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Quadratic Pencil of Schrödinger Operators with Spectral Singularities: Discrete Spectrum and Principal Functions
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
216, 303]320 Ž1997.
AY975689
Quadratic Pencil of Schrodinger Operators ¨ with Spectral Singularities: Discrete Spectrum and Principal Functions
¨ Elgiz Bairamov and Oner C ¸akar Ankara Uni¨ ersity, Mathematics Department, 06100 Bese¨ ler, Ankara, Turkey
˙
and A. Okay C ¸elebi Department of Mathematics, Middle East Technical Uni¨ ersity, Ankara 06531, Turkey Received September 11, 1996
In this article we investigated the spectrum of the quadratic pencil of Schro¨ dinger operators LŽ l. generated in L2 ŽRq . by the equation yy0 q w V Ž x . q 2 lU Ž x . y l2 x y s 0,
x g Rqs w 0, ` .
and the boundary condition y Ž 0 . s 0, where U, V are complex valued functions and U is absolutely continuous in each finite subinterval of Rq. Discussing the spectrum, we proved that LŽ l. has a finite number of eigenvalues and spectral singularities with finite multiplicities, if the conditions sup
exp Ž e'x . w < V Ž x . < q
e)0
0Fx-`
and lim U Ž x . s 0
xª`
hold. It is shown that the principal functions corresponding to eigenvalues of LŽ l. are in the space L2 ŽRq ., and the principal functions corresponding to spectral singularities are in another Hilbert space, which contains L2 ŽRq .. Some results about the spectrum of LŽ l. have also been applied to radial Klein]Gordon and one-dimensional Schrodinger equations. Q 1997 Academic Press ¨ 303 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
304
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
1. INTRODUCTION Let us consider the non-self-adjoint one-dimensional Schrodinger opera¨ tor L generated in L2 ŽRq . by the differential expression l Ž y . ' yy0 q V Ž x . y,
x g Rq
and the boundary condition y Ž0. s 0, where V is a complex valued function. The spectral analysis of L has been investigated by Naimark w12x. In this article, he has proved that some of the poles of the resolvent’s kernel of L are not the eigenvalues of the operator. He has also shown that those poles Žwhich are called spectral singularities by Schwartz w17x. are on the continuous spectrum. Moreover, he has shown that the spectral singularities play an important role in the discussion of the spectral analysis of L, and if the condition `
H0 < V Ž x .
e)0
holds, then the eigenvalues and the spectral singularities are of a finite number and each of them is of a finite multiplicity. The effect of spectral singularities in the spectral expansion of the operator L in terms of principal functions has been investigated in w9x. In w14x, the dependence of the structure of spectral singularities of L on the behavior of V at infinity has been considered. Some problems related to spectral analysis of differential and some other types of operators with spectral singularities have been discussed in w3]5, 10, 13, 15x. Let us consider the operator LŽ l. generated in L2 ŽRq . by the equation yy0 q V Ž x . q 2 lU Ž x . y l2 y s 0,
x g Rq ,
Ž 1.1.
and the boundary condition y Ž 0 . s 0,
Ž 1.2.
where U, V are complex-valued functions and U is absolutely continuous in each finite subinterval of Rq. If U ' 0, then the operator LŽ l. reduces to the operator L. If we choose V Ž x . s yU 2 Ž x ., then Ž1.1. will be reduced into the radial form of the Klein]Gordon equation. Some problems of the spectral theory of Ž1.1. and of the Klein]Gordon equation have been investigated by several authors w1, 2, 6, 7, 8, 11x with real functions U and V. We also want to note that, the finiteness of the number of the eigenvalues of LŽ l. has been given by the following technique in w7x. First it is proved that the set of eigenvalues of LŽ l. is bounded in the complex plane and is of countable, and their limit points lie on the real axis. Later, assuming that there are no limit points of eigenvalues, it has been
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determined that the eigenvalues are of finite number. But in w7x the functions U and V for which the set of eigenvalues has no limit points have not been investigated. Using the technique of the uniqueness of analytic functions, we proved that the eigenvalues and the spectral singularities are of finite number. In this paper, we discussed the spectrum of LŽ l. defined by Ž1.1. and Ž1.2., and proved that this operator has a finite number of eigenvalues and spectral singularities and that each of them is of a finite multiplicity, under the conditions sup
exp Ž e'x .
< V Ž x . < q < U9 Ž x . <
4 - `,
e ) 0,
0Fx-`
and lim U Ž x . s 0.
xª`
Afterward, the properties of the principal functions corresponding to the eigenvalues and the spectral singularities of LŽ l. are obtained. If U is absolutely continuous in every finite subinterval of Rq and satisfies sup
exp Ž e'x .
e)0
0Fx-`
and lim U Ž x . s 0,
xª`
then the eigenvalues and the spectral singularities of the Klein]Gordon equation have similar properties. In the particular case L of the operator LŽ l., the results we have obtained about the spectrum are better than the ones given by Lyance w9x and Naimark w12x, and are the same as the ones obtained by Pavlov w15x. In the sequel, we use the notations Cqs l < l g C, Im l ) 0 4 , R* s R _ 0 4 ,
Cys l < l g C, Im l - 0 4
Cqs l < l g C, Im l G 0 4 ,
Cys l < l g C, Im l F 0 4 Furthermore, sp Ž LŽ l.., ss s Ž LŽ l.., r Ž LŽ l.., and RŽ LŽ l.. will denote the eigenvalues, the spectral singularities, the resolvent set, and the resolvent of the operator LŽ l., respectively.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
306
2. PRELIMINARIES Let us suppose that the functions U and V satisfy the following conditions: `
`
H0
H0 x
< V Ž x . < q < U9 Ž x . < dx - `.
Ž 2.1.
Let us define the functions v , a , and b by `
vŽ x. s
Hx U Ž t . dt
a Ž x. s
Hx
b Ž x. s
Hx
`
`
< V Ž t . < q < U9 Ž t . < dt
Ž 2.2.
t < V Ž t . < q 2 < U Ž t . < dt
Jaulent and Jean w7x have obtained the following important result: if Ž2.1. holds, then Ž1.1. has the solution f Ž x, l . s e i v Ž x . e i l x q
`
Hx
A Ž x, t . e i l t dt
Ž 2.3.
for l g Cq and the solution g Ž x, l . s eyi v Ž x . eyi l x q
`
yi l t
Hx B Ž x, t . e
dt
Ž 2.4.
for l g Cy; moreover, the kernels AŽ x, t . and B Ž x, t . satisfy the inequality 1 xqt < A Ž x, t . < , < B Ž x, t . < F a exp b Ž x . 4 . Ž 2.5. 2 2 Therefore, the solutions f Ž x, l. and g Ž x, l. are analytic with respect to l, in Cq and Cy, respectively, and are continuous up to the real axis. f Ž x, l. and g Ž x, l. also satisfy the following asymptotic equalities w7x:
ž
/
f Ž x, l . s e i l x 1 q o Ž 1 . ,
l g Cq ,
xª`
f x Ž x, l . s e i l x i l q o Ž 1 . ,
l g Cq ,
xª`
g Ž x, l . s eyi l x 1 q o Ž 1 . ,
l g Cy ,
xª`
l g Cy ,
xª`
f Ž x, l . s e i v Ž x . e i l x q o Ž 1 . ,
l g Cq ,
< l< ª `
g Ž x, l . s eyi v Ž x . eyi l x q o Ž 1 . ,
l g Cy ,
< l< ª `
g x Ž x, l . s eyi l x yi l q o Ž 1 . ,
Ž 2.6. Ž 2.7.
From Ž2.3. and Ž2.4. we easily find
Ž 2.8.
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307
According to Ž2.6. and Ž2.7., the Wronskian of the solutions f Ž x, l. and g Ž x, l. is W w f , g x s lim W w f , g x s y2 i l xª`
Ž 2.9.
for l g R. So f Ž x, l. and g Ž x, l. provide the fundamental solutions of the equation Ž1.1. for l g R*. Let w Ž x, l. be the solution of Ž1.1. satisfying the initial conditions
w Ž 0, l . s 0,
w x Ž 0, l . s 1.
It is obvious that the solution w Ž x, l. exists, is unique, and is an entire function of l w7x.
3. EIGENVALUES AND SPECTRAL SINGULARITIES Let us define
r 1 Ž l . s l < l g Cq , F Ž l . / 0 4 ,
r 2 Ž l . s l < l g Cy , G Ž l . / 0 4
where F Ž l . [ f Ž 0, l . ,
G Ž l . [ g Ž 0, l . .
Using the standard techniques w13x, we can show that
r Ž L Ž l. . s r 1Ž l. j r 2 Ž l. and for l g r Ž LŽ l.. the resolvent of LŽ l. is the integral operator defined as R Ž L Ž l. . c Ž x . s
`
H0 R Ž x, t ; l. c Ž t . dt
for c g L2 ŽRq ., where the kernel RŽ x, t; l. Ži.e., the Green’s function for LŽ l.. is given by R Ž x, t ; l . s
½
R1 Ž x, t ; l . ,
l g r 1Ž l.
