AbstractIn this paper, using Livšic's theorem, we investigate the problem of completeness of the system of eigenfunctions and associated functions of dissipative operators generated by the Sturm–Liouville differential expression on the semi-axis in Weyl's limit-circle case

Bairamov, Elgiz

Krall, Allan M.

Elsevier - Publisher Connector 

Dissipative Operators Generated by the Sturm–Liouville Differential Expression in the Weyl Limit Circle Case 

Ž .Journal of Mathematical Analysis and Applications 254, 178190 2001
doi:10.1006jmaa.2000.7233, available online at http:www.idealibrary.com on
Dissipative Operators Generated by the
SturmLiouville Differential Expression in the Weyl
Limit Circle Case
Elgiz Bairamov
Department of Mathematics, Ankara Uniersity, 06100, Tandogan, Ankara, Turkey˘
E-mail: bairamov@science.ankara.edu.tr
and
Allan M. Krall
The Pennsylania State Uniersity, Uniersity Park, Pennsylania 16802
E-mail: krall@math.psu.edu
Submitted by William F. Ames
Received August 16, 2000
In this paper, using Livsic’s theorem, we investigate the problem of completenessˇ
of the system of eigenfunctions and associated functions of dissipative operators
generated by the SturmLiouville differential expression on the semi-axis in Weyl’s
limit-circle case.  2001 Academic Press
1. INTRODUCTION
Let us consider the differential expression
 l y  p x y  q x y , x R  0, ,Ž . Ž . Ž .Ž . .
1 Ž .where p, q are real-valued functions on R and p , q L R . We 1, loc 
Ž .denote by L the minimal symmetric operator in L R generated bymin 2 
Ž . Ž . Ž .l y . The operator L has defect index 1, 1 or 2, 2 . In this paper, wemin
always assume that the functions p and q are such that the operator Lmin
Ž .has the defect index 2, 2 ; in other words, Weyl’s limit circle case holds for
Ž .  .the differential expressions l y 1, p. 225 . We can give several sufficient
178
0022-247X01 $35.00
Copyright  2001 by Academic Press
All rights of reproduction in any form reserved.
DISSIPATIVE OPERATORS 179
Ž .conditions in which the operator L has the defect index 2, 2 . Such amin
 condition is given in 7, p. 196 .
 Ž .  1 Ž .  Assume that p, q, q  AC R , p p  L R , and that q and qloc  1 
 .have a constant sign in x , for sufficiently large x . Let q and0 0
  3Ž   .q O q as x , where 0  . If2
 12
q x dx ,Ž .H
Ž .then the operator L has the defect index 2, 2 .min
Let
 y L R : y , py AC R , l y  L R . 4Ž . Ž . Ž . Ž .2  loc  2 
Note that  is the domain of the maximal operator L generated bymax
Ž . l y and L  L .max min
For all y , y  we set1 2
  y , y  y x p x y x  p x y x y x ,Ž . Ž . Ž . Ž . Ž . Ž .Ž .Ž .1 2 1 2 1 2x
where the bar over a function denotes the complex conjugate.
Green’s formula
b b    l y y dx y l y dx y , y  y , y 1.1Ž . Ž .Ž .H H1 2 1 2 1 2 1 2b 0
0 0
  implies that, for all functions y , y , the limit y , y  lim y ,1 2 1 2  x 1
y exists and is finite.2 x
  Ž .Let W y , y denote the Wronskian of two solutions y  y x,  andx 1 2 1 1
Ž .y  y x,  of the equation2 2
 p x y  q x y  y , x R , 1.2Ž . Ž . Ž .Ž . 
   where  is a complex parameter. Then W y , y  y , y .x 1 2 1 2 x
Ž . Ž . Ž .Denote by  x,  and  x,  the solution of 1.2 satisfying the initial
conditions
 0,   cos  , p 0   0,   sin  ,Ž . Ž . Ž .
 0,  sin  , p 0  0,   cos  ,Ž . Ž . Ž .
Ž . Ž .where  is a some real number. The solutions  x,  and  x,  form a
Ž .fundamental system of solutions of 1.2 and they are entire functions of .
Also, they are real functions for real values of . Since the operator Lmin
