Spectrum of the Sturm Liouville operators with boundary conditions polynomially dependent on the spectral parameter

R E S E A R C H

Open Access

Spectrum of the Sturm-Liouville operators

with boundary conditions polynomially

dependent on the spectral parameter

Nihal Yokus

1*

_{and Turhan Koprubasi}

2

*_{Correspondence:}

[email protected] 1_{Department of Mathematics,}

Karamanoglu Mehmetbey University, Karaman, 70100, Turkey Full list of author information is available at the end of the article

Abstract

In this paper, we consider the operatorLgenerated inL2(R+) by the Sturm-Liouville

equation –y+q(x)y=

λ

2_y,x∈R

+= [0,∞), and the boundary condition

(

α

0+

α

1

λ

+

α

2

λ

2)y(0) – (

β

0+

β

1

λ

+

β

2

λ

2)y(0) = 0, whereqis a complex-valued

function,

α

i,

β

i∈C,i= 0, 1, 2, and

λ

is an eigenparameter. Under the conditions q,q∈AC(R+), limx→∞|q(x)|+|q(x)|= 0, supx∈R+[e

ε√x_{|q(x)|] <∞,}

_ε

_{> 0, using the} uniqueness theorems of analytic functions, we prove thatLhas a finite number of eigenvalues and spectral singularities with finite multiplicities.

MSC: 34B08; 34B09; 34B24

Keywords: Sturm-Liouville equations; eigenparameter; eigenvalues; spectral singularities

1 Introduction

Let us consider the non-selfadjoint Sturm-Liouville operatorLgenerated inL(R+) by

the differential expression

l(y) := –y+q(x)y, x∈R+, (.)

and the boundary conditiony() –hy() = , whereqis a complex-valued function and

h∈C. The spectrum and eigenfunction expansion ofLwere investigated by Naimark []. In this study, the spectrum ofLis investigated and it is shown that it is composed of the eigenvalues, a continuous spectrum, and spectral singularities. The spectral singularities are poles of the resolvent which are embedded in the continuous spectrum and are not eigenvalues.

The effect of the spectral singularities in the spectral expansion ofLin terms of the principal functions has been investigated in [–].

The spectral analysis of the non-selfadjoint operator, generated inL(R+) by (.) and

the integral boundary condition

_∞



A(x)y(x)dx+αy() –βy() = ,

whereA∈L(R+) is a complex-valued function, andα,β∈C, was investigated in detail

by Krall [, ].

Some problems of spectral theory of differential and other types of operators with spec-tral singularities were also studied in [–].

Note that in all the above articles, the boundary conditions are independent of the spec-tral parameter.

In , Fulton [], considered the Sturm-Liouville equation with one boundary condi-tion dependent on the spectral parameter and obtained asymptotic estimates of eigenval-ues or eigenfunctions. Since , one of such Sturm-Liouville equations with boundary condition dependent on the spectral parameter was discussed by a number of authors (see [–]).

LetLdenote the operator generated inL(R+) by

–y+q(x)y=λy, x∈R+, (.)

α+αλ+αλ

y() –β+βλ+βλ

y() = , (.)

whereqis a complex-valued function,αi,βi∈C,i= , , , with|α|+|β| = .

Differently from other studies in the literature, the specific feature of this paper, which is one of the articles having applicability in study areas such as physics, engineering, and mathematics, is the presence of the spectral parameter not only in the Sturm-Liouville equation but also in the boundary condition for a quadratic form.

In this article, we intend to investigate eigenvalues and the spectral singularities of the

L, which has a finite number of eigenvalues and spectral singularities with a finite multi-plicities, if the conditions

q,q∈AC(R+), lim

x→∞q(x)+q

_{(x)= ,}

sup

x∈R+

eε√x_{q(x)<∞, ε> ,}

hold, whereAC(R+) denotes the class of complex-valued absolutely continuous functions

onR+.

2 Jost solutions and Jost functions ofL

Let us suppose that

∞



xq(x)dx<∞. (.)

Bye(x,λ), we will denote the bounded solution of (.) satisfying the condition

lim

x→∞y(x,λ)e

–iλx_{= , forλ∈C}

+:={λ:λ∈C,Imλ≥}. (.)