R 2 Ž x, t ; l . ,
l g r 2 Ž l.
Ž 3.1.
in which R1 Ž x, t ; l . s R 2 Ž x, t ; l . s
1 F Ž l. 1 G Ž l.
½ ½
f Ž x, l . w Ž t , l . f Ž t , l . w Ž x, l .
0Ft-x xFt-`
Ž 3.2.
g Ž x, l . w Ž t , l . g Ž t , l . w Ž x, l .
0Ft-x x F t - `.
Ž 3.3.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
308
From the asymptotic equalities Ž2.6. and Ž2.7., we have f Ž ?, l . , f x Ž ?, l . g L2 Ž Rq .
Ž 3.4.
g Ž ?, l . , g x Ž ?, l . g L2 Ž Rq .
Ž 3.5.
for each l g Cq and for each l g Cy. LEMMA 3.1. Ža. sp Ž LŽ l.. s l < l g Cq, F Ž l. s 04 j l < l g Cy, GŽ l. s 04 , Žb. ss s Ž LŽ l.. s l < l g R*, F Ž l. s 04 j l < l g R*, GŽ l. s 04 . Proof. Ža. Obviously,
l < l g Cq , F Ž l . s 0 4 j l < l g Cy , G Ž l . s 0 4 ; sp Ž L Ž l . . . On the other hand, let us suppose that l0 g sp Ž LŽ l.. and examine the following cases: Ž1. Let l0 g Cq. Since l0 g sp Ž LŽ l.., then Ž1.1. has a solution y Ž x, l 0 . in L2 ŽRq . for l s l 0 that is nontrivial and y Ž0, l0 . s 0. Since W y Ž x, l 0 . , w Ž x, l0 . s 0, there exists a constant c / 0 such that y Ž x, l0 . s c w Ž x, l 0 .. Then we have W f Ž x, l0 . , y Ž x, l 0 . s cF Ž l0 . .
Ž 3.6.
But it is evident from Ž3.4. that W f Ž x, l0 . , y Ž x, l0 . s lim W f Ž x, l0 . , y Ž x, l 0 . s 0. xª`
From Ž3.6. and Ž3.7. we find F Ž l 0 . s 0. Ž2. Let l0 g Cy. In a similar way, we can prove that G Ž l0 . s 0. Ž3. Let l0 g R. In this case the general solution of Ž1.1. is y Ž x, l0 . s c1 f Ž x, l0 . q c 2 g Ž x, l 0 . for l s l0 . From Ž2.6. and Ž2.7. we get y Ž x, l0 . s c1 e i l 0 x q c 2 eyi l 0 x q o Ž 1 .
as x ª `.
Ž 3.7.
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309
That is, y Ž?, l0 . f L2 ŽRq .. Hence
sp Ž L Ž l . . l R s f . Combining Ž1., Ž2., and Ž3., we find
sp Ž L Ž l . . ; l < l g Cq , F Ž l . s 0 4 j l < l g Cy , G Ž l . s 0 4 . This completes the proof of Ža.. Žb. Spectral singularities are the poles of the kernel of the resolvent, but not the eigenvalues of the operator LŽ l.. From Ž3.1. ] Ž3.3. and part Ža. we obtain that the spectral singularities of LŽ l. are the zeros of F and G on the real axis. We can easily show that 0 f ss s Ž LŽ l.., which completes the proof of part Žb.. Note that from W f Ž x, l . , g Ž x, l . s F Ž l . g x Ž 0, l . y G Ž l . f x Ž 0, l . s y2 i l ,
l g R, we immediately get
l < l g R*, F Ž l . s 0 4 l l < l g R*, G Ž l . s 0 4 s f .
Ž 3.8.
To investigate the quantitative properties of the eigenvalues and the spectral singularities of LŽ l., we need to discuss the quantitative properties of the zeros of F and G in Cq and Cy, respectively. For the sake of simplicity, we will consider only the zeros of F in Cq. A similar procedure may also be employed for the zeros of G in Cy. Let us define Q1 s l < l g Cq , F Ž l . s 0 4 ,
Q2 s l < l g R, F Ž l . s 0 4 .
LEMMA 3.2. Ža. The set Q1 is bounded and has at most a countable number of elements, and its limit points can lie only in a bounded subinter¨ al of the real axis. Žb. The set Q2 is compact and its linear Lebesque measure is zero. Proof. From Ž2.8. we get F Ž l . s e i v Ž0. q o Ž 1 . , l g Cq , < l < ª `.
Ž 3.9.
The asymptotic equality Ž3.9. shows the boundedness of the sets Q1 and Q2 . From the analyticity of the function F in Cq, we get that Q1 has at most countable numbers of elements. By the uniqueness of analytic functions we find that the limit points of Q1 can lie only in a bounded
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
310
subinterval of the real axis. The closedness and the property of having linear Lebesgue measure zero of the set Q2 can be obtained from the uniqueness theorem of the analytic functions w16x. From Lemmas 3.1 and 3.2 we get the following. THEOREM 3.3. The sets of eigen¨ alues and spectral singularities of LŽ l. are bounded and are, at most, countable, and their limit points can lie only in a bounded subinter¨ al of the real axis if the condition Ž2.1. holds. Note. For the moment we want to start with ‘‘ Assumption III: F and G has no real zeros’’ in w7x, that is,
ss s Ž L Ž l . . s f . In this case, from Lemma 3.2 and Theorem 3.3 we may derive that the number of eigenvalues is finite, which corresponds to the technique given in w7x. But it has not been clarified for which functions U and V Assumption III holds. So Assumption III does not seem to be natural. Hence we will prove by another method that the number of the eigenvalues and the spectral singularities of LŽ l. are finite, without employing Assumption III. Up to now we have assumed that condition Ž2.1. holds. In the rest of the article the assumptions we will use are lim U Ž x . s 0,
xª`
sup
exp Ž e'x .
< V Ž x . < q < U9 Ž x . <
e ) 0.
4 - `,
0Fx-`
Ž 3.10. From Ž2.2. and Ž3.10. we have
a Ž x . F c 0 exp y
ž
e 2
'x
/
,
b Ž x . F c 0 exp y
ž
e 2
'x
/
,
Ž 3.11.
where c 0 ) 0 is constant. By Ž2.5. and Ž3.11. we get
e
xqt
2
2
ž ( /
< A Ž x, t . < , < B Ž x, t . < F c exp y
,
Ž 3.12.
where c ) 0. We have seen that whenever condition Ž2.1. holds, the function F is analytic in Cq and continuous up to the real axis. It is obvious from Ž3.12. that if the conditions Ž3.10. hold, then all derivatives of F are continuous up to the real axis. So the inequalities < F Ž r . Ž l . < F Mr ,
l g Cq ,
r s 0, 1, 2, . . .
Ž 3.13.
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311
hold, where `
Mr s 2 rq1 c
H0
t r exp y
½
H0
ž
e 2
't
/
r s 1, 2, . . .
dt,
Ž 3.14.
and `
M0 s exp y
5
ImU Ž t . dt q
`
H0
< A Ž 0, t . < dt,
provided that c ) 0 is a constant. Let us indicate the set of all limit points of Q1 and Q2 by Q3 and Q4 , respectively, and the set of all zeros of F with infinite multiplicity in Cq by Q5 . From the uniqueness theorem of analytic functions, it is obvious that Q3 ; Q2 , LEMMA 3.4.
Q4 ; Q2 ,
Q5 ; Q 2 .
Q 3 ; Q5 , Q 4 ; Q5 .
The proof of this lemma can be obtained by use of the continuity of all derivatives of F up to the real axis. We will use the following uniqueness theorem for the analytic functions on the upper half-plane to prove the next result. THEOREM 3.5 Žw14x.. Let us assume that the function g is analytic in Cq, all of its deri¨ ati¨ es are continuous up to the real axis, and there exist N ) 0 such that < g Ž r . Ž l . < F Mr ,
r s 0, 1, 2, . . . ,
l g Cq ,
< l < - 2 N, Ž 3.15.
and yN
Hy`
ln < g Ž x . < 1qx
2
dx - `,
`
HN
ln < g Ž x . < 1 q x2
dx - `
Ž 3.16.
hold. If the set P with linear Lebesque measure zero is the set of all zeros of the function g with infinity multiplicity and if h
H0
ln E Ž s . d m Ž Ps . s y`
holds, then g Ž l. ' 0, where EŽ s . s inf r Mr s rrr!, and r s 0, 1, 2, . . . , m Ž Ps . is the Lebesque measure of the s-neighborhood of P and h is an arbitrary positi¨ e constant.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
312 LEMMA 3.6.
Q5 s f .
