Green function for inhomogeneous Strum-Liouville differential equation proble

Simons, J.

English

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https://ntrs.nasa.gov/search.jsp?R=19700032839 2020-03-11T23:04:06+00:00Z
GREEN'S FUNCTIONS FOR STURM-LIOWTLLE PROBLEMS 
by 
Jack  Simons 
T h e o r e t i c a l  Chemistry I n s t i t u t e  
Univers i ty  of Wisconsin 
Madison, Wisconsin 53706 
ABSTRACT 
It i s  shown how t h e  Green's func t ion  f o r  any inhomogeneous 
Stum-L$ouvil le  (S-L) problem can b e  cons t ruc ted  once tyo  independent 
s o l u t i o n s  95 t h e  homoeeneous problem are knaw.  
is i l l u s t r a t e d  by consider ing t w o  s p e c i f i c  S-L problems. When, t h e  
parameter appearing i n  t h e  S-L equat ion  is equal  t o  an eigenvalue of 
t h e  homogeneous S-L problem an added d i f f i c u l t y  q i s e s .  
of cons t ruc t ing  a gene ra l i zed  Green's func t ion  which d e a l s  wi th  t h i s  
d i f f i c u l t  case is  a l s o  t r e a t e d  by example. 
The genera l  technique 
The method 
2 
I. General Sturm-Liouville Theory 
The inhomogeneous Sturm-Liouville (S-L) problem can b e  w r i t t e n  
1 
i n  t h e  fol lowing form 
wi th  t h e  boundary condi t ions  
The known func t ions  p ( x ) ,  q(x), gx), and f(x) are r e s t r i c t e d  only 
i n  t h a t  
(i) p(x)  L 0, dx) L 0 f o r  a 5 x b 
( i i )  g ( x )  i s  d i f f e r e n t i a b l e  f o r  a - -  < x < b 
'The cons tan t  X i s  a parameter of t h e  problem; cz( and 6 are e a n s t a n t s  
which s e r v e  t o  make the boundary cond i t ions  as gene ra l  as p o s s i b l e  
f o r  t h e  S-L problem. 
The Green's func t ion  6(x,y) is in t roduced  by seeking a s o l u t i o n  
of Eq.  (1) having t h e  s p e c i f i c  form given  below: 
r" ( 3 )  x-?a c x , v ) f W d y  + G<X,Y) m d y j  P
x: ')1 
For convenience E q .  (3)  i s  w r i t t e n  i n  shorthand as fol lows 
3 
b 
A 
S u b s t i t u t i n g  t h e  express ion  f o r  f ( y )  given i n  Eq. (1) i n t o  Eq. ( 4 )  r e s u l t s  
i n  
b 
Afte r  i n t e g r a t i o n  by p a r t s  Eq. (5)  becomes 
=-L L- 
97-7o.c 
c 
31-57 o+ 
Based on Eq. (6) w e  now p resen t  four  cond i t ions  which t h e  conrea t  
Green’e func t ion  must s a t i s f y  f o r  t h e  r i g h t  hand s i d e  of Eq. (3) t o  
c o r r e c t l y  y i e l d  t h e  func t ion  Y(x) . 
t h e  Green’s func t ion  t h a t  it s a t i s f i e s  t h e  homogeneous S-L equat iog  
We t ake  as t h e  f i r s t  ppoperty of 
i n  t h e  i n t e r v a l s  a < y < x < b ,  a < x < y b ,  i . e .  everwhere qxcept - - I
a$ t h e  po in t  x = y. With t h i s  t h e  i n t e g r a l  term i n  Eq. (6) vanishes  
i d e n t i c a l l y ,  The second proper ty  i s  t h a t  G(x,y) s a t i s f i e s  t h e  same 
boundary condi t ions  (Eq. ( 2 )  ) as Y a t  y = a and y = b .  This  impl tes  
t h a t  two of t h e  boundary terns appearing i n  Eq. (6) are equal  t o  zero: 
4 
Equation (6) i s  t h e r e f o r e  reduced t o  t h e  fol lowing s imple form: 
Use has been made of t h e  f a c t s  t h a t  p(x) and "(st) are conki-nuous 
func t ions  i n  t h e  range a 5 x 5 b.  