The solution e(x,λ) is called the Jost solution of (.). Under the condition (.), the solutione(x,λ) has the integral representation [, Chapter ]

e(x,λ) =eiλx+

∞

x

where the functionK(x,t) is the solution of the integral equation

K(x,t) =  

∞

x+t 

q(s)ds+ 

x+t 

x

t+s–x

t+x–s

q(s)K(s,u)du ds

+ 

∞

x+t 

t+s–x

s

q(s)K(s,u)du ds, (.)

andK(x,t) is continuously differentiable with respect to its arguments. We also have

K(x,t)<cw

x+t

 ,

Kx(x,t),Kt(x,t)≤

 

q

x+t

 +cw

x+t

 ,

(.)

wherew(x) =_x∞|q(s)|dsandc>  is a constant. Let

N+(λ) :=α+αλ+αλ

e(,λ) –β+βλ+βλ

e(,λ), λ∈C+, N–(λ) :=α+αλ+αλe(, –λ) –β+βλ+βλe(, –λ), λ∈C–,

(.)

whereC–={λ:λ∈C,Imλ≤}.

Therefore,N+_andN–_{are analytic inC}

+={λ:λ∈C,Imλ> }andC–={λ:λ∈C,Imλ<

}, respectively, and are continuous up to the real axis. The functionsN+_andN–_{are called}

Jost functions ofL.

3 Eigenvalues and spectral singularities ofL

We will denote the set of all eigenvalues and spectral singularities ofLbyσd(L) andσss(L),

respectively. It is evident that

σd(L) =

λ:λ∈C+,N+(λ) = 

∪λ:λ∈C–,N–(λ) = 

,

σss(L) =

λ:λ∈R∗,N+(λ) = ∪λ:λ∈R∗,N–(λ) = , (.)

λ:λ∈R∗,N+(λ) = ∩λ:λ∈R∗,N–(λ) = =∅, whereR∗=R\{}.

Definition  The multiplicity of a zero ofN+(orN–) inC+(orC–) is called the multiplicity

of the corresponding eigenvalue or spectral singularity ofL.

From (.) we find that, in order to investigate the quantitative properties of the eigen-values and the spectral singularities ofL, we need to discuss the quantitative properties of the zeros ofN+_andN–_inC

+andC–, respectively.

Let

M±_ :=λ:λ∈C_±,N±(λ) = , M±_ :=λ:λ∈R∗,N±(λ) = . (.) Let us denote the set of all limit points ofM+_ andM–

 byM+andM– and the set of all

zeros ofN+_andN–_{with infinite multiplicity inC}

It follows from the boundary uniqueness theorem of analytic functions that

M±_ ⊂M±_, M±_ ⊂M_±, M_±⊂M±_, (.)

and the linear Lebesgue measures ofM_±andM±_ are zero. Using (.) and (.), we get

σd(L) =M+∪M–, σss(L) =M+∪M–. (.)

Now, let us suppose that

q,q∈AC(R+), lim

x→∞q(x)+q

_{(x)= ,} ∞ 

xq(x)dx<∞. (.)

Theorem  Under condition(.)the functions N+_{and N}–_{have the following} represen-tations:

N+(λ) =iαλ+β+λ+δ+λ+ϕ++ ∞



f+(t)eiλt_{dt, λ∈C}

+, (.)

N–(λ) =iαλ+β–λ+δ–λ+ϕ–+ ∞



f–(t)e–iλtdt, λ∈C–, (.)

whereβ±,δ±,ϕ±∈C,and f±∈L(R+).

Proof Using (.), (.), and (.) we have (.), where

β+=iα–αK(, ) –β,

δ+=iα+iαKx(, ) –αK(, ) –β–iβK(, ),

ϕ+= –αKxt(, ) +iαKx(, ) –αK(, ) –β+βKt(, ),

f+(t) = –αKxtt(,t) +iαKxt(,t) +αKx(,t) +βKtt(,t) –iβKt(,t) –βK(,t). (.)

The following result is obtained in []:

Ktt(,t)≤c

tq

t

 +

q

t

 +tw

t

 +w

t

 . (.)