Proof. It is trivial from Lemma 3.2 and Ž3.13. that F satisfies the conditions Ž3.15. and Ž3.16.. Since the function F is not identically equal to zero, then by Theorem 3.5, Q5 satisfies the condition h
H0
ln E Ž s . d m Ž Q5, s . ) y`,
Ž 3.17.
where EŽ s . s inf r Mr s rrr!, m Ž Q5, s . is the Lebesque measure of the sneighborhood of Q5 , and the constant Mr is defined by Ž3.14.. Now we will obtain the following estimates for Mr : Mr s 2 rq1 c
`
H0
t r exp y
ž
e 2
't
/
dt
F Bb r r r r !,
Ž 3.18.
where B and b are constants depending on c and e . Substituting Ž3.18. in the definition of EŽ s ., we arrive at E Ž s . s inf r
Mr s r r!
F B inf b r s r r r 4 F B exp yby1 ey1 sy1 4 , r
or by Ž3.17., h
H0
1 s
d m Ž Q5, s . - `.
Ž 3.19.
The inequality Ž3.19. holds for an arbitrary s, if and only if m Ž Q5, s . s 0 or Q5 s f . LEMMA 3.7. The function F has a finite number of zeros with finite multiplicity in Cq. Proof. Lemmas 3.4 and 3.6 give Q3 s Q4 s f . So the bounded sets Q1 and Q2 have no limit points Žsee Lemma 3.2., i.e., the function F has only a finite number of zeros in Cq. Since Q5 s f , these zeros are of finite multiplicity. The discussions given above for F may be repeated for G. Hence we obtain the following. LEMMA 3.8. The function G has a finite number of zeros with finite multiplicity in Cy. DEFINITION 3.9. The multiplicity of a zero of F Žor G . in Cq Žor Cy . is called the multiplicity of the corresponding eigenvalue or spectral singularity of LŽ l..
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Lemmas 3.1, 3.7, and 3.8 give the following. THEOREM 3.10. The operator LŽ l. has a finite number of eigen¨ alues and spectral singularities, and each of them is of finite multiplicity if the condition Ž3.10. holds.
4. PRINCIPAL FUNCTIONS In this section we assume that Ž3.10. holds. Let l1q , . . . , lq and j F in Cq and G in Cy with multiplicities respectively. Similarly, let l1 , . . . , l p and m 1 , . . . , m q be the zeros of F and G on the real axis with multiplicities m1 , . . . , m p and n1 , . . . , n q , respectively. It is trivial that y my 1 , . . . , m k denote the zeros of q y m1 , . . . , mq and my j 1 , . . . , mk ,
½
n l
5
W f Ž x, l . , w Ž x, l . n
ls l q i
s
½
dn d ln
F Ž l.
5
ls lq i
G Ž l.
5
ls my l
s0
Ž 4.1.
s0
Ž 4.2.
for n s 0, 1, . . . , mq i y 1, i s 1, 2, . . . , j, and
½
n
5
W g Ž x, l . , w Ž x, l . ln
ls m y l
s
½
dn d ln
for n s 0, 1, . . . , my l y 1, l s 1, 2, . . . , k. If n s n s 0, we get q q w Ž x, lq i . s a 0 Ž l i . f Ž x, l i . , y y w Ž x, my l . s b 0 Ž m l . g Ž x, m l . ,
i s 1, 2, . . . , j,
Ž 4.3.
l s 1, 2, . . . , k.
Ž 4.4.
. / 0, b 0 Ž my . / 0. So a0 Ž lq i l THEOREM 4.1. The formulas
½
n l
w Ž x, l . n
5
n ls l q i
s
Ý ks0
k n a ny k f Ž x, l . k l k
ž / ½
5
ls lq i
Ž 4.5.
for n s 0, 1, . . . , mq i y 1, i s 1, 2, . . . , j, and
½
n
w Ž x, l . ln
5
ls m y i
s
n
Ý js0
j n bnyj g Ž x, l . j l j
ž/ ½
5
ls my i
,
Ž 4.6.
for n s 0, 1, . . . , my l y 1, l s 1, 2, . . . , k hold, where the constants y a0 , a1 , . . . , a n and b 0 , b1 , . . . , bn depend on lq i and m l , respecti¨ ely.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
314
Proof. Let us start with equality Ž4.5.. We will utilize mathematical induction. For n s 0, the proof is trivial from Ž4.3.. Let us assume that for Ž . 1 F n 0 F mq i y 2 the equality 4.5 holds, i.e.,
½
n0 l n 0
w Ž x, l .
5
n0 ls l q i
s
Ý ks0
k n0 q a n 0yk Ž l i . f Ž x, l . l k k
½
ž /
5
ls lq i
Ž 4.7.
Now we will prove that Ž4.5. also holds for n 0 q 1. If y Ž x, l. is a solution of the equation Ž1.1., then nrl n y Ž x, l. satisfies
½
y
d2 dx 2
q V Ž x . q 2 lU Ž x . y l2
s 2 ln
ny 1
5
n l n
y Ž x, l . q n Ž n y 1 .
l ny 1
y 2 nU Ž x .
ny 1 l ny 1
y Ž x, l .
ny 2 l ny2
y Ž x, l .
y Ž x, l .
Ž 4.8.
. and f Ž x, lq . Ž . Writing Ž4.8. for w Ž x, lq i i , and using 4.7 , we find
½
y
d2 dx
2
q q V Ž x . q 2 lq i U Ž x . y Ž li .
2
5
Fn 0q1 Ž x, lq i . s 0,
where Fn 0q1 Ž x, lq i . s
½
n 0q1 l n 0q1
w Ž x, l .
n 0q1
y
Ý ks1
ž
5 /
ls l q i
k n0 q 1 a n 0q1yk Ž lq f Ž x, l . . i l k k
½
5
ls l q i
.
From Ž4.1. we have W f Ž x,
lq i
. , Fn q1 Ž x, 0
lq i
. s
½
n 0q1 l n 0q1
W f Ž x, l . , w Ž x, l .
5
ls l q i
s 0.
. such that Hence there exists a constant a n 0q1Ž lq i q q Fn 0q1 Ž x, lq i . s a n 0 q1 Ž l i . f Ž x, l i . .
This shows that Ž4.5. holds for n s n 0 q 1. In a similar way, we can prove that Ž4.6. also holds.
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315
THEOREM 4.2.
½
n l n
w Ž ?, l .
5
ls l q i
n s 0, 1, . . . , mq i y 1,
g L2 Ž Rq . ,
i s 1, 2, . . . , j,
½
n ln
w Ž ?, l .
5
ls m y i
Ž 4.9.
n s 0, 1, . . . , my l y 1,
g L2 Ž Rq . ,
l s 1, 2, . . . , k.
Ž 4.10.
Proof. From Ž2.3. and Ž3.12. we obtain
½
n l
n
f Ž x, l .
5
ls l q i
F x n exp
½
`
H0
q c exp y
ž
e 2
'x
5
exp yxIm lq i 4
`
t n exp ytIm lq i 4 dt
/H
0
Ž4.9. using Ž4.5.. for n s 0, 1, . . . , mq i y 1, i s 1, 2, . . . , j, which gives Equation Ž4.10. may be derived analogously.
w Ž x,
lq i
w Ž x,
my l
.,
½
l
q
w Ž x, l .
5
,...,
ls l q i
½
mi l
y1
mq i y1
w Ž x, l .
5
ls lq i
w Ž x, l .
5
ls my l
and
.,
½
l
y
w Ž x, l .
5
ls m y l
,...,
½
ml l
y1
my l y1
are called the principal functions corresponding to eigenvalues l s lq i , Ž . i s 1, 2, . . . , j, and l s my , l s 1, 2, . . . , k of L l , respectively. l If l1 , . . . , l p and m 1 , . . . , m q are spectral singularities of LŽ l. Ži.e., the real zeros of F and G ., we then obtain
½
n l
W f Ž x, l . , w Ž x, l . n
5
ls l i
s
½
dn
F Ž l.
5
ls l i
G Ž l.
5
ls m l
d ln
s0
for n s 0, 1, . . . , m i y 1, i s 1, 2, . . . , p, and
½
n ln
W g Ž x, l . , w Ž x, l .
5
ls m l
s
½
dn d ln
s 0,
for n s 0, 1, . . . , n l y 1, l s 1, 2, . . . , q. In a way similar to that of the proof of Theorem 4.1, we have the following.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
316
Remark 4.3. The formulas
n
½
l
5
w Ž x, l . n
n ls l i
s
Ý ks0
k n d ny k Ž l i . f Ž x, l . k l k
½
ž /
5
Ž 4.11. ls l i
for n s 0, 1, . . . , m i y 1, i s 1, 2, . . . , p and
½
n
w Ž x, l . ln
5
ls m l
n
s
Ý ks0
j n hnyj Ž m l . g Ž x, l . j l j
½
ž/
5
, Ž 4.12. ls m l
for n s 0, 1, . . . , n l y 1, l s 1, 2, . . . , q hold. LEMMA 4.4.