Ž . Ž . Ž .has the defect index 2, 2 , the solutions  x,  and  x,  belong to
BAIRAMOV AND KRALL180
Ž . Ž . Ž . Ž . Ž . Ž . Ž .L R . Let z x   x, 0 and u x   x, 0 . So z x and u x are2 
Ž .solutions of the equation l y  0, satisfying the initial conditions.
z 0  cos  , p 0 z 0  sin  ,Ž . Ž . Ž .
u 0 sin  , p 0 u 0  cos  .Ž . Ž . Ž .
Ž .We have z, u L R ; moreover z, u. Consequently for each y2 
   the values y, z and y, u exist and are finite. 
Ž .Let D L denote the set of all functions y satisfying the boundary
conditions
    y 0 cos  p 0 y 0 sin  0, y , z  h y , u  0, 1.3Ž . Ž . Ž . Ž . 
  Ž .where h is a some complex number 4, 5 . In L R we define the2 
Ž .operator L with the domain D L and
Ly l y ,Ž .
Ž .for all yD L . We also denote by M the operator generated, in
Ž . Ž .L R , by the differential expression l y and the boundary conditions2 
    p 0 y 0  h y  0, y , z cos  y , u sin  0,Ž . Ž .  0 0
where h is a some complex number. In the following we will assume that0
Im h 0 and Im h  0. It is clear that the operators L and M are0
non-selfadjoint.
One general method of the spectral analysis of non-selfadjoint differen-
tial operators is the method of contour integration of the resolvent. It is
connected with a finite estimate of the resolvent on expanding contours
separating the spectrum. Since the resolvents of the operators L and M
have no fine estimate, this method cannot be applied to them.
In this paper, using Livsic’s theorem and the determinant of perturba-ˇ
tions connected with the operators L and M, we prove that the system
of all eigenvectors and associated vectors of this operators are complete
Ž .in L R .2 
2. PRELIMINARIES
Let A denote the linear non-selfadjoint operator in the Hilbert space H
Ž . Ž .with the domain D A . The element yD A , y 0 is called a root
vector of the operator A corresponding to the eigenvalue  , if all powers0
Ž .nof A are defined on this element and A  I y 0 for some integer0
n 0. The set of all root vectors of A corresponding to the same
DISSIPATIVE OPERATORS 181
eigenvalue  with the vector y 0 forms a linear set N and is called the0 0
root lineal. The dimension of the lineal N is called the algebraic0
multiplicity of the eigenvalue  . The root lineal N coincides with the0 0
linear span of all eigenvectors and associated vectors of A corresponding
to the eigenvalue  . Consequently the completeness of the system of all0
eigenvectors and associated vectors of A is equivalent to the completeness
of the system of all root vectors of this operator.
Ž . Ž .If Im Ay, y 	 0 for all yD A , then the operator A is called
Ž .dissipative. For a bounded operator A defined on all of H the condition
of dissipativeness is equivalent to the condition that its imaginary compo-
1 Ž .nent A  A A is nonnegative.J 2 i
We will denote the class of all nuclear and HilbertSchmidt operators
 Ž .4 Ž A.in H by 	 and 	 , respectively. Let 
 A be a sequence of all1 2 j j1
nonzero eigenvalues of A 	 , p 1, 2, arranged by considering alge-p
Ž . Ž .braic multiplicity and with decreasing modulus, where  A 
  is a sum
of algebraic multiplicities of all nonzero eigenvalues of A. If A 	 , then1
 Ž A. Ž .Ý 
 A is called the trace of A and is denoted by sp A.j1 j
 LIVSIC’S THEOREM 3, p. 226 . Let A be compact dissipatie operator andˇ
let A  	 . In order that the system of all root ectors of A be complete, it isJ 1
necessary and sufficient that
Ž . A
Im 
 A  sp A .Ž .Ý j J
j1
The determinant
Ž . A
det I 
A  1 

