Volume 2010, Article ID 306571,11pages doi:10.1155/2010/306571
Research Article
Jost Solution and the Spectrum of the Discrete
Dirac Systems
Elgiz Bairamov, Yelda Aygar, and Murat Olgun
Department of Mathematics, Ankara University, Tando˘gan, 06100 Ankara, Turkey
Correspondence should be addressed to Elgiz Bairamov,[email protected] Received 14 September 2010; Accepted 10 November 2010
Academic Editor: Raul F. Manasevich
Copyrightq2010 Elgiz Bairamov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We find polynomial-type Jost solution of the self-adjoint discrete Dirac systems. Then we investigate analytical properties and asymptotic behaviour of the Jost solution. Using the Weyl compact perturbation theorem, we prove that discrete Dirac system has the continuous spectrum filling the segment-2,2. We also study the eigenvalues of the Dirac system. In particular, we prove that the Dirac system has a finite number of simple real eigenvalues.
1. Introduction
Let us consider the boundary value problem BVP generated by the Sturm-Liouville equation
−yqxyλ2y, 0≤x <∞ 1.1
and the boundary condition
y0 0, 1.2
whereqis a real-valued function andλ∈ is a spectral parameter. The bounded solution of 1.1satisfying the condition
lim
x→ ∞yx, λe
−iλx1, λ∈
will be denoted byex, λ. The solutionex, λsatisfies the integral equation
ex, λ eiλx
∞
x
sinλt−x
λ qtet, λdt. 1.4
It has been shown that, under the condition
∞
0
xqxdx <∞, 1.5
the solutionex, λhas the integral representation
ex, λ eiλx
∞
x
Kx, teiλtdt, λ∈ , 1.6
where the functionKx, tis defined byq. The functionex, λis analytic with respect toλin
:{λ:λ∈ ,Imλ >0}, continuous , and
ex, λ eiλx1o1, λ∈ , x−→ ∞ 1.7
holds1, chapter 3.
The functionsex, λand eλ : e0, λare called Jost solution and Jost function of the BVP1.1and1.2, respectively. These functions play an important role in the solution of inverse problems of the quantum scattering theory1–4. In particular, the scattering date of the BVP1.1and1.2is defined in terms of Jost solution and Jost function. Letiλk,k
1,2,3, . . . , n, be the zeros of the Jost function, numbered in the order of increase of their moduli 0< λ1< λ2<· · ·< λnand
m−k1:
∞
0
e2x, iλkdx
1/2
. 1.8
The functions
Ex, λ:ex,−λ−sλex, λ, λ2∈0,∞,
Ex, iλk:mkex, iλk, k1,2, . . . , n,
1.9
are bounded solutions of the BVP1.1and1.2, wheresλ:e−λ/eλ is the scattering function1–4. Using1.7, we get that
Ex, λ e−iλx−sλeiλxo1, λ2∈0,∞, x−→ ∞,
Ex, iλk mke−λkx1o1, k1,2, . . . , n, x−→ ∞,
1.10
hold. The collection of quantities{sλ, λ ∈ ;λk, mk, k 1,2, . . . , n} that specify to as the behaviour of the radial wave functionsEx, λandEx, iλkat infinity is called the scattering
Let us consider the self-adjoint system of differential equations of first order
y2pxy1λy1, −y
1qxy2λy2, 0≤x <∞,
1.11
wherepandqare real-valued continuous functions. In the casepx Vxm,qx Vx− m, whereV is a potential function andmthe mass of a particle,1.11is called stationary Dirac system in relativistic quantum theory 5, chapter 7. Jost solution and the scattering theory of1.11have been investigated in6.
Jost solutions of quadratic pencil of Schr ¨odinger, Klein-Gordon, andq-Sturm-Liouville equations have been obtained in 7–9. In10–17, using the analytical properties of Jost functions, the spectral analysis of differential and difference equations has been investigated. Discrete boundary value problems have been intensively studied in the last decade. The modelling of certain linear and nonlinear problems from economics, optimal control theory, and other areas of study has led to the rapid development of the theory of difference equations. Also the spectral analysis of the difference equations has been treated by various authors in connection with the classical moment problemsee the monographs of Agarwal 18, Agarwal and Wong 19, Kelley and Peterson 20, and the references therein. The spectral theory of the difference equations has also been applied to the solution of classes of nonlinear discrete Korteveg-de Vries equations and Toda lattices21.