The remaining two r e s t r i c t i o n s  on G(x,y) are e a s i l y  i n f e r r e d  
from Eq. (9) .  C l e a r l y  i f  G(x,y) is a continuous func t ion  of y f o r  
- a l l  y and y i n  t h e  i n t e r v a l  [a ,b] ,  and i f  dG(x,X) is  a cont inuoqs 
func t ion  of y except  f o r  a jump d i s c o n t i n u i t y  of magnitude - 
t h e  p o i n t  y = x: 
dY 
a t  
P (X) 
then t h e  r i g h t  hand s i d e  of Eq. (9)  w i l l  y i e l d  t h e  d e s i r e d  func t ion  
5 
I n  review then ,  w e  see t h a t  t h e  c o r r e c t  Green's funCtion G(x,y) 
which when s u b s t i t u t e d  i n t o  Eq. (3 )  g ives  t h e  s o l u t i o n  Y(x) t o  t h e  
S-L problem, e x h i b i t s  t h e  fol lowing fou r  p rope r t i e s :  
(i) G(x,y) obeys t h e  homogeneous S-L equat ion,  Eq. ( 7 )  
( i i )  G(x,y) obeys t h e  same boundary condi t ion  as Y(y) (i.e. 
( i i i )  G(x,y) i s  continuous f o r  a l l  x and y i n  [a ,b]  
(iv) (x,y) i s  continuous f o r  a l l  x and y " i n  [a ,b]  eycept  
dY 
' f o r  a jump d i s c o n t i n u i t y  of magnitude -L/p(x) a t  t h e  
po in t  y = x ,  (Eq. (lo) ) .  
As w e  w i l l  soon demonstrate by example, t h e s e  f o u r  p r o p e r t i e s  are 
s u f f i c i e n t  t o  determine G(x,y) un le s s  t h e  parameter X i s  an eigenvalue 
of t h e  homogeneous S-L equat ion.  
t r e a t e d  a f t e r  w e  d i s c u s s  two i l l u s t r a t i v e  examples, 
This  more d i f f i c u l t  case w i l l  b e  
11. Two Simple Examples 
To understand how t h e  f o u r  p r o p e r t i e s  ( i )  - ( i v )  can b e  used t o  
determine t h e  Green 's  func t ion  G(x,y), l e t  us consider  a s p e c i f i c  
9-L problem 
w i t h  boundary condi t ions  
I n  t h e  i n t e r v a l  o < y < x b G(x,y) should s a t i s f y  - 
6 
Therefore  G(x,y) can b e  expanded i n  terms of t h e  two independent 
s o l u t i o n s  of t h e  homogeneous d i f f e r e n t i a l  equat ion:  
The func t ions  A(x) and D(x) are t o  b e  determined by us ing  t h e  four 
p r o p e r t i e s  of G(x,y) .  
s o l u t i o n s  of t h e  Bessel equat ion 
J (ky) and Yn(ky) are t h e  twp iQdependent n 
f o r  z = ky. 
w r i t t e n  as 
Likewise f o r  t h e  i n t e r v a l  o - < x < y - < b ,  G(x,y) can bq 
The boundary condi t ion  t h a t  G(x,o) be  f i n i t e  impl ies  that 
D(x) = o because Y (ky) is  n o t  f i n i t e  a t  y = 0 .  Qe second boundary 
condi t fon  G(x,b) = o t a k e s  t h e  form 
n 
Because G(x,y) is  continuous everywhere i n  [ a ,b ]  i t  must be  eontlnqouo 
a t  t h e  p o i n t  y = x f o r  a l l  x i n  [ a ,b ] .  
as 
This  proper ty  i s  e a s i l y  w r i t b e n ,  
7 
The condi t ion  t h a t  - dG have a jump d i s c o n t i n u i t y  a t  y = x l eads  t;o t h e  
equat ion  
dY 
where 
and 
Equations (15) - (2.7) can be  so lved  s imultaneously f 
func t ions  A(x),  B(x), and C(x). The r e s u l t i n g  s o l a t i o n s  are g iven  by: 
- I  
By using va r ious  p r o p e r t i e s  of t h e  J (kx) and Y (kx) t h e s e  express ions  
1. can be  s i m p l i f i e d  cons iderably  e Once t h i s  is done the  Green's func t ion  
G(x,y) g iven  i n  Eqs. (12) and (14) can be w r i t t e n  as fol lows:  
n n 
8 
for 
2 prov5ded t h a t  Jn(kb) # 0. 
eigenvalue of  t h e  homogeneous Bessel equat ion.  