½
n l n
w Ž ?, l .
5
ls l i
f L2 Ž Rq . ,
n s 0, 1, . . . ,
m i y 1,
i s 1, 2, . . . , p,
Ž 4.13.
and
½
n ln
w Ž ?, l .
5
ls m l
f L2 Ž Rq . ,
n s 0, 1, . . . ,
n l y 1,
l s 1, 2, . . . , q.
Ž 4.14.
The proof of this lemma may easily be obtained from Ž2.3., Ž2.4., Ž4.11., and Ž4.12.. Now let us introduce the Hilbert spaces:
½ ½
`
Hn s f :
H0
Hyn s g :
H0
`
2n Ž 1 q x . < f Ž x . < 2 dx - ` ,
5
Ž1 q x.
y2 n
n s 0, 1, . . .
< g Ž x . < 2 dx - ` ,
5
n s 0, 1, . . .
with 5 f 5 2n s
`
H0
2n Ž 1 q x . < f Ž x . < 2 dx ;
2 5 g 5yn s
`
H0
Ž1 q x.
y2 n
< g Ž x . < 2 dx,
respectively. It is evident that H0 s L2 Ž Rq . , Hn m L2 Ž Rq . m Hyn , Obviously Hyn is isomorphic to the dual of Hn .
n s 1, 2, . . . .
QUADRATIC PENCIL OF SCHRODINGER OPERATOR ¨
317
THEOREM 4.5.
½
n l n
5
w Ž ?, l .
ls l i
g HyŽ nq1. ,
n s 0, 1, . . . ,
m i y 1,
i s 1, 2, . . . , p,
Ž 4.15.
and
½
n ln
w Ž ?, l .
5
ls m l
n s 0, 1, . . . ,
g HyŽ nq1. ,
n l y 1,
l s 1, 2, . . . , q.
Ž 4.16.
Proof. From Ž2.3. we obtain
½
n l
n
f Ž x, l .
5
ls l i
½
`
F x n exp y
H0 ImU Ž t .
5
q
`
Hx < t <
n
< A Ž x, t . < dt.
By the definition of the space HyŽ nq1. and using Ž3.12. and Ž4.11., we arrive at Ž4.15.. In the same manner, we can show that Ž4.16. also holds.
w Ž x, l i . ,
½
l
w Ž x, l .
5
w Ž x, l .
5
ls l i
,...,
½
m iy1 l m iy1
w Ž x, l .
5
ls l i
w Ž x, l .
5
ls m l
and
w Ž x, m l . ,
½
l
ls m l
,...,
½
n ly1 l n ly1
are called the principal functions corresponding to the spectral singularities l s l i , i s 1, 2, . . . , p, and l s m l , l s 1, 2, . . . , q of LŽ l., respectively. Let us choose n 0 so that n 0 s max m1 , . . . , m p , n1 , . . . , n q 4 . Then Hn 0q1 m L2 Ž Rq . m HyŽ n 0q1. . By Theorem 4.5 we have the following.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
318 Remark 4.6.
½
n l n
w Ž ?, l .
5
ls l i
g HyŽ n 0q1. ,
n s 0, 1, . . . ,
m i y 1, i s 1, 2, . . . , p,
½
n
ln
w Ž ?, l .
5
ls m l
g HyŽ n 0q1. ,
n s 0, 1, . . . ,
n l y 1, l s 1, 2, . . . , q.
In a forthcoming article we will show that the spectral expansion of LŽ l. converges in the sense of HyŽ n 0q1. for every function in Hn 0q1. 5. KLEIN]GORDON EQUATION As we have pointed out previously, if we choose V Ž x . s yU 2 Ž x ., then the boundary value problem Ž1.1]1.2. is reduced to y0 q l y U Ž x .
2
y s 0,
x g Rq
y Ž 0 . s 0,
Ž 5.1. Ž 5.2.
where U is a complex-valued and absolutely continuous function in each finite subinterval of Rq. Equation Ž5.1. is called the Klein]Gordon s-wave equation for a particle of zero mass with static potential UŽ x .. It is trivial that the condition Ž3.10. assumes the form sup
exp Ž e'x .
e)0
0Fx-`
lim U Ž x . s 0
xª`
Ž 5.3.
for Ž5.1.. We can deduce the following from Theorem 3.10. THEOREM 5.1. The boundary ¨ alue problem Ž5.1., Ž5.2. has a finite number of eigen¨ alues and spectral singularities, and each of them is of finite multiplicity if the conditions Ž5.3. hold. Let us also note that if the function U is real and analytic and vanishes rapidly as x ª `, then the eigenvalues of Ž5.1., Ž5.2. have been discussed previously w2x.
QUADRATIC PENCIL OF SCHRODINGER OPERATOR ¨
319
¨ 6. ONE-DIMENSIONAL SCHRODINGER OPERATOR If UŽ x . ' 0 in Ž1.1., the operator LŽ l. is reduced to L, which was given in the Introduction. In this case the condition Ž3.10. will assume the form
exp Ž e'x . < V Ž x . < 4 - `,
sup
e ) 0.
Ž 6.1.
0Fx-`
Hence from Theorem 3.10 we get the following. THEOREM 6.1. Under the condition (6.1), the operator L has a finite number of eigen¨ alues and spectral singularities, and each of them is of finite multiplicity. The same result has been obtained by Lyance w9x and Naimark w12x under the stronger assumption `
H0
V Ž x . exp Ž e x . dx - `,
e ) 0.
The condition Ž6.1. has been obtained in Pavlov w15x.
ACKNOWLEDGMENT E. Bairamov would like to thank the Scientific and Technical Research Council of Turkey for financial support.
REFERENCES 1. H. Cornille, Existence and uniqueness of crossing symmetric NrD-type equations corresponding to the Klein]Gordon equation, J. Math. Phys. 11 Ž1970., 79]98. 2. A. Degasperis, On the inverse problem for the Klein]Gordon s-wave equation, J. Math. Phys. 11 Ž1970., 551]567. 3. M. G. Gasymov, Expansion in terms of a scattering theory problem for the non-selfadjoint Schrodinger equation, So¨ . Math. Dokl. 9 Ž1968., 390]393. ¨ 4. M. G. Gasymov and F. G. Maksudov, The principal part of the resolvent of non-selfadjoint operators in neighbourhood of spectral singularities, Funct. Anal. Appl. 6 Ž1972., 185]192. 5. S. V. Hruscev, Spectral singularities of dissipative Schrodinger operator with rapidly ¨ decreasing potential, Indiana Uni¨ . Math. J. 33 Ž1984., 313]338. 6. M. Jaulent, On an inverse scattering problem with an energy-dependent potential, Ann. Inst. Henri Poincare´ Sec. A 17 Ž1972., 363]378. 7. M. Jaulent and C. Jean, The inverse s-wave scattering problem for a class of potentials depending on energy, Comm. Math. Phys. 28 Ž1972., 177]220. 8. M. Jaulent and C. Jean, The inverse problem for the one dimensional Schrodinger ¨ equation with an energy-dependent potential I, II, Ann. Inst. Henri Poincare´ Sec. A 25 Ž1976., 105]118, 119]137.
320
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
9. V. E. Lyance, A differential operator with spectral singularities I, II, AMS Transl. 60 Ž1967., 185]225, 227]283. 10. F. G. Maksudov and B. P. Allakhverdiev, Spectral analysis of a new class of non-selfadjoint operators with continuous and point spectrum, So¨ . Math. Dokl. 30 Ž1984., 566]569. 11. F. G. Maksudov and G. Sh. Guseinov, On solution of the inverse scattering problem for a quadratic pencil of one-dimensional Schrodinger operators on the whole axis, So¨ . Math. ¨ Dokl. 34 Ž1987., 34]38. 12. M. A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of the second order on a semi-axis, AMS Transl. 16 Ž1960., 103]193. 13. M. A. Naimark, ‘‘Linear Differential Operators I, II,’’ Ungar, New York, 1968. 14. B. S. Pavlov, On the non-self-adjoint Schrodinger operator, in ‘‘Topics in Mathematics ¨ and Physics,’’ Vol. 1, pp. 87]114, Consultant Bureau, New York, 1967. 15. B. S. Pavlov, On separation conditions for spectral components of a dissipative operator, Math. USSR Iz¨ estiya 9, Ž1975., 113]137. 16. I. I. Privalov, The boundary properties of analytic functions, in ‘‘Hochshulbucher fur ¨ ¨ Math.,’’ Vol. 25, VEB Deutscher Verlag, Berlin, 1956. 17. J. T. Schwartz, Some non-self-adjoint operators, Comm. Pure Appl. Math. 13 Ž1960., 609]639.
216, 303]320 Ž1997.