 A , A 	 ,Ž . Ž .Ł j 1
j1
Ž .is called the characteristic determinant of A and is denoted by D 
 . TheA
Ž .characteristic determinant D 
 is an entire function of 
, since for anyA
A 	 ,1
Ž . A

 A  .Ž .Ý j
j1
For any A 	 , the regularized characteristic determinant is defined by2
Ž . A


 Ž A.jD˜ 
  1 

 A e . 2.1Ž . Ž . Ž .ÝA j
j1
BAIRAMOV AND KRALL182
If the operator I 
A has a bounded inverse defined on the whole
Žspace H, then the complex number 
 is called an F-regular point regular
.in the sense of Fredholm for A.
Let A and B be linear bounded operators in H and A B 	 . If the1
point 
 is an F-regular point of B, then
1 1I 
A I 
B  I 
 A B I 
B ,Ž . Ž . Ž . Ž .
Ž .Ž .1where 
 A B I 
B  	 . Consequently, the determinant1
1D 
  det I 
A I 
BŽ . Ž . Ž .AB
is meaningful and is called the determinant of perturbation of the operator
B by the operator K A B.
We will use the following two well-known theorems.
 THEOREM 2.1 3, p. 172 . If A, B 	 , A B 	 , and 
 is an2 1
F-regular point of B, then
D˜ 
Ž .A 
 spŽ BA.D 
  e .Ž .AB D˜ 
Ž .B
 THEOREM 2.2 3, p. 177 . Let A and B be bounded dissipatie operators
Ž .in particular, one of them or both may be self-adjoint and A B 	 .1
Ž .Then for any  0   the relation0 0 2
1
ilim log D  e  0Ž .AB
holds uniformly with respect to  in the sector

i :   e , 0  ,     .0½ 52
Let g be an entire function. If for each  0 there exists a finite
constant C  0, such that
  g  
 C e ,  C, 2.2Ž . Ž .
then g is called an entire function of order 
 1 of growth and minimal
Ž .type. It is evident from 2.2 that
1
lim sup log g  
 0. 2.3Ž . Ž .
  
DISSIPATIVE OPERATORS 183
Ž . Ž .It is known that each function g, having property 2.2 and g 0 1, has
the representation

g   lim 1 , 2.4Ž . Ž .Ł ž /r   
r jj
 and also the limit lim Ý 1 exists and is finite 6 .r  
 r jj
3. COMPLETENESS OF THE SYSTEM OF ROOT
VECTORS OF L AND M
 4LEMMA 3.1. Zero is not an eigenalue of L; i.e., ker L 0 .
Ž .Proof. Let yD L and Ly 0. Then
 p x y  q x y 0, 3.1Ž . Ž . Ž .Ž .
Ž .and the function y satisfies the boundary condition 1.3 . Therefore, there
Ž . Ž . Ž .exists the constants c and c , such that y x  c z x  c u x . Substi-1 2 1 2
Ž .tuting this in the boundary conditions 1.3 , we find c  c  0; conse-1 2
quently y 0.
From Lemma 3.1 we get that there exists the inverse operator L1. Let
Ž . Ž . Ž . Ž .us consider the functions u x and  x  z x  hu x . These functions
Ž . Ž .belong to L R . The first satisfies the boundary condition at zero in 1.32 
Ž . Ž .and the second at infinity. The functions u x and  x form a fundamen-
Ž .tal system of solutions of 3.1 .
Let K denote the integral operator defined by the formula

Kf G x , t f t dt , f L R , 3.2Ž . Ž . Ž . Ž .H 2 
0
where
u x  t , 0
 x
 t ,Ž . Ž .
G x , t  3.3Ž . Ž .½ u t  x , t
 x .Ž . Ž .
Since
  2
G x , t dx dt ,Ž .H H
0 0
we get that K 	 . It is easy to verify that K L1. Consequently, the2
root lineals of the operators L and K coincide and, therefore, the
Ž .completeness in L R of the system of all eigenvectors and associated2 
BAIRAMOV AND KRALL184
vectors of L is equivalent to the completeness of those for K. Since the
algebraic multiplicity of nonzero eigenvalues of a compact operator is
finite, each eigenvector of L may have only a finite number of linear
independent associated vectors.
Let
a    x ,  , z x , a    x ,  , u x ,Ž . Ž . Ž . Ž . Ž . Ž .1 2 
Ž . Ž .where  x,  the solution of 1.2 given in the introduction.
It is evident that
	 L   :  C, a   0 , 4Ž . Ž .d
Ž .where 	 L denotes the set of all eigenvalues of L andd
a   a   ha  . 3.4Ž . Ž . Ž . Ž .1 2
Ž .THEOREM 3.2. The functions a  , j 1, 2, are entire functions of orderj