Then from (.), (.), and (.), we get

_{Kxtt(,t)≤cq}t

 +t

q

t

 +t

_q

t



+tσ

t

 +tσ

t

 +δ

t

 +δ

t

 , (.)

where

δ(x) =

∞

x

q(s)ds, δ(x) = ∞

x

δ(t)dt

It follows from (.), (.), (.), and (.) thatf+_∈L(R

+). In a similar way we obtain

(.).

Theorem  Under the condition(.),we have:

(i) The set of eigenvalues ofLis bounded,has at most a countable number of elements,

and its limit points can lie only in a bounded subinterval of the real axis. (ii) The set of spectral singularities ofLis bounded andμ(σss(L)) = .

Proof From (.), (.), and (.), we have

N+(λ) =iαλ+β+λ+δ+λ+ϕ++o(), λ∈C+,|λ| → ∞, N–(λ) =iαλ+β–λ+δ–λ+ϕ–+o(), λ∈C–,|λ| → ∞.

(.)

Using (.), (.), and the uniqueness theorems of analytic functions [], we obtain (i)

and (ii).

Theorem  If

q,q∈AC(R+), lim

x→∞q(x)+q

_{(x)= ,} ∞ 

eεxq(x)<∞, ε> , (.)

then the operator L has a finite number of eigenvalues and spectral singularities,and each of them is of finite multiplicity.

Proof Using (.), (.), (.), (.), and (.) we find that

f+(t)≤ce–(ε)t_{, (.)}

wherec>  is a constant. By (.) and (.) we observe that the functionN+_{has an}

ana-lytic continuation to the half-planeImλ> –ε

. So we getM +

=∅. It follows from (.) that M+_=∅. Therefore the setsM_+andM+_have a finite number of elements with a finite mul-tiplicity. We obtain similar results for the setsM–

 andM–. From (.) we have the proof

of the theorem.

It is seen that the condition (.) guarantees the analytic continuation of the functions

N+_andN–_{from the real axis to the lower and upper half-planes, respectively. So the}

finite-ness of eigenvalues and spectral singularities ofLare achieved as a result of this analytic continuation.

Now let us suppose that

q,q∈AC(R+), lim

x→∞q(x)+q

_{(x)= , sup} x∈R+

eε√xq(x)<∞, ε> , (.)

which is weaker than (.).

It is evident that under the condition (.) the functionN+ _{is analytic inC}

+and

in-finitely differentiable on the real axis. ButN+_{does not have an analytic continuation from}

finiteness of eigenvalues and spectral singularities ofLcannot be proved in a way similar to Theorem .

Lemma  If(.)holds,then M+

=M–=∅.

Proof It follows from (.) and (.) that the functionN+ is analytic inC+, and all of

its derivatives are continuous up to the real axis. Moreover, by Theorem  for sufficiently largeT> , we have

_––T_∞ In_{ +}|N+_λ(λ)|dλ

<∞,

∞

T

In|N+_(λ)|

 +λ dλ <∞.

From (.), we obtain

_dλdnnN

+_(λ)≤A+

n, λ∈C+,|λ| ≤T,n∈N∪ {},

where

A+_n= nc ∞



tne–(ε)

√

t_{dt, n∈N∪ {}, (.)}

andc>  is a constant. Since the functionN+_{is not equal to zero identically, by Pavlov’s}

theorem [],M+

satisfies h



lnT+(s)dμM_+,s> –∞, (.)

whereT+_{(s) =inf} nA

+ nsn

n! ,μ(M +

,s) is the linear Lebesgue measure of ans-neighborhood of M+_, and the constantA+_nis defined by (.).

Now we obtain the following estimates forA+ n:

A+_n≤Aann!nn, (.)

whereAandaare constants depending oncandε. Substituting (.) in the definition of

T+_{(s), we arrive at}

T+(s)≤Ainf

n

ansnnn≤Aexp–a–e–s–.

Now by (.) we get

h





sdμ

M+_,s<∞. (.)

Inequality (.) holds for arbitrarysif and only ifμ(M+

,s) =  orM+=∅. In a similar way

we can prove thatM–

=∅.

Proof To be able to prove the theorem, we have to show that the functionsN+ _andN–

have a finite number of zeros with finite multiplicities inC+andC–, respectively. We give

the proof forN+_.