That is ,  provided t h a t  k is  no t  a n  
Thus w e  see t h a t  t h e  f o u r  p r o p e r t i e s  of G(x,y) d i scussed  above 
can b e  used t o  c a l c u l a t e  t h e  Green's func t ion  once w e  know two iplae- 
pendent s o l u t i o n s  of t h e  homogeneous S-L d i f f e r e n t i a l  equat ion.  Nstiee 
t h a t  nothing i s  s a i d  about boundary condi t ions ,  u n t i l  G:(x,y> is  qon- 
s ide red ,  t h a t  is  t h e  two independent s o l u t i o n s  of t h e  homogeneous 
equat ion  w i l l  n o t ,  i n  gene ra l ,  s a t i s f y  a l l  t h e  boundary condi t ions  
of t h e  problem. 
A s  a second example of t h e  g e n e r a l  method le t  us  cons ider  t h e  
2 fol lowing S-L problem 
wi th  boundary cond i t ions  
+(i-?l ==Q,  
I f  w e  d e f i n e  t h e  real  parameter n by 
then  two independent s o l u t i o n s  of t h e  homogeneous equat ion are g iven  
by D n b  and DWn - l ( i x  . These Weber func t ions  can b e  
expressed i n  terms of t h e  conf luent  hypergeometric func t ions  of t h e  
3 
9 
t h i r d  kind (V1 and V2) as3: 
and 
Because G(x,y) must a l s o  be  a s o l u t i o n  of t h e  homogeneous equat ion  
f o r  y < x and f o r  x < y s  w e  can immediately w r i t e  
and 
The condi t ions  t h a t  G(x, 9-00) - be zero imply t h a t  E(x) = 0 and A(x) = 0 
because D -n-1 ( i  00 f-) and Dn( j-"ZYj are un- 
bounded. Cont inui ty  of G(x,y) a t  t h e  po in t  y = x ( f o r  all x i n  
(- 08 00 ) ) r e q u i r e s  
The equat ion  f o r  t h e  jump d i s c o n t i n u i t y  of - dG at  y = x reads  as 
f o l laws  : 
dY 
By so lv ing  Eqs. (28) and (29) w e  arrive a t  t h e  fol lowing express ion  
f o r  B(x) and C(x):  
LO 
Defining A(x) as t h e  denominator appearing i n  Eqs. (30) and (31) t h e  
f ina l  express ion  for G(x,y) can be  w r i t t e n  as 
111. The Case When X is  an Eigenvalue 
The t rea tment  g iven  above is  only  v a l i d  i f  t h e  parameter E is 
- no t  an eigenvalue of t h e  homogeneous d i f f e r e n t i a l  equat ion  wi th  t h e  
same boundary condi t ions  as t h e  o r i g i n a l  S-L problem. I f  E 9 equal  
t o  some eigenvalue,  s ay  E where j is  a non-negative i n t e g e r  then  a 
gene ra l i zed  Green’s func t ion4 ,  which can s t i l l  b e  used i n  Eq. (2) t o  
j’ 
g ive  a s o l u t i o n  Y(x) of t h e  inhomogeneous equat ion ,  can be cons t ruc ted  
- i f  t h e  inhomogeneity f ( x )  i s  or thogonal  t o  t h e  normalized s o l u t i o n  
3 ’  U.(x) of t h e  homogeneous S-L problem belonging t o  t h e  e igenvalue  E 
i .e.  
