  Univers i ty  ~ 3 7  IC* 
z o  - [his  research  was supported i n  p a r t  by NASA Grant T7G-E: 39-Oljg-C$+l. 
https://ntrs.nasa.gov/search.jsp?R=19670001546 2020-03-16T17:44:56+00:00Z
3 
111 2 A . Y ( a . )  + a' K(x)Y(x)dx = 0. 
J =O J J  
o 
I n  add i t ion ,  S t a l l a r d  [ 91 has considered i n t e r f a c e  ci.ndiiLic;;s 
Y ( a  .+) - A .Y(a , - )  = 0. Krall [6] has genera l ized  t h e  d i f f e r e r i t i a l  opera tor  
J J J  
by inc lud ing  a boundary form wi th  t h e  d i f f e r e n t i a l  ope ra to r  L o  
A l l  of t h e s e  people have def ined  a n  "ad jo in t "  syster;, of  one f ' . m i  or 
another ,  which was u s u a l l y  motivated by t h e  p r o p e r t i e s  o f  t h e  Green 's  
rriatrjr or' +-Le or ig ina i .  system. (For t h e  except ion,  s ee  Whybwn [12].) 
The o r i g i n a l  system and the  "adjo in t"  a r e  mutual ly  cornpatably cr incar,- 
p a t a b l e .  If Gs(x , t )  i s  t h e  Green 's  func t ion  of t h e  o r i g i n a l  system and 
G ( x , k )  t h a t  of t h e  "adjoint" ,  then  Gs(x , t )  = - r ( t , x ) .  
a a 
A g e n e r a l i z a t i o n  of t h e s e  boundary cond i t ions  would be one of  t h e  forr!: 
b 
r m m 1 K(x) Y(x)dx +I A . Y ( a . + ) +  BiY(ai-)  = 0, which inc ludes  a l l  t h e  
i=o  1 1  a i = o  
o t h e r  condi t ions  as  s p e c i a l  cases .  We s h a l l  see t h a t  i f  t h e r e  are rri + 1 
p o i n t s  i n  ques t ion  ( m  i n t e r v a l s ) ,  t hen  m such condi t ions  a r e  necessary  
t o  achieve  t h e  systems p rev ious ly  mentioned. (This  has  alreac?y been done 
i q l i e i t l y ,  s ince ,  when i n t e r i o r  po in t  condi t ions  were employed, t h e  
ma t r i ces  involved were assumed t o  be continuous a t  t h o s e  i n t e r i o r  p i n t s -  
That is ,  Y(a. +) - Y(aj - )  = Q) 
J 
There are a l s o  systems f o r  which a Green 's  mat r ix  i s  impossible  t o  
de f ine .  These a l s o  have "adjo in t"  systems. Resul t s  of t h i s  kind coxerri i , , t l ,  
ntli o rde r  o rd ina ry  systems wi th  end p o i n t  cond i t ions  have been known f o r  
a long t i m e  [8]. A H i l b e r t  space s e t t i n g  i s  employed i n  de r iv ing  them. 
We b e l i e v e  t h a t  such a s e t t i n g  i s  indeed t h e  proper  one f o r  rratrix 
systems under boundary condi t ions  p rev ious ly  d iscussed .  We s h a i l  show t k t  
4 
-3- 
t h e  a d j o i n t  system i n  a H i l b e r t  space i s  p r e c i s e l y  t h e  one def ined  pre- -
v i o u s l y  by us ing  Green 's  mat r ices .  
systems when EO Green ' s ma t r ix  e x i s t s .  
I n  a d d i t i o n  w e  s h a l l  d e r i v e  adjoint, 
Since our  boundary condi t ions  
c o n t a i n  a l l  t h e  o t h e r s  as s p e c i a l  cases,  w e  can consider  a l l  such systems 
a t  oncz. 
The au thor  would l i k e  t o  express h i s  apprec i a t ion  t o  Professor. R. 2.  
Moyer f o r  h i s  many h e l p f u l  c r i t i c i s m s  and suggest ions,  which were made i n  
conversa t ion  wh i l e  t h i s  r e sea rch  was be ing  conducted. 
2. The D i f f e r e n t i a l  Operator  L. Let us cons ider  an interval [ao , am] 
which i s  subdivided i n t o  m s u b i n t e r v a l s  by a e . .  a < a <... < a l a  m - 1  (ao 1. n;' 
Def in i t i on .  L e t  H denote  t h e  Hi lbe r t  space of n x n ma t r i ces  
Y = (y .  .) wi th  inner  product 1 J  
< Y , Z >  = 1 (9 Z*Y dx) , 
0 
i j  
where * denotes  complex conjugate  t ranspose .  