Now let us consider the discrete Dirac system
Δyn2pnyn1λyn1,
−∇y1
n qnyn2λyn2, n∈ {1,2, . . .},
1.12
with the boundary condition
y01 0, 1.13
whereΔis the forward difference operator:Δunun1−unand∇is the backward difference
operator: ∇un un−un−1;pnand qnare real sequences. It is evident that1.12is the
discrete analogy of1.11. LetLdenote the operator generated in the Hilbert spacel2, 2
by the BVP1.12and1.13. The operatorLis self-adjoint, that is,L L∗. In the following, we will assume that, the real sequencespnandqnsatisfy
∞
n1
npnqn<∞. 1.14
In this paper, we find Jost solution of1.12and investigate analytical properties and asymptotic behaviour of the Jost solution. We also show that,σcL −2,2, whereσcL
denotes the continuous spectrum ofL, generated inl2,
2by1.12and1.13.
2. Jost Solution of
1.12
Ifpnqn0 for alln∈andλ−iz−iz
−1from1.12, we get
yn21−yn2 −iz−iz−1
yn1,
yn1−1−yn1 −iz−iz−1
yn2.
2.1
It is clear that
enz
⎛ ⎝en1z
en2z
⎞
⎠
z −i
z2n, n∈, 2.2
is a solution of 2.1. Now we find the solution fnz
f1 n
fn2
,n ∈ of 1.12for λ −iz−iz−1, satisfying the condition
fnz Io1enz, |z|1, n−→ ∞, 2.3
whereI1 0 0 1
.
Theorem 2.1. Under the condition1.14forλ −iz−iz−1and|z|1,1.12has the solution
fnz
fn1
fn2
,n∈, having the representation
fnz
⎛ ⎝fn1
fn2
⎞
⎠
I
∞
m1
Knmz2m
z −i
z2n, n∈, 2.4
f01z z
∞
m1
K011mz2m1−iK012mz2m
, 2.5
where
Knm
K11
nm K12nm
K21
nm K22nm
Proof. Substitutingfnz defined by2.4 and2.5into1.12and taking λ −iz−iz−1,
|z|1, we get the following:
pnz2n1−
∞
m1
K12nmz2m2n−1i
∞
m1
K22nm−pnK12nm−K11nm
z2m2n
∞
m1
pnK11nm−Knm21 Knm12
z2m2n1
i
∞
m1
K11nm−K22n1,m
z2m2n2
∞
m1
Kn211,mz2m2n3,
2.7
iqnz2ni
∞
m1
K12n−1,mz2m2n−2−
∞
m1
K11n−1,m−K22nm
z2m2n−1−i
∞
m1
K12nm−qnKnm22 −Knm21
z2m2n
∞
m1
qnK21nm−Knm11 K22nm
z2m2n1i
∞
m1
K21nmz2m2n2.
2.8
Using2.7and2.8,
Kn121 −
∞
kn1
pkqk
,
K11n1
∞
kn1 pkK12k1,
K22n1K11n−1,1
∞
kn
pkKk121,
K21n1Kn121pnK11n1
∞
kn1
qkK22k1pkK11k1
,
K12n2−
∞
kn1
pkKk111qkK22k1
,
K11
n2−K22n1,1
∞
kn1
pkKk122−qkK21k1
,
K22n2−K11n1
∞
kn
pkK12k2−qk1K21k1,1
,
Kn212K12n2
∞
kn
pkK11k2qk1Kk221,2
hold, wheren∈. Form≥3, we obtain
K12nmKn211,m−2−
∞
kn1
qkKk,m22 −1pkK11k,m−1
,
Knm11 −K22n1,m−1
∞
kn1
pkKkm12 −qkKk,m21 −1
,
K22
nm−K11n,m−1
∞
kn
pkKkm12 −qk1K21k1,m−1
,
Knm21 K12nmpnK11nm
∞
kn1
qkK22kmpkKkm11
.
2.10
By the condition 1.14, the series in the definition of Kijnm i, j 1,2 are absolutely
convergent. Therefore,Kijnm i, j 1,2can, by uniquely be defined bypn and qn, that is,
the system1.12forλ −iz−iz−1 and|z| 1, has the solution fnzgiven by2.4and
2.5.
By induction, we easily obtain that
Kijnm≤C
rnm/2
prqr, 2.11
wherem/2is the integer part ofm/2 andC > 0 is a constant. It follows from 2.4and 2.11that2.3holds.