J 
C34) 
Assuming t h a t  Eq. ( 3 4 )  i s  t r u e  f o r  t h e  s p e c i f i c  case of i n t e r e s t ,  
w e  then  proceed t o  cons t ruc t  G(x,y) by using t h e  same four  p r o p e r t i e s  
d i scussed  above wi th  one modi f ica t ion .  The requirement t h a t  G(x,y) 
s a t i s f y  t h e  homogeneous S-L d i f f e r e n t i a l  equat ion  f o r  x < y and f o r  
y K x i s  rep laced  by t h e  fol lowing proper ty :  
is  t h e  normal iza t ion  cons tan t  f o r  t h e  harmonic o s c i l l a t o r  e igenfunct ion ,  
and 
is t h e  j th Hemite polynomial. The func t ion  
To see t h a t  t h e  q u a n t i t y  
JF CX,Y) f CY) d y  
-0Q 
is  an accep tab le  s o l u t i o n  U(x> of t h e  S-L problem, w e  u s e  t h e  fol lowing 
i d e n t i y  f o r  f ( y ) :  
12 
t o  write 
I n t e g r a t i n g  by p a r t s  and using both Eq. (35) and t h e  boundary cond i t ions  
on G(x,y) and Y(y) , w e  ob ta in  
That Y(x) can, wi thout  l o s s  of g e n e r a l i t y ,  be  choeen t o  b e  orthogonal. 
t o  U.(x) i s  e a s i l y  shown: 
J 
, Replace Y(x) by Y(x) - cUj (x) ; then 
r 2  
Thus Y-cU. i s  an equa l ly  valid s o l u t i o n  of t h e  inhomogeneous 
J 
equat ion.  Hence w e  can choose t h e  cons tan t  c such t h a t  
M 
i .e.  w e  can choose t o  f i n d  a s o l u t i o n  of t h e  inhomogeneous S-L equat ion  
which is  or thogonal  t o  U 
j '  
13 
= Y W ,  ( 4 3 )  
which is  t h e  d e s i r e d  i d e n t i t y .  To a s s u r e  that  the func t ion  W(x) is  
or thogonal  t o  U.(x) i t  is s u f f i c i e n t  t o  c o n s t r a i n  t h e  Green's func t ion  
J 
G(x,y) t o  be or thogonal  t o  U, (x) : 
J 
This  c o n s t r a i n t  w i l l  s h o r t l y  b e  used along wi th  the o the r  €our 
p r o p e r t i e s  of G(x,y) t o  cons t ruc t  t h e  gene ra l i zed  Green's func t ion  
Before we can w r i t e  t h e  € o m  OS G(x,y) w e  must know t h e  gene ra l  
s o l u t i o n  of t h e  d i f f e r e n t i a l  equat ion  g iven  below: 
The s o l u t i o n  of Eq .  ( 4 5 )  r equ i r e s  t h e  knowledge of two independent 
s o l u t i o n s  of t h e  corresponding homogeneous equat ion  
C lea r ly  one s o l u t i o n  is  given by 
which i s  p ropor t iona l  t o  t h e  Weber func t ion  D (y J 
14 
A second s o l u t i o n  T j ( y m 9  of Eq. ( 4 6 )  can be cons t ruc ted  as: fol lows:  
Because both  Dn (y 9 and Dn(-y \?@) are s o l u t i o n s  of t h e  equat ion  
(499 
t h e  p a r t i c u l a r  combination given by 
37 -gs 
must a l s o  be a s o l u t i o n  BE E q .  (49).  
t h e  i n t e g e r  j 
Eq. (499, i .e.  Eq. ( 4 6 ) .  
Therefore  as w e  a l low n t o  approach 
t h e  s o l u t i o n  T .  (y 7”2”i53 obeys t h e  l i m i t i n g  va lue  of 
3 
To s i m p l i f y  t h e  express ion  f o r  T . ( y m  given i n  Eq. ( 4 8 )  l e t  
3 
us expand t h e  “n-dependence” of t h e  two Weber func t ions  D n ( y f T j  and 
Dn(-y i n  a Taylor  series about t h e  po in t  n = j: 
Using t h e  f a c t  t h a t  f o r  n equa l  t o  a non-negative i n t e g e r  D ( - z )  = 
(-UnD (z), we e a s i l y  o b t a i n  from Eqs.  (50) and (51) t h a t  
El 
n 
15 
These d e r r i v a t i v e s  are assumed t o  b e  known func t ions .  