Le t  P be an n x n matrix which i s  continuous i n  X. Let  A i j  and 
E be n x n mat r ices  of cons t an t s .  L e t  K ~ ( x )  be n x n mat.rices ir! ij 
#. We w i l l  cons ider  boundary condi t ions  of t h e  form 
Mi(Y) = (I KiYdx + L u i j (y )  , i = 1, ... h ,  
a J = O  
0 
( Y )  = A .  .Y(a 
i n d i c a t e s  t h e  l i m i t  of Y(x) as x approaches a from above or 'below. 
+) + B . . Y ( a j  - ) , and A = 0 , Bio = 0, 1 J  j 1 J  i m  where Ui j  
Y ( a j  +) - 
-4- 
D e f i n i t i o n .  The boundary condi t ions M . ( Y )  are sa id  t o  be accepta'zl-e if  
1 
whenever t h e r e  e x i s t  cons tan t  matr ices  CI such t h a t  
k 
C ; A i j  = 0 7 CiBij = 0 , j = 0, . o s n i ,  
i=l i=l 
k 
v 
t hen  L C iKi 
i=l 
D e f i n i t i o n  ., L e t  - -
1, Y i s  i n  
I 0 a l s o .  
Bo d.enote t h e  se t  of a l l  n x n mat r ices  Y satisfyii ig 
#. 
2.  Y i s  a b s o l u t e l y  continuous on each of t h e  open intervals 
(a j ,  aj+l) , j = O , a e >  m - 1. 
3. Y' + PY i s  i n  #. 
Def in i t i on .  - L e t  denote  t h e  se t  o f  a l l  n x n ma t r i ces  Y s a t i s fy i . ng  
2. M . ( Y )  = 0 ,  i = 1, ... 92-  
1 
- D e f i n i t i o n .  We de f ine  L by l e t t i n g  LY = Y '  + PY f o r  a l l  Y 0 ir,
Theorem 2.1, 
dense i n  W. 
If t h e  boundary condi t ions  a re  acceptab le ,  t hen  8 -_ i s  
Proof .  L e t  
0 
51 
If 
i s  or thogonal  t o  t h e  space K spanned by $*, K**. 
Clearly a0 i s  dense i n  3l - K. Therefore  8 i s  dense i n  - h. 
8 were not  dense i n  3, t hen  t h e r e  would e x i s t  an element K* i r l  H 
-> - 
A a /m 
such t h a t  K = 1 CiKi and 1 KY dx = 0 f o r  all Y i n  . But 
i=l a 
0 
A m 
t h e n  ci 1 U i j ( Y )  = 0. Since t h e  boundary c o n d i t i m s  a r e  
i=l j= l  
acceptab le ,  K* E 0, and w e  have arrived at a con t r ad ic t ion .  
Throughout t h e  remainder o f  t h i s  a r t i c l e  we w i l l  assume t h a t  t h e  
boundary cond i t ions  M. ( Y )  are acceptable .  
1 
3. The Adjoint  Operator L*. Various examples [51 
i n t e g r a l  terms i n  t h e  boundary condi t ions  a f f e c t  t h e  
o p e r a t o r -  We s h a l l  see p r e c i s e l y  why t h i s  i s  so as 
61 have shown t ,ha t  
form of  t h e  a d j o h t .  
ve d e r i v e  t h e  a d j o i n t  Lx-e 
Lemma 3.1. If Z i s  i n  t h e  domain of  L* , t hen  Z i s  abso lu te ly  
continuous i n  each of t h e  i n t e r v a l s  (a . ,  a ), j = 0, a .  .m-1. There exist 
l i n e a r  f u n c t i o n a l  matrices % ( Z ) ,  ... q A ( Z )  such t h a t  
j +1 
Proof .  L e t  Y be i n  H (Theorem 2.1) .  
0 
Then 
a a 
(L*Z) Y dx = 
a a 
im Z*(Y' + PY)dx . * 
0 
Im 
0 
Theref o r e  
. 