AY975689
Quadratic Pencil of Schrodinger Operators ¨ with Spectral Singularities: Discrete Spectrum and Principal Functions
¨ Elgiz Bairamov and Oner C ¸akar Ankara Uni¨ ersity, Mathematics Department, 06100 Bese¨ ler, Ankara, Turkey
˙
and A. Okay C ¸elebi Department of Mathematics, Middle East Technical Uni¨ ersity, Ankara 06531, Turkey Received September 11, 1996
In this article we investigated the spectrum of the quadratic pencil of Schro¨ dinger operators LŽ l. generated in L2 ŽRq . by the equation yy0 q w V Ž x . q 2 lU Ž x . y l2 x y s 0,
x g Rqs w 0, ` .
and the boundary condition y Ž 0 . s 0, where U, V are complex valued functions and U is absolutely continuous in each finite subinterval of Rq. Discussing the spectrum, we proved that LŽ l. has a finite number of eigenvalues and spectral singularities with finite multiplicities, if the conditions sup
exp Ž e'x . w < V Ž x . < q
e)0
0Fx-`
and lim U Ž x . s 0
xª`
hold. It is shown that the principal functions corresponding to eigenvalues of LŽ l. are in the space L2 ŽRq ., and the principal functions corresponding to spectral singularities are in another Hilbert space, which contains L2 ŽRq .. Some results about the spectrum of LŽ l. have also been applied to radial Klein]Gordon and one-dimensional Schrodinger equations. Q 1997 Academic Press ¨ 303 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
304
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
1. INTRODUCTION Let us consider the non-self-adjoint one-dimensional Schrodinger opera¨ tor L generated in L2 ŽRq . by the differential expression l Ž y . ' yy0 q V Ž x . y,
x g Rq
and the boundary condition y Ž0. s 0, where V is a complex valued function. The spectral analysis of L has been investigated by Naimark w12x. In this article, he has proved that some of the poles of the resolvent’s kernel of L are not the eigenvalues of the operator. He has also shown that those poles Žwhich are called spectral singularities by Schwartz w17x. are on the continuous spectrum. Moreover, he has shown that the spectral singularities play an important role in the discussion of the spectral analysis of L, and if the condition `
H0 < V Ž x .
e)0
holds, then the eigenvalues and the spectral singularities are of a finite number and each of them is of a finite multiplicity. The effect of spectral singularities in the spectral expansion of the operator L in terms of principal functions has been investigated in w9x. In w14x, the dependence of the structure of spectral singularities of L on the behavior of V at infinity has been considered. Some problems related to spectral analysis of differential and some other types of operators with spectral singularities have been discussed in w3]5, 10, 13, 15x. Let us consider the operator LŽ l. generated in L2 ŽRq . by the equation yy0 q V Ž x . q 2 lU Ž x . y l2 y s 0,
x g Rq ,
Ž 1.1.
and the boundary condition y Ž 0 . s 0,
Ž 1.2.
where U, V are complex-valued functions and U is absolutely continuous in each finite subinterval of Rq. If U ' 0, then the operator LŽ l. reduces to the operator L. If we choose V Ž x . s yU 2 Ž x ., then Ž1.1. will be reduced into the radial form of the Klein]Gordon equation. Some problems of the spectral theory of Ž1.1. and of the Klein]Gordon equation have been investigated by several authors w1, 2, 6, 7, 8, 11x with real functions U and V. We also want to note that, the finiteness of the number of the eigenvalues of LŽ l. has been given by the following technique in w7x. First it is proved that the set of eigenvalues of LŽ l. is bounded in the complex plane and is of countable, and their limit points lie on the real axis. Later, assuming that there are no limit points of eigenvalues, it has been
QUADRATIC PENCIL OF SCHRODINGER OPERATOR ¨
305
determined that the eigenvalues are of finite number. But in w7x the functions U and V for which the set of eigenvalues has no limit points have not been investigated. Using the technique of the uniqueness of analytic functions, we proved that the eigenvalues and the spectral singularities are of finite number. In this paper, we discussed the spectrum of LŽ l. defined by Ž1.1. and Ž1.2., and proved that this operator has a finite number of eigenvalues and spectral singularities and that each of them is of a finite multiplicity, under the conditions sup
exp Ž e'x .
< V Ž x . < q < U9 Ž x . <
4 - `,
e ) 0,
0Fx-`
and lim U Ž x . s 0.
xª`
Afterward, the properties of the principal functions corresponding to the eigenvalues and the spectral singularities of LŽ l. are obtained. If U is absolutely continuous in every finite subinterval of Rq and satisfies sup
exp Ž e'x .
e)0
0Fx-`
and lim U Ž x . s 0,
xª`
then the eigenvalues and the spectral singularities of the Klein]Gordon equation have similar properties. In the particular case L of the operator LŽ l., the results we have obtained about the spectrum are better than the ones given by Lyance w9x and Naimark w12x, and are the same as the ones obtained by Pavlov w15x. In the sequel, we use the notations Cqs l < l g C, Im l ) 0 4 , R* s R _ 0 4 ,
Cys l < l g C, Im l - 0 4
Cqs l < l g C, Im l G 0 4 ,
Cys l < l g C, Im l F 0 4 Furthermore, sp Ž LŽ l.., ss s Ž LŽ l.., r Ž LŽ l.., and RŽ LŽ l.. will denote the eigenvalues, the spectral singularities, the resolvent set, and the resolvent of the operator LŽ l., respectively.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
306
2. PRELIMINARIES Let us suppose that the functions U and V satisfy the following conditions: `
`
H0
H0 x
< V Ž x . < q < U9 Ž x . < dx - `.
Ž 2.1.
Let us define the functions v , a , and b by `
vŽ x. s
Hx U Ž t . dt
a Ž x. s
Hx
b Ž x. s
Hx
`
`
< V Ž t . < q < U9 Ž t . < dt
Ž 2.2.
t < V Ž t . < q 2 < U Ž t . < dt
Jaulent and Jean w7x have obtained the following important result: if Ž2.1. holds, then Ž1.1. has the solution f Ž x, l . s e i v Ž x . e i l x q
`
Hx
A Ž x, t . e i l t dt
Ž 2.3.
for l g Cq and the solution g Ž x, l . s eyi v Ž x . eyi l x q
`
yi l t
Hx B Ž x, t . e
dt
Ž 2.4.
for l g Cy; moreover, the kernels AŽ x, t . and B Ž x, t . satisfy the inequality 1 xqt < A Ž x, t . < , < B Ž x, t . < F a exp b Ž x . 4 . Ž 2.5. 2 2 Therefore, the solutions f Ž x, l. and g Ž x, l. are analytic with respect to l, in Cq and Cy, respectively, and are continuous up to the real axis. f Ž x, l. and g Ž x, l. also satisfy the following asymptotic equalities w7x:
ž
/
f Ž x, l . s e i l x 1 q o Ž 1 . ,
l g Cq ,
xª`
f x Ž x, l . s e i l x i l q o Ž 1 . ,
l g Cq ,
xª`
g Ž x, l . s eyi l x 1 q o Ž 1 . ,
l g Cy ,
xª`
l g Cy ,
xª`
f Ž x, l . s e i v Ž x . e i l x q o Ž 1 . ,
l g Cq ,
< l< ª `
g Ž x, l . s eyi v Ž x . eyi l x q o Ž 1 . ,
l g Cy ,
< l< ª `
g x Ž x, l . s eyi l x yi l q o Ž 1 . ,
Ž 2.6. Ž 2.7.
From Ž2.3. and Ž2.4. we easily find
Ž 2.8.
QUADRATIC PENCIL OF SCHRODINGER OPERATOR ¨
307
According to Ž2.6. and Ž2.7., the Wronskian of the solutions f Ž x, l. and g Ž x, l. is W w f , g x s lim W w f , g x s y2 i l xª`
Ž 2.9.
for l g R. So f Ž x, l. and g Ž x, l. provide the fundamental solutions of the equation Ž1.1. for l g R*. Let w Ž x, l. be the solution of Ž1.1. satisfying the initial conditions
w Ž 0, l . s 0,
w x Ž 0, l . s 1.
It is obvious that the solution w Ž x, l. exists, is unique, and is an entire function of l w7x.
3. EIGENVALUES AND SPECTRAL SINGULARITIES Let us define
r 1 Ž l . s l < l g Cq , F Ž l . / 0 4 ,
r 2 Ž l . s l < l g Cy , G Ž l . / 0 4
where F Ž l . [ f Ž 0, l . ,
G Ž l . [ g Ž 0, l . .
Using the standard techniques w13x, we can show that
r Ž L Ž l. . s r 1Ž l. j r 2 Ž l. and for l g r Ž LŽ l.. the resolvent of LŽ l. is the integral operator defined as R Ž L Ž l. . c Ž x . s
`
H0 R Ž x, t ; l. c Ž t . dt
for c g L2 ŽRq ., where the kernel RŽ x, t; l. Ži.e., the Green’s function for LŽ l.. is given by R Ž x, t ; l . s
½
R1 Ž x, t ; l . ,
l g r 1Ž l.