Given t h e  two s o l u t i o n s  D ,  (y and T o  (y- of Eq. (46) w e  
J J 
5 can immediately write down t h e  gene ra l  s o l u t i o n  of Eq. (45) 
where d, and 4, are a r b i t r a r y  cons t an t s  and 
The i n t e g r a l  appearing i n  Eq. (53) is an i n d e f i n i t e  i n t e g r a l .  For 
b r e v i t y  w e  w i l l  r e f e r  t o  t h i s  i n t e g r a l  as F (y) . 
j 
Knowing t h e  g e n e r a l  s o l u t i o n  of Eq. ( 4 5 ) ,  w e  can now write t h e  
form which G(x,y) must have i f  i t  is  to be  8 s o l u t i o n  of Eq. (35):  
The cond i t ions  t h a t  G(x3 C - 08) be zero imply t h a t  
- = - b 'W (57) 
16 
and 
Cont inui ty  of C(x,y) a t  t h e  po in t  y = x f o r  a11 x i n  (- 
r e q u i r e s  
00 , 00 ) 
The equat ion  f o r  t h e  jump d i s c o n t i n u i t y  of dG a t  y = x reads  as follows: 
dY 
Notice t h a t  Eqs. (59) and (60) can - n o t  be  so lved  f o r  both unknown 
funct ions  A(x) and C(x); t h e  d i f f e r e n c e  A(x) - C(x> arises i n  both 
equat ions.  
By using Eqs. (57) - (59) w e  can w r i t e  t h e  previous express ion  
f o r  G(x,y) as: 
1 7  
G ($,$I = C C X )  DJ' (2 Q) - e LJj ($1 5 (89) 
"f- - 2 r  (3) E& /"ob- x < y -  d 
The remaining unknown C(x) i s  t o  be  determined from Eq. (44):  
a0 
-a fGcr,rl% Cyld a x  + J Glx ,g l -v jcy )d3  = 0 1 
and t h e  requirement4 t h a t  G(x,y) b e  symmetric 
G C X , ~ I  = G(y,)o . (63) 
From Eqs. (61) and (62) i t  i s  clear t h a t  t h e  proper  choice of C(x) 
is 
where t h e  cons tan t  o( arises from 
t o  Dj(x 
d 
t 44)  
t h e  f a c t  t h a t  U (x) is  p ropor t iona l  
j 
and 4;: can be  determined from Eq. ( 4 4 ) .  
The f i n a l  express ion  f o r  G(x,y) i s  then  g iven  by 
18 
Hence the most 
y = + m  ( E =  E . )  
J - 
general solution of Eq. (24) which vanishes at 
can be written as follows: 
The constant a is arbitrary. 1 
19 
ACKNOWLEDGMENTS 
The au tho r  thanks Professors  Saul. T. Epstein and Joseph 0. Hirsch- 
f e l d e r  f o r  h e l p f u l  d i scuss ions .  
20 
REFERENCES AND NOTES 
1. The t rea tment  presented  h e r e  i s  taken Largely from Dettman's book 
Mathematical Methods i n  Physics  and Engineering, McGraw-Hil l ,  1962 a 
This  i s  not o r i g i n a l  research .  
Th i s  equat ion  arises i n  a quantum-mechanical t rea tment  of t h e  one- 
dimensional harmonic o s c i l l a t o r .  
For a d i scuss ion  of Weber functions and conf luent  hypergeometric 
func t ions  of t h e  t h i r d  k ind  see Morse and Feshbaah's book Methods 
of T h e o r e t i c a l  Physics ,  M c G r a w - H i l l ,  1953. 
See page 198 of Reference 1 f o r  t h e  l o g i c a l  development of the 
theory  of gene ra l i zed  Green's func t ions .  
2. 
3. 
4 .  
5 .  See page 529 of Vol. I of Reference 4. 


Green's functions for Sturm-Liouville problems

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