 1 of growth and minimal type.
Proof. We set
 a    x ,  , z x  p b  b ,  z b   b ,  z b ,Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .b , 1 b
 a    x ,  , u x  p b  b ,  u b   b ,  u b .Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .b , 2 b
Ž . Ž .Since for arbitrary fixed b, the functions  b,  and  b,  are entire
1 Ž .functions of  of order , consequently, the functions a  , j 1, 2,b, j2
Ž .have the same property. Now we prove that the entire function a b, j
Ž .converges to a  as b , uniformly in  in each compact set of thej
complex plane C.
Ž . Ž .Let y y x,  be a solution of 1.2 ; then
   y y , u z y , x u. 3.5Ž .x x
If we define
   f x ,   y , z , f x ,   y , u ,Ž . Ž .x x1 2
  Ž . Ž .then following 2 , we get that f x,  and f x,  satisfy a system of first1 2
order differential equations
 
f x ,    y x ,  z x , f x ,    y x ,  u x , x R .Ž . Ž . Ž . Ž . Ž . Ž .1 2  x  x
DISSIPATIVE OPERATORS 185
Ž .Using 3.5 we obtain

f x ,   Q x f x ,  , x R , 3.6Ž . Ž . Ž . Ž . x
where
f x ,  2Ž . z x u x z xŽ . Ž . Ž .1f x ,   , Q x  ,Ž . Ž . 2ž / ž /f x , Ž . u x z x u xŽ . Ž . Ž .2
Ž . Ž .and the elements of Q x are in L R . For1 
w1w ,wž /2
	 	    we put w  w  w and the norm of a square 2
 2 matrix will be1 2
	 	 	 Ž .	 Ž .denoted by . . The inclusion Q x  L R holds.1 
Ž . Ž . Ž .If y x,    x,  , then the system 3.6 is equivalent to the integral
equation
x
f x ,   f b ,    Q t f t ,  dt , x R , 3.7Ž . Ž . Ž . Ž . Ž .H 
b
where
a  a Ž . Ž .b , 1 11f b ,   , f 0,   , f ,   .Ž . Ž . Ž .ž /0ž / ž /a  a Ž . Ž .b , 2 2
Ž .From 3.7 we find that
x
 f x ,  
 f b ,  exp  Q t dt ;Ž . Ž . Ž .Hž /b
hence
 
   f ,   f b ,  
  Q t dt exp  Q t dt , 3.8Ž . Ž . Ž . Ž . Ž .H Hž / ž /b 0

 f ,  
 f b ,  exp  Q t dt . 3.9Ž . Ž . Ž . Ž .Hž /b
Ž . Ž . Ž .It follows from 3.8 that a  converges to a  as b , uniformly inb j j
Ž . in a compact set. Consequently a  , j 1, 2, are entire functions.j
BAIRAMOV AND KRALL186
Ž .For b 0, from 3.9 we get

 f ,  
 exp  Q t dt ;Ž . Ž .Hž /b
Ž .hence a  are of not higher than the first order. Since, for arbitrary fixedj
1Ž .b, the functions a  , j 1, 2, are entire functions of  of order , fromb, j 2
Ž . Ž .3.9 we obtain that the entire functions a  , j 1, 2, are of minimalj
type.
Ž .Using Green’s formula 1.1 as b , we have

