Hacettepe Journal of Mathematics and Statistics
Volume 45 (4) (2016), 999 1005
Investigation of spectral analysis of matrix
quantum dierence equations with spectral
singularities
Yelda Aygar∗†
Abstract
In this paper, we investigate the Jost solution, the continuous spectrum,
the eigenvalues and the spectral singularities of a nonselfadjoint matrixvalued q -dierence equation of second order with spectral singularities.
Quantum dierence equation, Discrete spectrum, Spectral theory,
Spectral singularity, Eigenvalue.
2000 AMS Classication: 39A05, 39A70, 47A05, 47A10, 47A55.
Keywords:
Received : 17.02.2015 Accepted : 02.07.2015
Doi : 10.15672/HJMS.20164513107
1. Introduction
Spectral analysis of nonselfadjoint dierential equations including SturmLiouville,
Schrödinger and KleinGordon equations has been treated by various authors since 1960
[23, 9, 11, 22, 12]. Study of spectral theory of nonselfadjoint discrete Schrödinger and
Dirac equations were obtained in [1, 20, 8, 10, 7]. Also, spectral analysis of these equations
in self-adjoint case is well-known [4, 5]. In addition to dierential and discrete equations,
spectral theory of q -dierence equations has been investigated in recent years [2, 3],
and important generalizations and results were given for dynamic equations including
q -dierence equations as a special case in [14, 13].
Some problems of spectral theory of dierential and dierence equations with matrix
coecients were studied in [15, 24, 18, 6]. But spectral analysis of the matrix q -dierence
equations with spectral singularities has not been investigated yet.
In this paper, we let q > 1 and use the notation q N0 := {q n : n ∈ N0 }, where N0 denotes
N
m
the set of nonnegative integers. Let us introduce the Hilbert space
P `2 (q , C ) consisting
m
N
of all vector sequences y ∈ C , (y = y(t), t ∈ q ), such that t∈qN µ(t)ky(t)k2Cm < ∞
P
with the inner product hy, ziq := t∈qN µ(t) (y(t), z(t))Cm , where Cm is m-dimensional
(m < ∞) Euclidean space, µ(t) = (q − 1)t for all t ∈ q N , and k · kCm and (·, ·)Cm denote
∗University
of Ankara, Faculty of Science, Department of Mathematics, 06100, Ankara,
Turkey, Email: [email protected]
†Corresponding Author.
1000
the norm and inner product in Cm , respectively. We denote by L the operator generated
in `2 (q N , Cm ) by the q -dierence expression
t
t
(ly)(t) := qA(t)y(qt) + B(t)y(t) + A
y
, t ∈ qN ,
q
q
and the boundary condition y(1) = 0, where A(t), t ∈ q N0 and B(t), t ∈ q N are linear
operators (matrices) acting in Cm . Throughout the paper, we will assume that A(t) is
invertible and A(t) 6= A∗ (t) for all t ∈ q N0 . Furthermore B(t) 6= B ∗ (t) for all t ∈ q N ,
where ∗ denotes the adjoint operator. It is clear that L is a nonselfadjoint operator in
`2 (q N , Cm ). Related to the operator L, we will consider the matrix q -dierence equation
of second order
t
t
(1.1)
qA(t)y(qt) + B(t)y(t) + A
y
= λy(t), t ∈ q N ,
q
q
where λ is a spectral parameter.
The set up of this paper is summarized as follows: Section 2 discusses the Jost solution
of (1.1) and contains analytical properties and asymptotic behavior of this solution. In
Section 3, we give the continuous spectrum of L, by using the Weyl compact perturbation
theorem. In Section 4, we investigate the eigenvalues and the spectral singularities of L.
In particular, we prove that L has a nite number of eigenvalues and spectral singularities
with a nite multiplicity.
L
2. Jost solution of
We assume that the matrix sequences {A(t)} and {B(t)}, t ∈ q N satisfy
X
(2.1)
(kI − A(t)k + kB(t)k) < ∞,
t∈q N
where k · k denotes the matrix norm in Cm and I is identity matrix. Let F (·, z), denotes
the matrix solution of the q -dierence equation
√
t
t
y
= 2 q cos zy(t), t ∈ q N ,
(2.2)
qA(t)y(qt) + B(t)y(t) + A
q
q
satisfying the condition
(2.3)
ln t z
i ln
q
lim F (t, z)e
t→∞
p
µ(t) = I,
z ∈ C+ := {z ∈ C : Im z ≥ 0}.
The solution F (·, z) is called the Jost solution of (2.2).