R 2 Ž x, t ; l . ,
l g r 2 Ž l.
Ž 3.1.
in which R1 Ž x, t ; l . s R 2 Ž x, t ; l . s
1 F Ž l. 1 G Ž l.
½ ½
f Ž x, l . w Ž t , l . f Ž t , l . w Ž x, l .
0Ft-x xFt-`
Ž 3.2.
g Ž x, l . w Ž t , l . g Ž t , l . w Ž x, l .
0Ft-x x F t - `.
Ž 3.3.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
308
From the asymptotic equalities Ž2.6. and Ž2.7., we have f Ž ?, l . , f x Ž ?, l . g L2 Ž Rq .
Ž 3.4.
g Ž ?, l . , g x Ž ?, l . g L2 Ž Rq .
Ž 3.5.
for each l g Cq and for each l g Cy. LEMMA 3.1. Ža. sp Ž LŽ l.. s l < l g Cq, F Ž l. s 04 j l < l g Cy, GŽ l. s 04 , Žb. ss s Ž LŽ l.. s l < l g R*, F Ž l. s 04 j l < l g R*, GŽ l. s 04 . Proof. Ža. Obviously,
l < l g Cq , F Ž l . s 0 4 j l < l g Cy , G Ž l . s 0 4 ; sp Ž L Ž l . . . On the other hand, let us suppose that l0 g sp Ž LŽ l.. and examine the following cases: Ž1. Let l0 g Cq. Since l0 g sp Ž LŽ l.., then Ž1.1. has a solution y Ž x, l 0 . in L2 ŽRq . for l s l 0 that is nontrivial and y Ž0, l0 . s 0. Since W y Ž x, l 0 . , w Ž x, l0 . s 0, there exists a constant c / 0 such that y Ž x, l0 . s c w Ž x, l 0 .. Then we have W f Ž x, l0 . , y Ž x, l 0 . s cF Ž l0 . .
Ž 3.6.
But it is evident from Ž3.4. that W f Ž x, l0 . , y Ž x, l0 . s lim W f Ž x, l0 . , y Ž x, l 0 . s 0. xª`
From Ž3.6. and Ž3.7. we find F Ž l 0 . s 0. Ž2. Let l0 g Cy. In a similar way, we can prove that G Ž l0 . s 0. Ž3. Let l0 g R. In this case the general solution of Ž1.1. is y Ž x, l0 . s c1 f Ž x, l0 . q c 2 g Ž x, l 0 . for l s l0 . From Ž2.6. and Ž2.7. we get y Ž x, l0 . s c1 e i l 0 x q c 2 eyi l 0 x q o Ž 1 .
as x ª `.
Ž 3.7.
QUADRATIC PENCIL OF SCHRODINGER OPERATOR ¨
309
That is, y Ž?, l0 . f L2 ŽRq .. Hence
sp Ž L Ž l . . l R s f . Combining Ž1., Ž2., and Ž3., we find
sp Ž L Ž l . . ; l < l g Cq , F Ž l . s 0 4 j l < l g Cy , G Ž l . s 0 4 . This completes the proof of Ža.. Žb. Spectral singularities are the poles of the kernel of the resolvent, but not the eigenvalues of the operator LŽ l.. From Ž3.1. ] Ž3.3. and part Ža. we obtain that the spectral singularities of LŽ l. are the zeros of F and G on the real axis. We can easily show that 0 f ss s Ž LŽ l.., which completes the proof of part Žb.. Note that from W f Ž x, l . , g Ž x, l . s F Ž l . g x Ž 0, l . y G Ž l . f x Ž 0, l . s y2 i l ,
l g R, we immediately get
l < l g R*, F Ž l . s 0 4 l l < l g R*, G Ž l . s 0 4 s f .
Ž 3.8.
To investigate the quantitative properties of the eigenvalues and the spectral singularities of LŽ l., we need to discuss the quantitative properties of the zeros of F and G in Cq and Cy, respectively. For the sake of simplicity, we will consider only the zeros of F in Cq. A similar procedure may also be employed for the zeros of G in Cy. Let us define Q1 s l < l g Cq , F Ž l . s 0 4 ,
Q2 s l < l g R, F Ž l . s 0 4 .
LEMMA 3.2. Ža. The set Q1 is bounded and has at most a countable number of elements, and its limit points can lie only in a bounded subinter¨ al of the real axis. Žb. The set Q2 is compact and its linear Lebesque measure is zero. Proof. From Ž2.8. we get F Ž l . s e i v Ž0. q o Ž 1 . , l g Cq , < l < ª `.
Ž 3.9.
The asymptotic equality Ž3.9. shows the boundedness of the sets Q1 and Q2 . From the analyticity of the function F in Cq, we get that Q1 has at most countable numbers of elements. By the uniqueness of analytic functions we find that the limit points of Q1 can lie only in a bounded
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
310
subinterval of the real axis. The closedness and the property of having linear Lebesgue measure zero of the set Q2 can be obtained from the uniqueness theorem of the analytic functions w16x. From Lemmas 3.1 and 3.2 we get the following. THEOREM 3.3. The sets of eigen¨ alues and spectral singularities of LŽ l. are bounded and are, at most, countable, and their limit points can lie only in a bounded subinter¨ al of the real axis if the condition Ž2.1. holds. Note. For the moment we want to start with ‘‘ Assumption III: F and G has no real zeros’’ in w7x, that is,
ss s Ž L Ž l . . s f . In this case, from Lemma 3.2 and Theorem 3.3 we may derive that the number of eigenvalues is finite, which corresponds to the technique given in w7x. But it has not been clarified for which functions U and V Assumption III holds. So Assumption III does not seem to be natural. Hence we will prove by another method that the number of the eigenvalues and the spectral singularities of LŽ l. are finite, without employing Assumption III. Up to now we have assumed that condition Ž2.1. holds. In the rest of the article the assumptions we will use are lim U Ž x . s 0,
xª`
sup
exp Ž e'x .
< V Ž x . < q < U9 Ž x . <
e ) 0.
4 - `,
0Fx-`
Ž 3.10. From Ž2.2. and Ž3.10. we have
a Ž x . F c 0 exp y
ž
e 2
'x
/
,
b Ž x . F c 0 exp y
ž
e 2
'x
/
,
Ž 3.11.
where c 0 ) 0 is constant. By Ž2.5. and Ž3.11. we get
e
xqt
2
2
ž ( /
< A Ž x, t . < , < B Ž x, t . < F c exp y
,
Ž 3.12.
where c ) 0. We have seen that whenever condition Ž2.1. holds, the function F is analytic in Cq and continuous up to the real axis. It is obvious from Ž3.12. that if the conditions Ž3.10. hold, then all derivatives of F are continuous up to the real axis. So the inequalities < F Ž r . Ž l . < F Mr ,
l g Cq ,
r s 0, 1, 2, . . .
Ž 3.13.
QUADRATIC PENCIL OF SCHRODINGER OPERATOR ¨
311
hold, where `
Mr s 2 rq1 c
H0
t r exp y
½
H0
ž
e 2
't
/
r s 1, 2, . . .
dt,
Ž 3.14.
and `
M0 s exp y
5
ImU Ž t . dt q
`
H0
< A Ž 0, t . < dt,
provided that c ) 0 is a constant. Let us indicate the set of all limit points of Q1 and Q2 by Q3 and Q4 , respectively, and the set of all zeros of F with infinite multiplicity in Cq by Q5 . From the uniqueness theorem of analytic functions, it is obvious that Q3 ; Q2 , LEMMA 3.4.
Q4 ; Q2 ,
Q5 ; Q 2 .
Q 3 ; Q5 , Q 4 ; Q5 .
The proof of this lemma can be obtained by use of the continuity of all derivatives of F up to the real axis. We will use the following uniqueness theorem for the analytic functions on the upper half-plane to prove the next result. THEOREM 3.5 Žw14x.. Let us assume that the function g is analytic in Cq, all of its deri¨ ati¨ es are continuous up to the real axis, and there exist N ) 0 such that < g Ž r . Ž l . < F Mr ,
r s 0, 1, 2, . . . ,
l g Cq ,
< l < - 2 N, Ž 3.15.
and yN
Hy`
ln < g Ž x . < 1qx
2
dx - `,
`
HN
ln < g Ž x . < 1 q x2
dx - `
Ž 3.16.
hold. If the set P with linear Lebesque measure zero is the set of all zeros of the function g with infinity multiplicity and if h
H0
ln E Ž s . d m Ž Ps . s y`
holds, then g Ž l. ' 0, where EŽ s . s inf r Mr s rrr!, and r s 0, 1, 2, . . . , m Ž Ps . is the Lebesque measure of the s-neighborhood of P and h is an arbitrary positi¨ e constant.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
312 LEMMA 3.6.
Q5 s f .