- 6- 
Since Y vanishes  a t  a ... a i n t e g r a t i n g  by p a r t s ,  
0' m' 
a 
i 
a 
0 
Z"Y' dx = - 
a m 
a 
0 
[f ( L * Z - P Z ) d t  1" Y'dx 
1 
Heme 
a 
[ 1 + f ( L * Z - P . z ) d t  Y'dx = 0. 
a 
JY'dx = 0 .  s 
0 I* 
We m Q s t  t h e r e f o r e  f i n d  all t h o s e  func t ions  J such t h a t  
a 
0 a 
dx = 0. L e t  Ji be d i f f e r e n t i a b l e  except a t  a O 9 .  . a  m' Suppo s e \ 
a 
0 
+ J ill W .  Upon i n t e g r a t i o n  by p a r t s ,  Ji acd suppose 
a 
J i Y '  dx -+ 0. Thus t h e r e  e x i s t s  a W such s" 
J! 
f J i ' Y  dx = - 
a a 
0 0 
WY dx = 0. However, t h e  d i € f e r e n t i a t i o n  I t h a t  l i m  Ji = W and 
0 
I ope ra to r  i s  c losed .  So i f  Ji + J and Ji -+ W, t hen  J' e x i s t s  and 
X 
.J I = W. J = s W d t .  
Conversely, s ince  J K.Y dx = 0, it i s  easy t o  show t h a t  
1 
-7- 
X X 
Thus Z + (L*Z - PxZ)dt i s  a l i n e a r  combination of  K *(it, . . . 1 
That is, t h e r e  e x i s t  l i n e a r  func t iona l  mat r ices  r$, (Z), . . e $& Z) such t h a t  
i=l 
Z i s  e a s i l y  seen t o  be abso lu te ly  continuous (except  p o s s i b l y  at. 
a -.  .,am), and 
0' 
We have L* r ep resen ted  i n  terms of +3 parameters  $,( Z )  ... $&Z)* - 
We s t i l l  need t o  f i n d  boundary condi t ions for Z, as w e l l  as see  when 
t h e s e  parameters  can be el iminated.  
-- Lemma 3.2. If Z i s  i n  t h e  domain of L*, t hen  Z sa t isf ies  t h e  fo l lowing  
eaua t  ions .  
i=l 
j = 1, ... m. 
Proof. We e a s i l y  compute 
4 
-a- 
a 
I [(L*Z)*Y - Z*(LY) dx m j - j=l a j-1 
m 
j =1 
m a m 
qZ)* u. .w j =J = 1 -Z*Y 1 + i= 
j =o a j= l  j-1 
Since Y(a +) and Y(a - )  may be a r b i t r a r y ,  
3 j 
.92 
i=l 
- 9- 
k 
= 0, 
i=l 
j = 1, ... m. 
The converse of t h e s e  c a l c u l a t i o n s  i s  t r i v i a l .  Hence w e  a r e  led t o  
t h e  fol lowing.  
Theorem 3.10 The domain of L*, a*, c o n s i s t s  of a l l  t hose  n x n 
ma t r i ces  Z which sat isfv 
1. z i s  in%,. 
2. There e x i s t  paramet r ic  l i n e a r  mat r ix  f u n c t i o n a l s  
% ( Z )  . * .  $$Z) such t h a t  
i=l 
k 
j = 1, ... m. 
If Z i s  i n  &*. t hen  
62 
L*Z = - Z '  + P Z  - 1 Ki* q i ( Z ) .  
i=l 
There are 2m equat ions  f o r  t h e  92 l i n e a r  f u n c t i o n a l s  @-p 0 e . , " 3 R .  
It i s  sometimes p o s s i b l e  t o  achieve two express ions  f o r  each. Subs t i t u -  
t i o n  i n t o  t h e  express ion  f o r  L* , e l imina te s  them t h e r e .  Equating t h e  
d i f f e r e n t  express ions  f o r  each gives  boundary condi t ions .  
b 
-10- 
4. Different ia l -Boundary Operators.  
cond i t ions  determining a i s  d i f f e r e n t  from zero, 
a d J o i n t .  I n  o rde r  t o  have s e l f a d j o i n t n e s s  as a p o s s i b i l i t y ,  t h e  form of 
t h e  ope ra to r  L must be extended so t h a t  it resembles t h e  form of  t h e  
a d j o i n t s  w e  have j u s t  found. 
extend those  r e s u l t s  t o  t h e  present  s i t u a t i o n .  