2.1. Theorem. Assume
Then
(2.4)
(2.1). Let the solution F (·, z) be the Jost solution of (2.2).
ln t z
i ln
q
e
F (t, z) = p
µ(t)
X
I+
s∈[qt,∞)∩q N
r
s
qt
sin
ln s−ln t
ln q
sin z
z
H(s),
where
s
s
H(s) := I − A
F
, z − B(s)F (s, z) + q[I − A(s)]F (qs, z).
q
q
Proof. Using (2.2), we obtain
(2.5)
F
√
t
+ qF (qt) − 2 q cos zF (t) = H(t).
q
1001
Since
ln t z
exp i ln
q
√
µ(t)
equation
F
I and
ln t z
exp −i ln
q
√
µ(t)
I are linearly independent solutions of the homogeneous
√
t
+ qF (qt) − 2 q cos zF (t) = 0,
q
we get the general solution of (2.5) by
−i ln t z
i ln t z
(2.6)
e ln q
e ln q
α+ p
β
F (t, z) = p
µ(t)
µ(t)
s
X
µ(s) 1
p
+
q
µ(t)
s∈[qt,∞)∩q N
sin
ln s−ln t
ln q
sin z
z
H(s),
where α and β are constants in Cm . Using (2.1), (2.3), and (2.6), we nd α = I and
β = 0. This completes the proof, i.e., F (t, z) satises (2.4).
2.2. Theorem. Assume
(2.7)
(2.1). Then the Jost solution F (·, z) has a representation
X
ln r z
e
i
I +
K(t, r)e ln q , t ∈ q N0
F (t, z) = T (t) p
µ(t)
N
r∈q
ln t z
i ln
q
where z ∈ C+ , T (t) and K(t, r) are expressed in terms of {A(t)} and {B(t)}.
Proof. If we put F (·, z) dened by (2.7) into (2.2), then we have the relations
t
1
K(t, q) − K( , q) = √ T −1 (t)B(t)T (t),
q
q
t 2
1
K( , q ) − K(t, q 2 ) = T −1 (t) T (t) − A2 (t)T (t) − √ B(t)T (t)K(t, q) ,
q
q
1
t
K(t, rq 2 ) − K( , rq 2 ) = T −1 (t) A2 (t)T (t)K(qt, r) + √ B(t)T (t)K(t, qr) − K(t, r),
q
q
A(t)T (t) = T (qt),
and using these relations, we obtain
Y
X
1
T (t) =
[A(p)]−1 , K(t, q) = − √
T −1 (p)B(p)T (p),
q
N
N
p∈[t,∞)∩q
p∈[qt,∞)∩q
X
1
2
−1
2
K(t, q ) =
T (p) − √ B(p)T (p)K(p, q) + (I − A (p))T (p) ,
q
p∈[qt,∞)∩q N
X
T −1 (p) I − A2 (p) T (p)K(qp, r)
K(t, rq 2 ) = K(qt, r) +
p∈[qt,∞)∩q N
1
−√
q
X
T −1 (p)B(p)T (p)K(p, qr),
p∈[qt,∞)∩q N
for r ∈ q N and t ∈ q N0 . Due to the condition (2.1), the innite product and the series in
the denition of T (t) and K(t, r) are absolutely convergent.
Note that, in analogy
to the SturmLiouville
equation the function
rz
P
i ln
T (1)
ln
q
√
F (1, z) :=
I + r∈qN K(1, r)e
is called the Jost function.
µ(1)
2.3. Theorem. Assume
(2.8)
X ln t
(kI − A(t)k + kB(t)k) < ∞.
ln q
N
t∈q
1002
Then the Jost solution F (·, z) is continuous in C+ and analytic with respect to z in
C+ := {z ∈ C : Im z > 0}.
Proof. Using the equalities for K(t, r) given in Theorem 2.2 and mathematical induction,
we get
(2.9)
X
kK(t, r)k ≤ C
"
b ln r c
tq 2 ln q ,∞
p∈
(kI − A(p)k + kB(p)k) ,
!
∩q N
where C > 0 is a constant and b 2lnlnrq c is the integer part of
we get that the series
X
X ln r
i ln r z
i ln r z
K(t, r)e ln q ,
K(t, r)e ln q
ln
q
N
N
r∈q
ln r
.
2 ln q
From (2.8) and (2.9),
r∈q
are absolutely convergent in C+ and in C+ , respectively. This completes the proof.
2.4. Theorem. Under the condition
(2.8), the Jost solution satises
ln t z
i ln
q
(2.10)
e
F (t, z) = p
(2.11)
e ln q
F (t, z) = T (t) p
(I + o(1)) , t ∈ q N0 , Im z → ∞.
µ(t)
µ(t)
(I + o(1)) , z ∈ C+ , t → ∞,
i ln t z
Proof. It follows from the denition of T (t), (2.8), and (2.9) that
(2.12)
lim T (t) = I,
t→∞
and
(2.13)
X
rz
i ln
ln q
K(t, r)e
= o(1), z ∈ C+ , t → ∞.
r∈q N
From (2.7), (2.12), and (2.13), we get (2.10). Using (2.8) and (2.9), we have
X
i ln r z
(2.14)
K(t, r)e ln q = o(1), z ∈ C+ , Im z → ∞.
r∈q N
From (2.7) and (2.14), we get (2.11).
3. Continuous spectrum of
L
Let L1 and L2 denote the q -dierence operators generated in `2 (q N , Cm ) by the q dierence expressions
t
(l1 y)(t) = qy(qt) + y
q
and
t
t
(l2 y)(t) = q [A(t) − I] y(qt) + B(t)y(t) + A
−I y
q
q
with the boundary condition y(1) = 0, respectively. It is clear that L = L1 + L2 .