Proof. It is trivial from Lemma 3.2 and Ž3.13. that F satisfies the conditions Ž3.15. and Ž3.16.. Since the function F is not identically equal to zero, then by Theorem 3.5, Q5 satisfies the condition h
H0
ln E Ž s . d m Ž Q5, s . ) y`,
Ž 3.17.
where EŽ s . s inf r Mr s rrr!, m Ž Q5, s . is the Lebesque measure of the sneighborhood of Q5 , and the constant Mr is defined by Ž3.14.. Now we will obtain the following estimates for Mr : Mr s 2 rq1 c
`
H0
t r exp y
ž
e 2
't
/
dt
F Bb r r r r !,
Ž 3.18.
where B and b are constants depending on c and e . Substituting Ž3.18. in the definition of EŽ s ., we arrive at E Ž s . s inf r
Mr s r r!
F B inf b r s r r r 4 F B exp yby1 ey1 sy1 4 , r
or by Ž3.17., h
H0
1 s
d m Ž Q5, s . - `.
Ž 3.19.
The inequality Ž3.19. holds for an arbitrary s, if and only if m Ž Q5, s . s 0 or Q5 s f . LEMMA 3.7. The function F has a finite number of zeros with finite multiplicity in Cq. Proof. Lemmas 3.4 and 3.6 give Q3 s Q4 s f . So the bounded sets Q1 and Q2 have no limit points Žsee Lemma 3.2., i.e., the function F has only a finite number of zeros in Cq. Since Q5 s f , these zeros are of finite multiplicity. The discussions given above for F may be repeated for G. Hence we obtain the following. LEMMA 3.8. The function G has a finite number of zeros with finite multiplicity in Cy. DEFINITION 3.9. The multiplicity of a zero of F Žor G . in Cq Žor Cy . is called the multiplicity of the corresponding eigenvalue or spectral singularity of LŽ l..
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313
Lemmas 3.1, 3.7, and 3.8 give the following. THEOREM 3.10. The operator LŽ l. has a finite number of eigen¨ alues and spectral singularities, and each of them is of finite multiplicity if the condition Ž3.10. holds.
4. PRINCIPAL FUNCTIONS In this section we assume that Ž3.10. holds. Let l1q , . . . , lq and j F in Cq and G in Cy with multiplicities respectively. Similarly, let l1 , . . . , l p and m 1 , . . . , m q be the zeros of F and G on the real axis with multiplicities m1 , . . . , m p and n1 , . . . , n q , respectively. It is trivial that y my 1 , . . . , m k denote the zeros of q y m1 , . . . , mq and my j 1 , . . . , mk ,
½
n l
5
W f Ž x, l . , w Ž x, l . n
ls l q i
s
½
dn d ln
F Ž l.
5
ls lq i
G Ž l.
5
ls my l
s0
Ž 4.1.
s0
Ž 4.2.
for n s 0, 1, . . . , mq i y 1, i s 1, 2, . . . , j, and
½
n
5
W g Ž x, l . , w Ž x, l . ln
ls m y l
s
½
dn d ln
for n s 0, 1, . . . , my l y 1, l s 1, 2, . . . , k. If n s n s 0, we get q q w Ž x, lq i . s a 0 Ž l i . f Ž x, l i . , y y w Ž x, my l . s b 0 Ž m l . g Ž x, m l . ,
i s 1, 2, . . . , j,
Ž 4.3.
l s 1, 2, . . . , k.
Ž 4.4.
. / 0, b 0 Ž my . / 0. So a0 Ž lq i l THEOREM 4.1. The formulas
½
n l
w Ž x, l . n
5
n ls l q i
s
Ý ks0
k n a ny k f Ž x, l . k l k
ž / ½
5
ls lq i
Ž 4.5.
for n s 0, 1, . . . , mq i y 1, i s 1, 2, . . . , j, and
½
n
w Ž x, l . ln
5
ls m y i
s
n
Ý js0
j n bnyj g Ž x, l . j l j
ž/ ½
5
ls my i
,
Ž 4.6.
for n s 0, 1, . . . , my l y 1, l s 1, 2, . . . , k hold, where the constants y a0 , a1 , . . . , a n and b 0 , b1 , . . . , bn depend on lq i and m l , respecti¨ ely.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
314
Proof. Let us start with equality Ž4.5.. We will utilize mathematical induction. For n s 0, the proof is trivial from Ž4.3.. Let us assume that for Ž . 1 F n 0 F mq i y 2 the equality 4.5 holds, i.e.,
½
n0 l n 0
w Ž x, l .
5
n0 ls l q i
s
Ý ks0
k n0 q a n 0yk Ž l i . f Ž x, l . l k k
½
ž /
5
ls lq i
Ž 4.7.
Now we will prove that Ž4.5. also holds for n 0 q 1. If y Ž x, l. is a solution of the equation Ž1.1., then nrl n y Ž x, l. satisfies
½
y
d2 dx 2
q V Ž x . q 2 lU Ž x . y l2
s 2 ln
ny 1
5
n l n
y Ž x, l . q n Ž n y 1 .
l ny 1
y 2 nU Ž x .
ny 1 l ny 1
y Ž x, l .
ny 2 l ny2
y Ž x, l .
y Ž x, l .
Ž 4.8.
. and f Ž x, lq . Ž . Writing Ž4.8. for w Ž x, lq i i , and using 4.7 , we find
½
y
d2 dx
2
q q V Ž x . q 2 lq i U Ž x . y Ž li .
2
5
Fn 0q1 Ž x, lq i . s 0,
where Fn 0q1 Ž x, lq i . s
½
n 0q1 l n 0q1
w Ž x, l .
n 0q1
y
Ý ks1
ž
5 /
ls l q i
k n0 q 1 a n 0q1yk Ž lq f Ž x, l . . i l k k
½
5
ls l q i
.
From Ž4.1. we have W f Ž x,
lq i
. , Fn q1 Ž x, 0
lq i
. s
½
n 0q1 l n 0q1
W f Ž x, l . , w Ž x, l .
5
ls l q i
s 0.
. such that Hence there exists a constant a n 0q1Ž lq i q q Fn 0q1 Ž x, lq i . s a n 0 q1 Ž l i . f Ž x, l i . .
This shows that Ž4.5. holds for n s n 0 q 1. In a similar way, we can prove that Ž4.6. also holds.
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315
THEOREM 4.2.
½
n l n
w Ž ?, l .
5
ls l q i
n s 0, 1, . . . , mq i y 1,
g L2 Ž Rq . ,
i s 1, 2, . . . , j,
½
n ln
w Ž ?, l .
5
ls m y i
Ž 4.9.
n s 0, 1, . . . , my l y 1,
g L2 Ž Rq . ,
l s 1, 2, . . . , k.
Ž 4.10.
Proof. From Ž2.3. and Ž3.12. we obtain
½
n l
n
f Ž x, l .
5
ls l q i
F x n exp
½
`
H0
q c exp y
ž
e 2
'x
5
exp yxIm lq i 4
`
t n exp ytIm lq i 4 dt
/H
0
Ž4.9. using Ž4.5.. for n s 0, 1, . . . , mq i y 1, i s 1, 2, . . . , j, which gives Equation Ž4.10. may be derived analogously.
w Ž x,
lq i
w Ž x,
my l
.,
½
l
q
w Ž x, l .
5
,...,
ls l q i
½
mi l
y1
mq i y1
w Ž x, l .
5
ls lq i
w Ž x, l .
5
ls my l
and
.,
½
l
y
w Ž x, l .
5
ls m y l
,...,
½
ml l
y1
my l y1
are called the principal functions corresponding to eigenvalues l s lq i , Ž . i s 1, 2, . . . , j, and l s my , l s 1, 2, . . . , k of L l , respectively. l If l1 , . . . , l p and m 1 , . . . , m q are spectral singularities of LŽ l. Ži.e., the real zeros of F and G ., we then obtain
½
n l
W f Ž x, l . , w Ž x, l . n
5
ls l i
s
½
dn
F Ž l.
5
ls l i
G Ž l.
5
ls m l
d ln
s0
for n s 0, 1, . . . , m i y 1, i s 1, 2, . . . , p, and
½
n ln
W g Ž x, l . , w Ž x, l .
5
ls m l
s
½
dn d ln
s 0,
for n s 0, 1, . . . , n l y 1, l s 1, 2, . . . , q. In a way similar to that of the proof of Theorem 4.1, we have the following.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
316
Remark 4.3. The formulas
n
½
l
5
w Ž x, l . n
n ls l i
s
Ý ks0
k n d ny k Ž l i . f Ž x, l . k l k
½
ž /
5
Ž 4.11. ls l i
for n s 0, 1, . . . , m i y 1, i s 1, 2, . . . , p and
½
n
w Ž x, l . ln
5
ls m l
n
s
Ý ks0
j n hnyj Ž m l . g Ž x, l . j l j
½
ž/
5
, Ž 4.12. ls m l
for n s 0, 1, . . . , n l y 1, l s 1, 2, . . . , q hold. LEMMA 4.4.