a    x ,  , z x 1   x ,  z x dx , 3.10Ž . Ž . Ž . Ž . Ž . Ž .H1 
0

a    x ,  , u x    x ,  u x dx . 3.11Ž . Ž . Ž . Ž . Ž . Ž .H2 
0
Ž . Ž . Ž . Ž .From 3.4 , 3.10 , and 3.11 we find that a 0 1.
We will use the well-known formula
   y , u , y , z1 1x x y , y  det , 0
 x
 , 3.12Ž .1 2 x ž /   u , y , z , y2 2x x
 where y , y  4, 5 .1 2
THEOREM 3.3. The operator L is dissipatie.
Ž . Ž .Proof. Let yD L ; then by Green’s formula 1.1 we get
   Ly , y  y , Ly  y , y  y , y . 3.13Ž . Ž . Ž . 0
Ž . Ž .From the boundary condition 1.3 and 3.12 we obtain that
2     y , y  0, y , y  2 i Im h y , u ,Ž .0  
Ž .or by 3.13
2 Im Ly , y  Im h y , u 	 0. 3.14Ž . Ž . Ž .
Consequently, the operator L is dissipative.
It follows from Theorem 3.3 that all the eigenvalues of L lie in the
closed upper half-plane Im 	 0.
THEOREM 3.4. The operator L has no real eigenalue.
DISSIPATIVE OPERATORS 187
Ž .Proof. Let  be a real eigenvalue of L and let the function  x 0 0
Ž . Ž . x,  be the corresponding eigenfunction. Since Im L ,  0 0 0
Ž 	 	 2 . Ž .  Im    0, from 3.14 we find that  , u  0. By the boundary0 0 0 
Ž .   Ž . Ž .condition 1.3 we have  , z  0. Let  x   x,  ; then using0  0 0
Ž .3.12 we get
     1W  ,  W  ,    , 0 0 0  0 0 0 0 
         , u z ,    , z u ,   0,0 0 0 0   
which is a contradiction whence the result.
Ž . Ž . Ž . Ž .Since  x  z x  hu x , setting h h  ih we get from 3.2 in1 2
Ž .view of 3.3 that K K  iK , where1 2
 
K f G x , t f t dt , K f G x , t f t dt ,Ž . Ž . Ž . Ž .H H1 1 2 2
0 0
and
u x z t  h u t , 0
 x
 t ,Ž . Ž . Ž .1
G x , t Ž .1 ½ u t z x  h u x , t
 x
 ,Ž . Ž . Ž .1
G x , t h u x u t , h  Im h 0.Ž . Ž . Ž .2 2 2
Ž .The operator K is the self-adjoint HilbertSchmidt operator in L R1 2 
Ž .and K is the self-adjoint one-dimensional operator in L R , and2 2 
Ž . Ž .K f , f 
 0 for all f L R .2 2 
Ž .Let L denote the operator generated in L R by the differential1 2 
Ž .expression l y and the boundary condition
    y 0 cos  p 0 y 0 sin  0, y , z  h y , u  0.Ž . Ž . Ž .  1
It is easy to verify that K is the inverse of L : K  L1.1 1 1 1
Let TK and T T  iT , where T K , T K .1 2 1 1 2 2
We will denote by  and  the eigenvalues of the operators L and L ,j k 1
Ž .respectively. Then the eigenvalues of T are 1 and the eigenvaluesj
Ž .of T are 1 . Since L is a self-adjoint operator, therefore Im   01 k 1 k
for all k.
Ž .THEOREM 3.5. Ý Im 1  sp T .j j 2
Proof. Using Theorem 2.1 for A T and B T we obtain1
D˜ 
Ž .T1 i
 sp T2D 
  e , 3.15Ž . Ž .T T1 D˜ 
Ž .T
BAIRAMOV AND KRALL188
Ž .and by 2.1 we get

 


  
j k˜ ˜D 
  1 e , D 
  1 e .Ž . Ž .Ł ŁT T1 ž /ž / j kj k
We set
a 
  a 
  ha 
 , d 
  a 
  h a 
 ,Ž . Ž . Ž . Ž . Ž . Ž .1 2 1 1 2
Ž . Ž . Ž . Ž .where the functions a 
 and a 
 are given by 3.10 and 3.11 . The1 2
Ž .eigenvalues of K and K coincide with the root of the functions a 
 and1
Ž . Ž . Ž .d 
 , respectively. Theorem 3.2 shows that the functions a 
 and d 

are entire functions of order 
 1 of growth and minimal type and
Ž . Ž .a 0  d 0 1; therefore

