Singular nonself-adjoint differential operator of second order eigenvalue

Krall, A. M.

English

NASA Technical Reports Server

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4 
On The Eig68values of a Singular 
of Second Order 
Nonself-Adjoint Differential Operator 
by 
GPO PRICE S 
CFSTI PMICEf8) $ 
Miorofidrr, (MF) 
(1663 -8s 
>r .  The nonself-adjoint operator #y = y"+q(x)y, where 
q(x) = ql(x)+iq2(x) , ql(x) and q2(x) are real valued, 
limit q2(x) = 6 , limit q2(x) = y , 
an interval (a,b) in [l]. From h a nonself-adjoint 
operator L was defined i n , L  (a,b), The spectrum and 
adjoint of L were found, and a "spectral resolution" 
was considered over 
r a  r b  
2 
I 
was derived. 
If r is an arbitrary point in (a,b) , it was shown 
that when A = g+iv , v # y , Ly = Ay has a solution 
g(x,A) in L (r,b) , and when v # 6 , By = Ay has a 
solution h(x,A) in L (a,r). A is an eigenvalue - _ . of L 
if and only if the Wronskian of $(.,A) and h ( x , h )  is 
zero. -This paper wishes to 
characterize these eigenvalues, the zeros of W[$,h]. 
2 
2 
I 
i - 
I 
* McAllister Building, The Pennsylvania State University, I 
University Park, Pennsylvania. '. 1 
This work was supported in part by NASA Grant No. It&?. -39- ;I>? - c j 4  
-__..._ _- - . - 
https://ntrs.nasa.gov/search.jsp?R=19660007803 2020-03-16T23:05:14+00:00Z
c& 
4 In  what follows every expression is  a functibn of the 
complex variable A .  In  the  i n t e r e s t  of notat ional  c l a r i t y ,  
i t  has been suppressed. 
. 
Let  e(x)  and @(x) be the solutions of a y  = hy 
satisfying e ( r )  = 1 , e ' ( r )  = o , Q(r) = o , +'(r)  = -1. 
We choose s i n  [ r , b )  such tha t  Iq2(x)-yl<lv-yl/2 f o r  a l l  
x i n  '[s,b] when y i s  f i n i t e l o r  such t h a t  lq2(x)-vl > E f o r a  
x in [s ,b]  for some E. > 0 when y i s  i n f i n i t e .  
Ylb(x) = 
- 
and y2b % 
of nested c i r c l e s  C(@). i s  found a l l  
3r l i m i t  point given by 
where zb is any real  number. +(x)  i s  then given by 
$(x)  = ~ ( x ) [ - $ ' ( s )  + Mq(S)I + O ( X ) [ ~ ' ( S )  + M@(S)I. 
(See [l] and [2] . )  
Similarly w e  choose t i n  ( a , r )  such t h a t  
lq2(x)-61 < Iv-61/2 f o r  all x i n  ( a , t ]  when' 6 is 
f i n i t e l o r  such t ha t  Iq2(x) -v I  > E for a l l '  x i n  (a , t ]  
f o r  some E > 0 when 6 is i n f i n i t e .  
We define yla(x) = 
. r 
1 
i 
' ?  
: 
, .  
! 
a l l  containing a l i m i t  c i r c l e  o r  l i m i t  point  givgnlby 
3 
where za is  any r e a l  number. h (x )  i s  then given by 
h:(x) = -e(x)[Q'( t )  + m + ( t ) ]  + $(x ) [e ' ( t )  + me(t)] .  
(Again see 111 and 121 ) 
Proof. We observe tha t  W[S,h] is  independent of x. 
Computing w[$,h'] a t  x = r we see 
w[$,hl = [ @ ' ( s )  + M$(s) l [e ' ( t )  + m e ( t ) l  - 
- [e'+) + M S ( S ) ] [ ~ ' ( ~ )  + m $ ( t ) l .  
I n se r t ing  the  expressions fo r  m and M and taking l i m i t s  
completes the  proof. 
- Theorem 1. - If Da(u) - and Db(p) approach f i n i t e  l i m i t s  as 
@+a - and B4bl then WE$, r?] = 0 i f  and only i f  
l i m i t  
a-a 
B-b 
. .. 
I 
Note t h a t  t h i s  last equation i n  no way depends upon the 
poin ts  x = s o r  s = t .  
In  the l i m i t  point cases the value of M (or  m) i s  
independent of the choice of zb (or za) .  Only i n  the  l i m i t  
c i r c l e  case8 does M (or m) vary w i t h  the choice of Z b  
I 
(0' 'a)* 
Theorem 2. If the l i m i t  point case holds a t  . a  and b, 
then *&?der the mnd i t i cns  of tkrecrem 1 
W[$,h] = 0 i f  and only i f  
Proof. Multiply the expression i n  theorem one out and 
c o l l e c t  the coef f ic ien ts  of zazb, za and Zb together . 
Since za and Z b  a re  a rb i t r a ry  the r e s u l t  follows. 
Theorem 3.  If the l i m i t  point case holdd at b,and the l i m i t  
c i r c l e  case holds a t  a, then under the conditions of 
theorem 1, W[$,h] = 0 if  and only i f  
f . . .  
8 
A similar statement is valid if the limit circle case 
holds at b )  and the limit point cast holds at a. 
These results are valid only in nonself -aa3oint problems. 
They are not applicable in the self-ad joint case ,since ,in 
that instance,all eigenvalues lie on the real axis,There the 
existence of $ and h cannot be established in general. 
~ .. 
. P C  
Q 
J 
113 
E21 
References 
l t  
Allan M. K r a l l ,  On the expansion problem f o r  nonself- 11 ad jo in t  ordinary d i f f e r e n t i a l  operators of second order, 
submitted f o r  publication..  
d i f f e r e n t i a l  operators of second order, 
Akademii  Nauk, t o  appear. 
11 , On nonself-adjoint ordinary 
Doklady 
! 
. .  
... . 


On the Eigenvalues of a Singular Nonself- Adjoint Differential Operator of Second Order

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