If any of t h e  K ' s  i n  t h e  boundary 
L cannot be self- 
A beginning s t e p  i s  bound i n  [6] We 
Def in i t i on .  L e t  C i j  and Dij, i = 1,. .. &. L e t  J ~ ~ ( Y )  = 1 
i =O 
be i n  8, i = 1, ...L 
J Hi 
[ C i . Y ( a .  +) + 
We def ine  L, by l e t t i n g  $Y = Y '  + PY + 
i j Y ( a ,  - ) I  , i = 1 ,... &. L e t  
-e, 
Hiqi(Y) f o r  a l l  Y i n # .  
IJ i=l 
Lemma 4.1. If Z i s  i n  t h e  domain of  L b*, t h e n  Z i s  a b s o l u t e l y  
cont inuous i n  each of t h e  i n t e r v a l s  
exis t  l i n e a r  f u n c t i o n a l  mat r ices  
( a . ,  a.+l) , j = 0, ..., m-1. There 
$, ( Z ) ,  . . . & (  Z )  such t h a t  
.B.1 
Lb*Z = -Z '  + P Z  - Ki* $&Z).  
i=l 
Th.e proof i s  i d e n t i c a l  t o  t h a t  of Lemma 3a1e 
- Lemma 4$2. If Z i s  i n  t h e  domain of Lb* , t hen  Z s a t i s f i e s  t he  
f ollowing equat ions  e 
Z*Hidx D i j  = 0, 
.B.1 
-z*(a - >  + q ( z ) *  B~~ 
j 
i=l i=l a 
0 
-11- 
j = 1, ... m. 
The proof i s  only  s l i g h t l y  more complicated than  that of Lema 3 * z 0  
Again t h e  converse of  t h e s e  s ta tements  i s  trivial., and w e  are l.ed to 
t h e  fo l lowing .  
Theorem 4.1.  The domain of Lh* ,&*, c o n s i s t s  of a l l  those n x n 
ma t r i ces  Z which s a t i s f y  
1. z i s  i n  % . 
0 
2. There e x i s t  parametric l i n e a r  mat r ix  func- t iona ls  
%(Z), . . . @+&Z) such t h a t  
./ 
i=l i=l a 
0 
j = 1, ... m. 
If Z i s  m ab*, then 
~ ~ * z  = -z’ + P Z  - 1 
i=l 
K * @.(Z). 
1 i 
-12- 
5.  Some Examples. 
1, Consider t h e  system LY = Y ’  + PY, wi th  endpoint boundary cond i t ion  
A Y ( a o )  + BY(al )  = 0. 
boundary condi t ions  -Z(a,) + gX$(Z) = 0, Z(ao)  + A+$l(Z) = 0. If 
A or  B has  an inverse ,  % ( Z )  can be e l imina ted  t o  g ive  t h e  a d j o i n t  
t h e  form u s u a l l y  found. 
The a d j o i n t  i s  then  def ined  by L*Z = - Z ’  + F z  wi th  
2# Consider t h e  system LY = Y ’  + PY wi th  boundary cond i t ions  
m 
A . Y ( a . )  = 0, Y ( a j  - )  - Y ( a j  +) = 0, j = 1 ,... m-1. The a d j o i n t  
J J  
j =1 
opera to r  i s  t h e  same as before .  L*Z = -Z‘ + P Z -  The boundary condi t ions  
are 
- z b j  -) - @j+l (z)  = o j = 1, ... m-1, 
-Z(a, -) - A * %(z) = 0, 
Z(ao +) + A * %(z) = o 
j 
m 
and 
0 
Z b j  +) + A * @,(Z) + Qj+l ( z )  = 0, j = 1, ... m-1. 
If w e  agree  t h a t  Z(ao -) = 0 and 
Gm(Z) can be el iminated,  g iv ing  
Z(am +) = 0, t hen  @2(Z)  through 
Z(a.  +) - Z(aj -) = -A.* q ( Z ) ,  j = O,.,. m. 
J J 
3. If t h e  M po in t  boundary condi t ion  i n  t h e  prev ious  example i s  rep laced  
a m 
by 1 A . Y ( a . 1  + f KIYdx = 0, t h e  boundary va lues  f o r  Z are  t h e  
J J  
j =1 a 
0 
same, but  L*Z = -Z‘  + PXZ - Kl* @l( Z )  
'. 