½
n l n
w Ž ?, l .
5
ls l i
f L2 Ž Rq . ,
n s 0, 1, . . . ,
m i y 1,
i s 1, 2, . . . , p,
Ž 4.13.
and
½
n ln
w Ž ?, l .
5
ls m l
f L2 Ž Rq . ,
n s 0, 1, . . . ,
n l y 1,
l s 1, 2, . . . , q.
Ž 4.14.
The proof of this lemma may easily be obtained from Ž2.3., Ž2.4., Ž4.11., and Ž4.12.. Now let us introduce the Hilbert spaces:
½ ½
`
Hn s f :
H0
Hyn s g :
H0
`
2n Ž 1 q x . < f Ž x . < 2 dx - ` ,
5
Ž1 q x.
y2 n
n s 0, 1, . . .
< g Ž x . < 2 dx - ` ,
5
n s 0, 1, . . .
with 5 f 5 2n s
`
H0
2n Ž 1 q x . < f Ž x . < 2 dx ;
2 5 g 5yn s
`
H0
Ž1 q x.
y2 n
< g Ž x . < 2 dx,
respectively. It is evident that H0 s L2 Ž Rq . , Hn m L2 Ž Rq . m Hyn , Obviously Hyn is isomorphic to the dual of Hn .
n s 1, 2, . . . .
QUADRATIC PENCIL OF SCHRODINGER OPERATOR ¨
317
THEOREM 4.5.
½
n l n
5
w Ž ?, l .
ls l i
g HyŽ nq1. ,
n s 0, 1, . . . ,
m i y 1,
i s 1, 2, . . . , p,
Ž 4.15.
and
½
n ln
w Ž ?, l .
5
ls m l
n s 0, 1, . . . ,
g HyŽ nq1. ,
n l y 1,
l s 1, 2, . . . , q.
Ž 4.16.
Proof. From Ž2.3. we obtain
½
n l
n
f Ž x, l .
5
ls l i
½
`
F x n exp y
H0 ImU Ž t .
5
q
`
Hx < t <
n
< A Ž x, t . < dt.
By the definition of the space HyŽ nq1. and using Ž3.12. and Ž4.11., we arrive at Ž4.15.. In the same manner, we can show that Ž4.16. also holds.
w Ž x, l i . ,
½
l
w Ž x, l .
5
w Ž x, l .
5
ls l i
,...,
½
m iy1 l m iy1
w Ž x, l .
5
ls l i
w Ž x, l .
5
ls m l
and
w Ž x, m l . ,
½
l
ls m l
,...,
½
n ly1 l n ly1
are called the principal functions corresponding to the spectral singularities l s l i , i s 1, 2, . . . , p, and l s m l , l s 1, 2, . . . , q of LŽ l., respectively. Let us choose n 0 so that n 0 s max m1 , . . . , m p , n1 , . . . , n q 4 . Then Hn 0q1 m L2 Ž Rq . m HyŽ n 0q1. . By Theorem 4.5 we have the following.
BAIRAMOV, C ¸AKAR, AND C¸ELEBI
318 Remark 4.6.
½
n l n
w Ž ?, l .
5
ls l i
g HyŽ n 0q1. ,
n s 0, 1, . . . ,
m i y 1, i s 1, 2, . . . , p,
½
n
ln
w Ž ?, l .
5
ls m l
g HyŽ n 0q1. ,
n s 0, 1, . . . ,
n l y 1, l s 1, 2, . . . , q.
In a forthcoming article we will show that the spectral expansion of LŽ l. converges in the sense of HyŽ n 0q1. for every function in Hn 0q1. 5. KLEIN]GORDON EQUATION As we have pointed out previously, if we choose V Ž x . s yU 2 Ž x ., then the boundary value problem Ž1.1]1.2. is reduced to y0 q l y U Ž x .
2
y s 0,
x g Rq
y Ž 0 . s 0,
Ž 5.1. Ž 5.2.
where U is a complex-valued and absolutely continuous function in each finite subinterval of Rq. Equation Ž5.1. is called the Klein]Gordon s-wave equation for a particle of zero mass with static potential UŽ x .. It is trivial that the condition Ž3.10. assumes the form sup
exp Ž e'x .
e)0
0Fx-`
lim U Ž x . s 0
xª`
Ž 5.3.
for Ž5.1.. We can deduce the following from Theorem 3.10. THEOREM 5.1. The boundary ¨ alue problem Ž5.1., Ž5.2. has a finite number of eigen¨ alues and spectral singularities, and each of them is of finite multiplicity if the conditions Ž5.3. hold. Let us also note that if the function U is real and analytic and vanishes rapidly as x ª `, then the eigenvalues of Ž5.1., Ž5.2. have been discussed previously w2x.
QUADRATIC PENCIL OF SCHRODINGER OPERATOR ¨
319
¨ 6. ONE-DIMENSIONAL SCHRODINGER OPERATOR If UŽ x . ' 0 in Ž1.1., the operator LŽ l. is reduced to L, which was given in the Introduction. In this case the condition Ž3.10. will assume the form
exp Ž e'x . < V Ž x . < 4 - `,
sup
e ) 0.
Ž 6.1.
0Fx-`
Hence from Theorem 3.10 we get the following. THEOREM 6.1. Under the condition (6.1), the operator L has a finite number of eigen¨ alues and spectral singularities, and each of them is of finite multiplicity. The same result has been obtained by Lyance w9x and Naimark w12x under the stronger assumption `
H0
V Ž x . exp Ž e x . dx - `,
e ) 0.
The condition Ž6.1. has been obtained in Pavlov w15x.
ACKNOWLEDGMENT E. Bairamov would like to thank the Scientific and Technical Research Council of Turkey for financial support.
REFERENCES 1. H. Cornille, Existence and uniqueness of crossing symmetric NrD-type equations corresponding to the Klein]Gordon equation, J. Math. Phys. 11 Ž1970., 79]98. 2. A. Degasperis, On the inverse problem for the Klein]Gordon s-wave equation, J. Math. Phys. 11 Ž1970., 551]567. 3. M. G. Gasymov, Expansion in terms of a scattering theory problem for the non-selfadjoint Schrodinger equation, So¨ . Math. Dokl. 9 Ž1968., 390]393. ¨ 4. M. G. Gasymov and F. G. Maksudov, The principal part of the resolvent of non-selfadjoint operators in neighbourhood of spectral singularities, Funct. Anal. Appl. 6 Ž1972., 185]192. 5. S. V. Hruscev, Spectral singularities of dissipative Schrodinger operator with rapidly ¨ decreasing potential, Indiana Uni¨ . Math. J. 33 Ž1984., 313]338. 6. M. Jaulent, On an inverse scattering problem with an energy-dependent potential, Ann. Inst. Henri Poincare´ Sec. A 17 Ž1972., 363]378. 7. M. Jaulent and C. Jean, The inverse s-wave scattering problem for a class of potentials depending on energy, Comm. Math. Phys. 28 Ž1972., 177]220. 8. M. Jaulent and C. Jean, The inverse problem for the one dimensional Schrodinger ¨ equation with an energy-dependent potential I, II, Ann. Inst. Henri Poincare´ Sec. A 25 Ž1976., 105]118, 119]137.
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9. V. E. Lyance, A differential operator with spectral singularities I, II, AMS Transl. 60 Ž1967., 185]225, 227]283. 10. F. G. Maksudov and B. P. Allakhverdiev, Spectral analysis of a new class of non-selfadjoint operators with continuous and point spectrum, So¨ . Math. Dokl. 30 Ž1984., 566]569. 11. F. G. Maksudov and G. Sh. Guseinov, On solution of the inverse scattering problem for a quadratic pencil of one-dimensional Schrodinger operators on the whole axis, So¨ . Math. ¨ Dokl. 34 Ž1987., 34]38. 12. M. A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of the second order on a semi-axis, AMS Transl. 16 Ž1960., 103]193. 13. M. A. Naimark, ‘‘Linear Differential Operators I, II,’’ Ungar, New York, 1968. 14. B. S. Pavlov, On the non-self-adjoint Schrodinger operator, in ‘‘Topics in Mathematics ¨ and Physics,’’ Vol. 1, pp. 87]114, Consultant Bureau, New York, 1967. 15. B. S. Pavlov, On separation conditions for spectral components of a dissipative operator, Math. USSR Iz¨ estiya 9, Ž1975., 113]137. 16. I. I. Privalov, The boundary properties of analytic functions, in ‘‘Hochshulbucher fur ¨ ¨ Math.,’’ Vol. 25, VEB Deutscher Verlag, Berlin, 1956. 17. J. T. Schwartz, Some non-self-adjoint operators, Comm. Pure Appl. Math. 13 Ž1960., 609]639.