https://doi.org/10.1007/s40995-018-00672-3
RESEARCH PAPER
Spectral Properties of Discrete Klein–Gordon s‑Wave Equation
with Quadratic Eigenparameter‑Dependent Boundary Condition
Nihal Yokus1 · Nimet Coskun1
Received: 26 June 2018 / Accepted: 23 December 2018 / Published online: 19 January 2019 © Shiraz University 2019
Abstract
In this study, we consider the spectral analysis of the boundary value problem (BVP) consisting of the discrete Klein–Gordon equation and the quadratic eigenparameter-dependent boundary condition. Presenting the Jost solution and Green’s function, we investigate the finiteness and other spectral properties of the eigenvalues and spectral singularities of this BVP under certain conditions.
Keywords Eigenparameter · Spectral analysis · Eigenvalues · Spectral singularities · Discrete equation · Klein–Gordon equation
1 Introduction
Spectral theory is an extremely rich field of mathematics which has a number of significant application areas from quantum physics to engineering. So, various problems of spectral analysis of differential and difference operators have been investigated in detail (Levitan and Sargsjan 1991; Naimark 1960, 1968).
Investigation of the spectral properties of the some basic differential operators can be traced back to Naimark (1960). He studied the spectrum of the Sturm–Liouville equation considering the boundary value problem (BVP)
where h ∈ ℂ and q is a complex valued function. He showed that the spectrum of this BVP is composed of eigenvalues, spectral singularities and continuous spectrum. He also proved that these eigenvalues and spectral singularities are of finite number with finite multiplicity under certain conditions.
Levitan and Sargsjan (1991) have presented basic prob-lems in the spectral analysis of Sturm–Liouville and one-dimensional Dirac-type operators in their books. In Baira-mov et al. (1997), spectrum of the quadratic pencil of the Schrödinger operator and principal functions correspond-ing to the spectral scorrespond-ingularities were discussed. Bairamov and Coskun (2005) investigated the structure of the discrete spectrum of the system of non-selfadjoint difference equa-tions of first order using the uniqueness theorems of analytic functions. Adıvar (2010) determined the resolvent and Jost solutions of the Schrödinger and Klein–Gordon-type differ-ence operators.
In recent years, spectral analysis of the boundary value problems with eigenparameter-dependent boundary condi-tion has become an attractive research area since they arise in models of certain physical problems such as vibration of a string, quantum mechanics and geophysics (Koprubasi 2014; Koprubasi and Yokus 2014; Bairamov and Yokus 2009; Kir et al. 2005). In such problems, the structure of the Green’s function and resolvent operator differ drastically based upon the eigenparameter-dependent boundary condition.
Note that the equation
is called the Klein–Gordon s-wave equation for a particle of zero mass with static potential Q (Bairamov 2004).
Spectral theory of Klein–Gordon s-wave equation has become main research topic of the papers (Bairamov et al.
1997; Bairamov 2004; Degasperis 1970). In particular, { −y��+ q(x)y − 𝜆2y= 0, x ∈ ℝ +, y�(0) − hy(0) = 0, y��+ [𝜆 − Q(x)]2y= 0, x ∈ ℝ+, * Nihal Yokus nyokus@kmu.edu.tr Nimet Coskun cannimet@kmu.edu.tr
1 Department of Mathematics, Karamanoglu Mehmetbey
Bairamov et al. (1997) obtained the conditions for the poten-tial, under which there exists a finite number of eigenvalues and spectral singularities with finite multiplicities.
The present paper was motivated by the above-mentioned studies.
In this paper, we consider the following BVP
where (an),(vn), n ∈ ℕ are complex sequences, an≠ 0 for
all n ∈ ℕ ∪ {0}, 𝛾0𝛽1− 𝛾1𝛽0≠ 0, ||𝛾2|| +||𝛽2|| ≠ 0 and 𝛾2 ≠ −𝛽1
a0
for 𝛾i, 𝛽i∈ ℂ , i = 0, 1, 2 and 𝜆 is an eigenparameter.
The specific feature of this study is the presence of the spectral parameter not only in the difference equation but also in the boundary condition, which is quadratic.