3.1. Lemma. The operator L1 is self-adjoint in `2 (qN , Cm ).
1003
Proof. Since
√
kL1 ykq ≤ 2 qkykq
for all y ∈ `2 (q N , Cm ), L1 is bounded in the Hilbert space `2 (q N , Cm ), and since
X
t
hl1 y, ziq =
µ(t)(z(t))∗ qy(qt) + y
q
t∈q N
∗
X
t
=
µ(t) qz(qt) + z
y(t) = hy, l1 ziq ,
q
N
t∈q
the operator L1 is self-adjoint in `2 (q N , Cm ).
√ √
3.2. Theorem. Assume (2.8). Then σc (L) = [−2 q, 2 q], where σc (L) denotes the
continuous spectrum of L.
Proof. It is easy to see that L1 has no eigenvalues, so the spectrum of the operator L1
consists only its continuous spectrum and
√ √
σ(L1 ) = σc (L1 ) = [−2 q, 2 q],
where σ(L1 ) denotes the spectrum of the operator L1 . Using (2.8), we nd that L2 is
compact operator in `2 (q N , Cm )[21]. Since L = L1 + L2 and L1 = (L1 )∗ , we obtain that
√ √
σc (L) = σc (L1 ) = [−2 q, 2 q]
by using Weyl's theorem of a compact perturbation [19, p.13].
4. Eigenvalues and spectral singularities of
L
If we dene
(4.1)
f (z) := detF (1, z), z ∈ C+ ,
then the function f is analytic in C+ , f (z) = f (z + 2π) and is continuous in C+ . Let
us dene the semi-strips P0 = {z ∈ C+ : 0 ≤ Re z ≤ 2π} and P = P0 ∪ [0, 2π]. We will
denote the set of all eigenvalues and spectral singularities of L by σd (L) and σss (L), respectively. From the denitions of eigenvalues and spectral singularities of nonselfadjoint
operators[22, 23], we have
√
(4.2)
σd (L) = {λ ∈ C : λ = 2 q cos z, z ∈ P0 , f (z) = 0} ,
√
(4.3)
σss (L) = {λ ∈ C : λ = 2 q cos z, z ∈ [0, 2π], f (z) = 0} \{0}.
4.1. Theorem. Assume
(2.8). Then
i) the set σd (L) is bounded and countable, and its limit points lie only in the interval
√ √
[−2 q, 2 q],
√ √
ii) σss (L) ⊂ [−2 q, 2 q] and the linear Lebesgue measure of the set σss (L) is zero.
Proof. In order to investigate the quantitative properties of eigenvalues and spectral
singularities of L, it is necessary to discuss the quantitative properties of zeros of f in P
from (4.2) and (4.3). Using (2.11) and (4.1), we get
1
(4.4)
f (z) = det T (1)
[I + o(1)], Im z > 0, z ∈ P0 , Im z → ∞,
µ(1)
where detT (1) 6= 0. From (4.4), we get the boundedness of zeros of f in P0 . Since f is
a 2π -periodic function and is analytic in C+ , we obtain that f has at most a countable
number of zeros in P0 . By the uniqueness of analytic functions, we nd that the the
√ √
limit points of zeros of f in P0 can lie only in [0, 2π]. We get σss (L) ⊂ [−2 q, 2 q] using
1004
(4.3). Since f (z) 6= 0 for all z ∈ C+ , we get that the linear Lebesgue measure of the set
of zeros of f on real axis is not positive, by using the boundary uniqueness theorem of
analytic functions [17], i.e., the linear Lebesgue measure of the σss (L) is zero.
4.2. Denition. The multiplicity of a zero of f in P is called the multiplicity of the
corresponding eigenvalue or spectral singularity of L.
4.3. Theorem. If, for some ε > 0,
(4.5)
n ln t
o
ε
sup e ln q (kI − A(t)k + kB(t)k) < ∞
t∈q N
then the operator L has a nite number of eigenvalues and spectral singularities, and each
of them is of nite multiplicity.
Proof. Since F (1, z) =
T (1)
√
q−1
that
(4.6)
ε ln r
−4
ln q
kK(1, r)k ≤ De
P
i ln r z
I + r∈qN K(1, r)e ln q , using (2.9) and (4.5), we get
, r ∈ qN ,
where D > 0 is a constant. From (4.1) and (4.6), we obtain that the function f has an
analytic continuation to the half-plane Im z > − 4ε . Because the series
X
rz
ln r i ln
e ln q
iK(1, r)
ln
q
N
r∈q
is uniformly convergent in Im z > − 4ε . Since f is a 2π periodic function, the limit points
of its zeros in P cannot lie in [0, 2π]. Using Theorem 4.1, we nd that the bounded
sets σd (L) and σss (L) have no limit points, i.e., the sets σd (L) and σss (L) have a nite
number of elements. From the analyticity of f in Im z > − 4ε , we get that all zeros of f
in P have a nite multiplicity.
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