Theorem 2.2. The solutionfnzhas an analytic continuation from{z:|z|1}toD:{z:|z|<
1} \ {0}.
Proof. From1.14and2.11, we obtain that the series∞m1Knmeimzand
∞
m1mKnmeimzare
uniformly convergent inD. This shows that the solutionfnzhas an analytic continuation
from{z:|z|1}toD.
The functionsfnzandf01zare called Jost solution and Jost function of the BVP
1.12and1.13, respectively. It follows from Theorem2.2that Jost solution and Jost function are analytic inDand continuous onD:{z:|z| ≤1} \ {0}.
Theorem 2.3. The following asymptotics hold:
fnz
⎛ ⎝fn1z
fn2z
⎞
⎠ Io1
z −i
Proof. From2.4, we get that
fn1z z2n1
∞
m1
Knm11z2m−i
∞
m1
K12nmz2m−1
z2n1, z∈D,
fn1zz−2n−11
∞
m1
Knm11z2m−i
∞
m1
Knm12z2m−1, z∈D.
2.13
Using2.11and2.13, we obtain
fn1zz−2n−1≤1
∞
m1
K11nm
∞
m1
K12nm
≤1c
∞
m1
kn|m/2|
pkqk≤1c ∞
kn1
k−n
m1
pkqk
≤1c
∞
kn1
k−npkqk≤1c
∞
kn1
kpkqk.
2.14
So we have
fn1z z2n11o1, z∈D, n−→ ∞, 2.15
by2.14. In a manner similar to2.15, we get
fn2z −iz2n1o1, z∈D, n−→ ∞. 2.16
From2.15and2.16, we obtain2.12.
3. Continuous and Discrete Spectrum of the BVP
1.12
and
1.13
Let2, C
2denote the Hilbert space of all complex vector sequences
y
⎧ ⎨ ⎩
⎛ ⎝yn1
yn2
⎞ ⎠
⎫ ⎬ ⎭
n∈
3.1
with the norm
y2 ∞
n1
yn1
2
yn2
2
Theorem 3.1. σcL −2,2.
Proof. LetL0denote the operator generated in2, C
2by the BVP
Δyn2λyn1,
−∇y1
n λyn2,
y010.
3.3
We also define the operatorPin2, C
2by the following:
P
⎛ ⎝yn1
yn2
⎞
⎠:
pn 0
0 qn
⎛ ⎝yn1
yn2
⎞
⎠. 3.4
It is clear thatP P∗and
LL0P, 3.5
whereLdenotes the operator generated in2, C
2by the BVP1.12and1.13. It follows
from1.14that the operatorPis compact in2, C
2. We easily prove that
σcL0 −2,2. 3.6
Using the Weyl theorem22of a compact perturbation, we obtain
σcL σcL0 −2,2. 3.7
Since the operatorLis selfadjoint, the eigenvalues ofLare real. From the definition of the eigenvalues, we get that
σdL
λ:λ−iz−iz−1, iz∈−1,0∪0,1, f01z 0 , 3.8
whereσdLdenotes the set of all eigenvalues ofL.
Definition 3.2. The multiplicity of a zero of the functionf01 is called the multiplicity of the corresponding eigenvalue ofL.
Theorem 3.3. Under the condition 1.14, the operator L has a finite number of simple real eigenvalues.
Letz0be one of the zeros off01. Now we show that
d dzf
1 0 z
zz0
/
0. 3.9
Letfnz
fn1z
fn2z
be the Jost solution of1.12that is,
fn21z−fn2z pnfn1z −iz−iz−1
fn1z,
fn1−1z−fn1z qnfn2z −iz−iz−1
fn2z.
3.10
Differentiating3.10with respect toz, we have
d dzf
2
n1z− d dzf
2
n z pn d
dzf
1
n z −iz−iz−1
d
dzf
1
n z−i
1−z−2fn1z,
d dzf
1
n−1z− d dzf
1
n z qn d
dzf
2
n z −iz−iz−1
d
dzf
2
n z−i
1−z−2fn2z.