The remainder of the manuscript is organized as follows: In Sect. 2, we present the Jost solution and Green’s function of the BVP (1.1), (1.2). Section 3 deals with the eigenvalues and spectral singularities of this BVP. In the last part, we investigate the quantitative properties of these eigenvalues and spectral singularities under the condition
2 Solution of (
1.1
), (
1.2
)
Let the conditionis satisfied. It is known from Adıvar (2010) that Eq. (1.1) has the solution
for 𝜆 = 2 cos(z
2
)
, z ∈ ℂ+. Note that the expressions of Knm
and 𝛼n can be written uniquely in terms of
(
an) and (vn). Moreover,
is satisfied for [|||m2||| ]
which is the integer part of m
2 , and C > 0
is a constant. Hence, fn(z) is analytic with respect to z in ℂ
+ ∶= {z ∶ z ∈ ℂ, Imz > 0} and continuous in Imz = 0.
We define the function 𝜙 using the boundary condition (1.2) and (2.2) as (1.1) △(an−1△ yn−1)+(vn− 𝜆)2yn= 0, n ∈ ℕ, (1.2) ( 𝛾0+ 𝛾1𝜆 + 𝛾2𝜆2)y 1+ ( 𝛽0+ 𝛽1𝜆 + 𝛽2𝜆2)y 0 = 0 ∑ n∈ℕ exp(𝜖n𝛿)(| |1 − an|| +||vn|| ) < ∞ for 𝜖 > 0 and1 2 ≤ 𝛿 ≤ 1. (2.1) ∑ n∈ℕ n(||1 − an|| +||vn||)< ∞, (2.2) fn(z) = 𝛼neinz ( 1+ ∞ ∑ m=1 Knmeimz2 ) (2.3) | |Knm|| ≤ C ∞ ∑ r=n+[|||m2||| ] (| |1 − ar|| +||vr|| ) ,
The function 𝜙 is analytic in ℂ+, continuous in ℂ+, and 𝜙(z) = 𝜙(z + 4𝜋).
Let ̂𝜑(𝜆) ={𝜑̂n(𝜆)} , and n ∈ ℕ ∪ {0} be the solution of (1.1) subject to the initial conditions
If we define
then 𝜑 is entire function and
Let us define the semi-strips P0 ∶= {z ∶ z ∈ ℂ, z = 𝜉 + i𝜏,
−𝜋≤ 𝜉 ≤ 3𝜋, 𝜏 > 0} and P ∶= P0∪ [−𝜋, 3𝜋]. For all z ∈ P and 𝜙(z) ≠ 0, we define
The function Gnm is introduced as the Green’s function of
the BVP (1.1), (1.2). Obviously, for g =(gm), m ∈ ℕ ∪ {0},
is the resolvent of the BVP (1.1), (1.2).
3 Eigenvalues and Spectral Singularities
of (
1.1
), (
1.2
)
Let us denote the set of eigenvalues and spectral singularities of the BVP (1.1), (1.2) by 𝜎d and 𝜎ss, respectively. From (2.4)
and definition of the eigenvalues and the spectral singulari-ties, we have (2.4) 𝜙(z) =(𝛾0+ 𝛾1𝜆 + 𝛾2𝜆2)f 1(z) + ( 𝛽0+ 𝛽1𝜆 + 𝛽2𝜆2)f 0(z). (2.5) ̂ 𝜑0(𝜆) = 𝛾0+ 𝛾1𝜆 + 𝛾2𝜆2, ̂𝜑 1(𝜆) = − ( 𝛽0+ 𝛽1𝜆 + 𝛽2𝜆2). (2.6) 𝜑(z) = ̂𝜑(2 cosz 2 ) ={𝜑̂n(2 cosz 2 )} , n∈ ℕ ∪ {0}, 𝜑(z) = 𝜑(z + 4𝜋). Gnm(z) ∶= { −𝜑m(z)fn(z) 𝜙(z) , m≤ n, −𝜑n(z)fm(z) 𝜙(z) , m > n. (2.7) (Rg)n∶= ∞ ∑ m=0 Gnm(z)gm, n∈ ℕ ∪ {0}, (3.1) 𝜎d={𝜆 ∶ 𝜆 = 2 cosz 2, z∈ P0, 𝜙(z) = 0 } , (3.2) 𝜎ss={𝜆 ∶ 𝜆 = 2 cosz 2, z∈ [−𝜋, 3𝜋], 𝜙(z) = 0 } �{0, 𝜋, 2𝜋}.