It follows from (.) and Lemma  thatM_+=∅. So the bounded setsM+_ andM+_ have no limit points,i.e., the functionN+has only a finite number of zeros inC+. SinceM+=∅,

these zeros are of finite multiplicity.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Karamanoglu Mehmetbey University, Karaman, 70100, Turkey.}2_{Department of}

Mathematics, Kastamonu University, Kastamonu, 37100, Turkey.

Acknowledgements

The authors would like to express their thanks to the reviewers for their helpful comments and suggestions.

Received: 5 September 2014 Accepted: 14 January 2015 References

1. Naimark, MA: Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint operator of second order on a semi-axis. Am. Math. Soc. Transl. (2)16, 103-193 (1960)

2. Lyance, VE: A differential operators with spectral singularities I. Am. Math. Soc. Transl. (2)60, 185-225 (1967) 3. Lyance, VE: A differential operators with spectral singularities II. Am. Math. Soc. Transl. (2)60, 227-283 (1967) 4. Adıvar, M: Quadratic pencil of difference equations: Jost solutions, spectrum, and principal vectors. Quaest. Math.33,

305-323 (2010)

5. Krall, AM: A nonhomogeneous eigenfunction expansion. Trans. Am. Math. Soc.117, 352-361 (1965)

6. Krall, AM: The adjoint of differential operator with integral boundary conditions. Proc. Am. Math. Soc.16, 738-742 (1965)

7. Krall, AM, Bairamov, E, Cakar, O: Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary conditions. J. Differ. Equ.151, 252-267 (1999)

8. Krall, AM, Bairamov, E, Cakar, O: Spectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities. Math. Nachr.231, 89-104 (2001)

9. Bairamov, E, Cakar, O, Krall, AM: An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities. J. Differ. Equ.151, 268-289 (1999)

10. Bairamov, E, Cakar, O, Krall, AM: Non-selfadjoint difference operators and Jacobi matrices with spectral singularities. Math. Nachr.229, 5-14 (2001)

11. Bairamov, E, Celebi, AO: Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators. Q. J. Math.

50, 371-384 (1999)

12. Adıvar, M, Bairamov, E: Spectral singularities of the nonhomogeneous Sturm-Liouville equations. Appl. Math. Lett.15, 825-832 (2002)

13. Fulton, CT: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb. A77, 293-308 (1977)

14. Chernozhukova, AY, Freiling, G: A uniqueness theorem for inverse spectral problems depending nonlinearly on the spectral parameter. Inverse Probl. Sci. Eng.17, 777-785 (2009)

15. Freiling, G, Yurko, VA: Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter. Inverse Probl.26(6), 055003 (2010)

16. Binding, PA, Browne, PJ, Watson, BA: Transformations between Sturm-Liouville problems with eigenvalue dependent and independent boundary conditions. Bull. Lond. Math. Soc.33, 749-757 (2001)

17. Binding, PA, Browne, PJ, Watson, BA: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter I. Proc. Edinb. Math. Soc.33, 749-757 (2001)

18. Binding, PA, Browne, PJ, Watson, BA: Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter II. Comput. Appl. Math.148, 147-168 (2002)

19. Code, WJ, Browne, PJ: Sturm-Liouville problems with boundary conditions depending quadratically on the eigenparameter. J. Math. Anal. Appl.309, 729-742 (2005)

20. Yokus, N: Principal functions of non-selfadjoint Sturm-Liouville problems with eigenvalue-dependent boundary conditions. Abstr. Appl. Anal.2011, Article ID 358912 (2011)

21. Bairamov, E, Koprubasi, T: Eigenparameter dependent discrete Dirac equations with spectral singularities. Appl. Math. Comput.215, 4216-4220 (2010)

22. Marchenko, VA: Sturm-Liouville Operators and Applications. Birkhäuser, Basel (1986)

23. Bairamov, E, Yokus, N: Spectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions. Abstr. Appl. Anal.2009, Article ID 28959 (2009)

24. Dolzhenko, EP: Boundary value uniqueness theorems for analytic functions. Math. Notes26(6), 437-442 (1979) 25. Pavlov, BS: On separation conditions for spectral components of a dissipative operator. Math. USSR, Izv.9, 113-137