4 -13- 
4. Consider t h e  system LY = Y' + PY w i t h  boundary condi t ions  
AoY(ao  +) + BmY(am -) = 0, 
The a d j o i n t  ope ra to r  i s  
A . Y ( a . + )  + B . Y ( a . - )  = 0, j = 1, ... In-1. 
J J  J J  
L*Z = -Z' + P Z .  The a d j o i n t  boundary condi t ions  
are 
and 
z ( a j  +) + A ~ *  r i j j + l ( ~ )  = 0, j = 0, ... m-1. 
Again, if t h e  v a r i o u s  A ' s  o r  B ' s  have inverses ,  t h e  pa rane te r s  Q j ( Z )  
can be e l imina ted .  
5. Consider t h e  system. 
L ~ Y  = Y '  + PY + H~ [CY(a ) + D y (a l ) l ,  
0 
1 a 
I- 
A Y ( a  ) + BY(al )  + 
0 
a 
0 
The a d j o i n t  system i s  
Lb*Z = -z '  + P Z  - $*rij1(Z) , 
Hl*Zdx = 0, J -Z(a,) + B*rij,(Z) - P a 
0 
a 
Z(ao) + A*qj,(Z) - C* f Hl*Zdx = 0. 
a 
0 
Under t h e  cond i t ions  given i n  [61, t h i s  a d j o i n t  system i s  equiva len t  t o  t he  
one de f ined  i n  [61. We no te  t h a t  i n  a l l  of t h e s e  examples, t h e r e  were m 
8 -14- 
'boundary condi t ions  for t h e  m i n t e r v a l s  (m + 1 p o k t s )  under discussion.  
I n  a l l  of these exanrples Green's mat r ices  have been derP,i-ed, w-d, with 
t h e  exeep-hiorj of t h e  las t  one, each has an eigenfunct ion expansion a s s o c i a t e d  
w i t h  it. It s e e m  reasonable  t o  expect t h a t  t h i s  w i l l  always be t r u e .  
Solving a nonhomogeneous equat ion i n  each of m i n t e r v a l s  produces m 
arbitrary cons tan ts  of i n t eg ra t ion .  In  order  t o  spec i fy  them, 11: q u a -  
t i o n s  are needed e More equat ions overdetermine t he s y s t  ern. Fewer 
underdetermine it. 
References 
1. G. D. Birkhoff, "Boundary value and expacsion Froblems of ord inary  
d i f f e r e n t i a l  equations,  I t  Trans. Amer. Math. Soe. Vol. 9 (1908) 
PP. 373-395. 
2. G. D. Birkhoff and R. E. Langer, "The boundary problems and develop- 
ments assoc ia ted  with a system of ordinary d i f f e r e n t i a l  equations of 
t h e  f i r s t  order ,"  Proc. Amer. Acad. Arts Sci .  No. 2, Vol. 58 (1923) 
pp. 51-128. 
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l i n e a r e s  o rd ina i r e s , "  J. Math. Fures Appl. Vol. 5 (1909) pp. 6:1-125. 
4. R. H. Cole,"The expansion problem with boundary condi t ions a t  a 
f i n i t e  set of po in t s , "  Canad. J. Math. Vol. 13 (1961) pp. 462-479. 
5 .  , "General boundary condi t ions for an ordinary l i n e a r  
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pp* 521-550- 
6. A. M. Kral l ,  "Nonhomogeneous d i f f e r e n t i a l  operators ,  Kich. blath. J, 
V O ~ .  12 (1965) pp. 247-255. 
7. R. E. Langer, "The boundary problem of an ordinary l i n e a r  d i f f e r e n t i a l  
system i n  t h e  complex domain," Trans. Amer. Math. SOC. Vol. $6 
(i939) pp. 151-190. 
8. M. A. Naimark, "Lineare Dif fe ren t ia lopera toren ,  Akad. Verlag, 
Ber l in  1960. 
9. F. W. Sta l l a rd ,  "Di f f e ren t i a l  systems with i n t e r f a c e  conditions,  
Oak Ridge Natl .  Lab. Dept. OFNL 1876 (1955) e 
. 
c 
References - 2 
10. C. E. Wilder, "Expansion problems of o rd ina ry  l i n e a r  d i f f e y e n t i a l  
equat ions  w i t h  a u x i l i a r y  condi t ions  a t  more than  two p o i n t s ,  " 
A m e r .  Math. SOC. Vol. 18 (1917) pp. 415-442. 
Trans 
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Differential operators and their adjoints under integral and multiple point boundary conditions

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