Using (2.2) and (2.4), we obtain
Let
Then, the function 𝜓 is analytic in ℂ+, continuous in ℂ+,
and 𝜙(z) =(𝛾0+ 𝛾1(ei2z + e−i z 2 ) + 𝛾2(eiz+ 2 + e−iz)) × [ 𝛼1eiz+ 𝛼1eiz ∞ ∑ m=1 K1meim2z ] +(𝛽0+ 𝛽1(eiz2+ e−i z 2 ) + 𝛽2(eiz+ 2 + e−iz)) × [ 𝛼0+ 𝛼0 ∞ ∑ m=1 K0meim2z ] = 𝛼0𝛽2e−iz+ 𝛼0𝛽1e−iz2 +(2𝛼 0𝛽2+ 𝛼0𝛽0 ) +(𝛼1𝛾1+ 𝛼0𝛽1)eiz2+(𝛼 0𝛽2+ 2𝛼1𝛾2+ 𝛼1𝛾0 ) eiz +(𝛼1𝛾1)ei3z2 +(𝛼 1𝛾2 ) e2iz + ∞ ∑ m=1 ( 𝛼0𝛽2Kom)eiz ( m 2−1 ) + ∞ ∑ m=1 ( 𝛼0𝛽1Kom)eiz ( m 2− 1 2 ) + ∞ ∑ m=1 ( 2𝛼0𝛽2Kom+ 𝛼0𝛽0Kom+ 𝛼1𝛾2K1m)eizm2 + ∞ ∑ m=1 ( 𝛼1𝛾1K1m+ 𝛼0𝛽1K0m)eiz ( m 2+ 1 2 ) + ∞ ∑ m=1 ( 𝛼1𝛾0K1m+ 2𝛼1𝛾2K1m+ 𝛼0𝛽2Kom)eiz (m 2+1 ) + ∞ ∑ m=1 ( 𝛼1𝛾1K1m)eiz ( m 2+ 3 2 ) + ∞ ∑ m=1 ( 𝛼1𝛾2K1m)eiz ( m 2+2 ) . (3.3) 𝜓(z) = eiz𝜙(z). 𝜓(z) = 𝜓(z + 4𝜋), (3.4) 𝜓(z) = 𝛼0𝛽2+ 𝛼0𝛽1e i2z +(2𝛼0𝛽2+ 𝛼0𝛽0 ) eiz+(𝛼1𝛾1+ 𝛼0𝛽1 ) ei3z2 +(𝛼0𝛽2+ 2𝛼1𝛾2+ 𝛼1𝛾0 ) e2iz+(𝛼1𝛾1 ) ei5z2 +(𝛼 1𝛾2 ) e3iz + ∞ ∑ m=1 ( 𝛼0𝛽2Kom ) eizm2 + ∞ ∑ m=1 ( 𝛼0𝛽1Kom ) eiz ( m 2+ 1 2 ) + ∞ ∑ m=1 ( 2𝛼0𝛽2Kom+ 𝛼0𝛽0Kom+ 𝛼1𝛾2K1m ) eiz ( m 2+1 ) + ∞ ∑ m=1 ( 𝛼1𝛾1K1m+ 𝛼0𝛽1K0m ) eiz ( m 2+ 3 2 ) + ∞ ∑ m=1 ( 𝛼1𝛾0K1m+ 2𝛼1𝛾2K1m+ 𝛼0𝛽2Kom ) eiz ( m 2+2 ) + ∞ ∑ m=1 ( 𝛼1𝛾1K1m ) eiz ( m 2+ 5 2 ) + ∞ ∑ m=1 ( 𝛼1𝛾2K1m ) eiz ( m 2+3 ) . From (3.1)–(3.3), we get
Definition 3.1 The multiplicity of a zero of 𝜓 in P is called the multiplicity of the corresponding eigenvalue or spectral singularity of the BVP (1.1), (1.2).
We can see from (3.5) and (3.6) that we need to inves-tigate the zeros of 𝜓 in P in order to study the quantitative properties of eigenvalues and spectral singularities of the BVP (1.1), (1.2).
Let us define the sets
Let us also introduce all limit points of M1 by M3 and the set
of all zeros of 𝜓 with infinite multiplicity by M4.
It is obvious from (3.5)–(3.7) that
4 Quantitative Properties of Eigenvalues
and Spectral Singularities
Theorem 4.1 If (2.1) holds, then
(i) M1 is bounded and countable,
(ii) M1∩ M3 = �, M1∩ M4= �,
(iii) M2 is compact and 𝜇(M2) = 0 , where 𝜇 is Lebesgue measure in the real axis,
(iv) M3⊂ M2, M4 ⊂ M2; 𝜇(M3) = 𝜇(M4) = 0,
(v) M3⊂ M4.