3.11
Using3.10and3.11, we obtain
⎡
⎣dfn1z
dz f
2
n1z−f
1
n z
dfn21z dz
⎤
⎦−
⎡
⎣dfn1−1z dz f
2
n z−fn1−1zdf
2
n z
dz
⎤ ⎦
i1−z−2 fn1z
2
fn2z
2
3.12
or
−df
1 0 z
dz f
2
1 z i
1−z−2
∞
n1
fn1z
2
fn2z
2
. 3.13
It follows from3.13that
d dzf
1 0 z
zz0
/
0, 3.14
Letδdenote the infimum of distances between two neighboring zeros off01. We show thatδ > 0. Otherwise, we can take a sequence of zeroszkandwkof the functionf01, such
that
lim
k→ ∞zk−wk 0. 3.15
It follows from2.4that, for largep∈,
∞
np
%
fn1zkfn1wk fn2zkfn2wk
&
≥M 3.16
holds, whereM >0. From the equation
∞
n1
%
fn1zkfn1wk fn2zkfn2wk
&
0, 3.17
we get
lim inf
k→ ∞
∞
np
%
fn1zkfn1wk fn2zkfn2wk
&
≤0. 3.18
There is a contradiction comparing3.16and3.18. Soδ > 0 andf01 function has only a finite number of zeros.
References
1 V. A. Marchenko,Sturm-Liouville Operators and Applications, vol. 22 ofOperator Theory: Advances and Applications, Birkh¨auser, Basel, Switzerland, 1986.
2 B. M. Levitan,Inverse Sturm-Liouville Problems, VSP, Zeist, The Netherlands, 1987.
3 K. Chadan and P. C. Sabatier,Inverse Problems in Quantum Scattering Theory, Springer, New York, NY, USA, 1977.
4 V. E. Zaharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevski˘ı,Theory of Solutions, Plenum Press, New York, NY, USA, 1984.
5 B. M. Levitan and I. S. Sargsjan,Sturm-Liouville and Dirac Operators, vol. 59 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.
6 M. G. Gasymov and B. M. Levitan, “Determination of the Dirac system from the scattering phase,”
Doklady Akademii Nauk SSSR, vol. 167, pp. 1219–1222, 1966.
7 M. Jaulent and C. Jean, “The inverses-wave scattering problem for a class of potentials depending on energy,”Communications in Mathematical Physics, vol. 28, pp. 177–220, 1972.
8 E. Bairamov and ¨O. Karaman, “Spectral singularities of Klein-Gordons-wave equations with an integral boundary condition,”Acta Mathematica Hungarica, vol. 97, no. 1-2, pp. 121–131, 2002.
9 M. Adıvar and M. Bohner, “Spectral analysis ofq-difference equations with spectral singularities,”
Mathematical and Computer Modelling, vol. 43, no. 7-8, pp. 695–703, 2006.
10 A. M. Krall, “A nonhomogeneous eigenfunction expansion,”Transactions of the American Mathematical Society, vol. 117, pp. 352–361, 1965.
12 V. E. Lyance, “A differential operator with spectral singularities, II,”AMS Translations, vol. 2, no. 60, pp. 227–283, 1967.
13 E. Bairamov and A. O. C¸ elebi, “Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators,”The Quarterly Journal of Mathematics. Oxford Second Series, vol. 50, no. 200, pp. 371–384, 1999.
14 A. M. Krall, E. Bairamov, and ¨O. C¸ akar, “Spectrum and spectral singularities of a quadratic pencil of a Schr ¨odinger operator with a general boundary condition,”Journal of Differential Equations, vol. 151, no. 2, pp. 252–267, 1999.
15 E. Bairamov, ¨O. C¸ akar, and A. M. Krall, “An eigenfunction expansion for a quadratic pencil of a Schr ¨odinger operator with spectral singularities,”Journal of Differential Equations, vol. 151, no. 2, pp. 268–289, 1999.
16 E. Bairamov, ¨O. C¸ akar, and A. M. Krall, “Non-selfadjoint difference operators and Jacobi matrices with spectral singularities,”Mathematische Nachrichten, vol. 229, pp. 5–14, 2001.
17 A. M. Krall, E. Bairamov, and ¨O. C¸ akar, “Spectral analysis of non-selfadjoint discrete Schr ¨odinger operators with spectral singularities,”Mathematische Nachrichten, vol. 231, pp. 89–104, 2001.
18 R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of
Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
19 R. P. Agarwal and P. Y. J. Wong,Advanced Topics in Difference Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
20 W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Har-court/Academic Press, San Diego, Calif, USA, 2nd edition, 2001.
21 M. Toda,Theory of Nonlinear Lattices, vol. 20 ofSpringer Series in Solid-State Sciences, Springer, Berlin, Germany, 1981.