 

a 
  1 , d 
  1 ,Ž . Ž .Ł Ł ž /ž / j kj k
Ž .by 2.4 . So
˜ 
Ý jŽ1 j. ˜ 
ÝkŽ1 k .D 
 a 
 e , D 
 d 
 e ,Ž . Ž . Ž . Ž .T T1
Ž .and from 3.15 we find
d 
 1 1Ž .
D 
  exp 
  
  i
 sp T .Ž . Ý ÝT T 21 ž /a 
  Ž . j kj k
Ž .If we take 
 it 0 t  , then we get
1 1 1
log D it  log d it  log a itŽ . Ž . Ž .T T1t t t
1
 Im  sp T . 3.16Ž .Ý 2 jj
Ž .From Theorem 2.2 and 2.3 we obtain that
1
lim log D it  0, 3.17Ž . Ž .T T1tt
and
1 1
lim sup log a it 
 0, lim sup log d it 
 0. 3.18Ž . Ž . Ž .
t tt t
DISSIPATIVE OPERATORS 189
On the other hand, for t 0,
2 2it it
1 	 1, 1 	 1,
 j k
 Ž .   Ž . and we have a it 	 1, d it 	 1 for all t 0. Consequently,
1 1
log a it 	 0, log d it 	 0,Ž . Ž .
t t
Ž .and from 3.18 it follows that
1 1
lim log a it  lim log d it  0. 3.19Ž . Ž . Ž .
t tt t
Ž . Ž . Ž .Hence we get, by 3.16 , 3.17 , and 3.19 that
1
Im   sp T .Ý 2ž / jj
Thus the operator T carries out all the conditions of Livsic theorem onˇ
completeness. Hence we have the following.
COROLLARY 3.6. The system of all root ectors of the dissipatie operator
Ž . Ž .T also of K is complete in L R .2 
Ž .Since the completeness in L R of the system of all eigenvectors and2 
associated vectors of L is equivalent to the completeness of those for K ,
from Corollary 3.6 we get
THEOREM 3.7. The system of all eigenectors and associated ectors of L
Ž .is complete in L R .2 
Analogously for the operator M we obtain
THEOREM 3.8. The system of all root ectors of the dissipatie operator M
Ž .is complete in L R .2 
REFERENCES
1. E. A. Coddington and N. Levinson, ‘‘Theory of Ordinary Differential Equations,’’ Mc-
Graw-Hill, New York, 1955.
Ž .2. C. T. Fulton, Parametrization of Titchmarsh’s m  -functions in the limit circle case,
Ž .Trans. Amer. Math. Soc. 229 1977 , 5163.
BAIRAMOV AND KRALL190
3. I. C. Goldberg and M. G. Krein, ‘‘Introduction to the Theory of Linear Nonselfadjoint
Operators,’’ Am. Math. Soc., Providence, 1969.
4. A. M. Krall and A. Zettl, Singular self-adjoint Sturm-Liouville problems, J. Differential
Ž .Integral Equations 1 1988 , 423432.
5. A. M. Krall and A. Zettl, Singular self-adjoint Sturm-Liouville problems II: Interior
Ž .singular points, SIAM J. Math. Anal. 19 1988 , 11351141.
6. M. G. Krein, On the indeterminate case of the SturmLiouville boundary problem in the
Ž . Ž . Ž .interval 0, , Iz . Akad. Nauk SSSR Ser. Mat. 16, No. 4 1952 , 293324 in Russian .
7. M. A. Naimark, ‘‘Linear Differential Operators II,’’ Ungar, New York, 1968.


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Dissipative Operators Generated by the Sturm–Liouville Differential Expression in the Weyl Limit Circle Case

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Elsevier - Publisher Connector

Elsevier - Publisher Connector