Proof Using (2.3) and (3.4), we obtain the asymptotic
Boundedness of the set M1 is achieved as a consequence
of (4.1). The function 𝜓 is analytic in ℂ+ and is a 4𝜋
peri-odic function. So, M1 has at most a countable number of
(3.5) 𝜎d={𝜆 ∶ 𝜆 = 2 cosz 2, z∈ P0, 𝜓(z) = 0 } , (3.6) 𝜎ss= { 𝜆 ∶ 𝜆 = 2 cos z 2, z∈ [−𝜋, 3𝜋], 𝜓(z) = 0 } �{0, 𝜋, 2𝜋}. (3.7) M1∶={z∶ z ∈ P0, 𝜓(z) = 0}, M2∶={z ∶ z ∈ [−𝜋, 3𝜋]�{0, 𝜋, 2𝜋}, 𝜓(z) = 0}. (3.8) 𝜎d={𝜆 ∶ 𝜆 = 2 cosz 2, z∈ M1 } , (3.9) 𝜎ss={𝜆 ∶ 𝜆 = 2 cosz 2, z∈ M2 } . (4.1) 𝜓(z) = { 𝛼0𝛽2+ O(e− 𝜏 2 ) , 𝛽2≠ 0, z ∈ P, 𝜏 → ∞, 𝛼0𝛽1ei2z + O(e−𝜏), 𝛽 2= 0, z ∈ P, 𝜏 → ∞.
elements. This proves (i). (ii)–(iv) is obtained from the boundary uniqueness theorems of analytic functions (Dolz-henko 1979). From the continuity of all derivatives of 𝜓 on
[−𝜋, 3𝜋], we find (v). □
As a consequence of Theorem 4.1, (3.8) and (3.9), we get the following theorem.
Theorem 4.2 If (2.1) is satisfied, then
(i) the set of eigenvalues of the BVP (1.1), (1.2) is
bounded, has at most a countable number of ele-ments, and its limit points can lie only in [−2, 2].
(ii) 𝜎ss ⊂ [−2, 2] and 𝜇(𝜎ss) = 0.
Assume
for the complex sequences (an),(vn) and for some 𝜀 > 0. Theorem 4.3 If (4.2) holds, then the BVP (1.1), (1.2) has a
finite number of eigenvalues and spectral singularities with a finite multiplicity.
Proof Using (2.3), we get the inequality
where B > 0 is a constant. From the expression of the func-tion 𝜓(z) in (3.4), its analytic continuation can be obtained to the half-plane Imz > −𝜀
2. Since 𝜓 is 4𝜋 periodic function,
the accumulation points of its zeros cannot be on the interval [−𝜋, 3𝜋]. We have already got that the bounded sets M1 and
M2 have a finite number of elements. If we use the
analytic-ity of 𝜓 in Imz > −𝜀
2 , we get that all zeros of 𝜓 in P have a
finite multiplicity. Hence, finiteness of the eigenvalues and spectral singularities of the BVP (1.1), (1.2) is obtained.
□ Let us consider the condition
We can see that analytic continuation of 𝜓 is achieved from real axis to lower half-plane for the condition (4.2). How-ever, this is not the case for the condition (4.3). Therefore, we need to use a different technique to prove the finiteness of the eigenvalues and spectral singularities of the BVP (1.1), (1.2). We will employ the following theorem and investigate if the conditions of Pavlov’s theorem (Pavlov 1967) is valid or not for our case.
(4.2) sup n∈ℕ [ exp(𝜀n)(||1 − an|| +||vn||)]< ∞, | |Knm|| ≤ B exp [ −𝜀 4(n + m) ] , n, m∈ ℕ (4.3) sup n∈ℕ [ exp(𝜀n𝛿)(||1 − an|| +||vn|| )] < ∞, 𝜀 > 0, 1 2 ≤ 𝛿 < 1.
Theorem 4.4 Let us assume that the function u is analytic
in ℂ+ , all of its derivatives are continuous up to the real axis and there exists T > 0 such that
and
If the set Q with linear Lebesgue measure zero is the set of all zeros of the function u is with infinite multiplicity, and if
where F(s) = infn(Cnsn∕n!) , n = 0, 1, 2, … , 𝜇(Qs) is the
lin-ear Lebesgue measure of s-neighborhood of Q and h is an arbitrary positive constant, then u(z) = 0.
The function 𝜓 is analytic in ℂ+ and infinitely
differenti-able on the real axis under condition (4.3). Using Theo-rem 4.4, (2.3) and (3.4), we get
where
The following estimation is obtained for 𝜂k
where D and d are constants depending on C, 𝜀 and 𝛿.
Theorem 4.5 If (4.3) satisfies, then M4 = �.
Proof From Theorem 4.4, we see that | | |u (n) (z)|||≤ Cn, n= 0, 1, … , z ∈ ℂ+,|z| < 2T, | | | | | | | T ∫ −∞ ln|u(x)| 1+ x2 dx | | | | | | | < ∞, | | | | | | | ∞ ∫ T ln|u(x)| 1+ x2 dx | | | | | | | < ∞. h ∫ 0 ln F(s)d𝜇(Qs) = −∞, | | |𝜓 (k) (z)|||≤ 𝜂k, k∈ ℕ ∪ {0}, 𝜂k= 4kC ∞ ∑ m=1 mkexp(−𝜀m𝛿). (4.4) 𝜂k≤ 4kC ∞ � 0 xkexp(−𝜀x𝛿)dx≤ Ddkk!kk1−𝛿 𝛿 , (4.5) 𝜔 ∫ 0 ln t(s)d𝜇(Gs) > −∞,
where t(s) = inf
k 𝜂ksk
k! , k ∈ ℕ ∪ {0}, 𝜇(M4,s) is the Lebesgue
measure of the s-neighborhood of M4 and 𝜂k is defined by
(4.4).
Now we have
by (4.4). From (4.5) and (4.6), we get
Since 𝛿
1−𝛿 ≥ 1, (4.7) holds for arbitrary s if and only if 𝜇(M4,s) = 0 or M4= �. □
Theorem 4.6 If (4.3) holds, then the BVP (1.1), (1.2) has a
finite number of eigenvalues and spectral singularities, and each of them is of finite multiplicity.
Proof We need to show that the function 𝜓(z) has a finite number of zeros with a finite multiplicities in P. Theo-rem 4.1 and Theorem 4.5 imply that M3 = M4= �. Thus, the
bounded sets M1 and M2 do not have accumulation points,
i.e., 𝜓(z) has only finite number of zeros in P. Since M4= �,
these zeros are of finite multiplicity. □
Acknowledgements The authors would like to express their thanks to the reviewers for their helpful comments and suggestions.
(4.6) t(s)≤ D exp{−1− 𝛿 𝛿 e −1d−1−𝛿𝛿 s−1−𝛿𝛿 } , (4.7) 𝜔 ∫ 0 s− 𝛿 1−𝛿d𝜇(M 4,s) < ∞.
References
Adıvar M (2010) Quadratic pencil of difference equations: Jost solutions, spectrum, and principal vectors. Quaest Math 33(3):305–323
Bairamov E (2004) Spectral properties of the nonhomogeneous Klein– Gordon s-wave equations. Rocky Mt J Math 34:1–11
Bairamov E, Coskun C (2005) The structure of the spectrum of a sys-tem of difference equations. Appl Math Lett 18(4):387–394 Bairamov E, Yokus N (2009) Spectral singularities of Sturm-Liouville
problems with eigenvalue-dependent boundary conditions. Abstr Appl Anal. https ://doi.org/10.1155/2009/28959 6
Bairamov E, Çakar Ö, Celebi AO (1997) Quadratic pencil of schrödinger operators with spectral singularities: discrete spec-trum and principal functions. J Math Anal Appl 216(1):303–320 Degasperis A (1970) On the inverse problem for the Klein–Gordon
s-wave equation. J Math Phys 11(2):551–567
Dolzhenko EP (1979) Boundary-value uniqueness theorems for ana-lytic functions. Math Notes Acad Sci USSR 25(6):437–442 Kir E, Bascanbaz-Tunca G, Yanik C (2005) Spectral properties of a
non selfadjoint system of differential equations with a spectral parameter in the boundary condition. Proyecciones (Antofagasta) 24(1):49–63
Koprubasi T (2014) Spectrum of the quadratic eigenparameter depend-ent discrete Dirac equations. Adv Differ Equ 2014(1):148 Koprubasi T, Yokus N (2014) Quadratic eigenparameter dependent
discrete Sturm–Liouville equations with spectral singularities. Appl Math Comput 244:57–62
Levitan BM, Sargsjan IS (1991) Sturm–Liouville and Dirac opera-tors. Translated from the Russian mathematics and its applications (Soviet series), vol. 59
Naimark MA (1960) Investigation of the spectrum and the expansion in eigenfunctions of a nonselfadjoint differential operator of the second order on a semi-axis. Am Math Soc Transl 2(16):103–193 Naimark MA (1968) Linear differential operators: part II: linear dif-ferential operators in Hilbert space with additional material by the author. F. Ungar Publishing Company, New York
Pavlov BS (1967) The non-self-adjoint schrodinger operator I, II, III, topics in math. Phys Consult Bur NY 1968:1969
