Excerpt
Chapter 1
Introduction
1.1
Background
Sturm-Liouville operators and Jacobi matrices have been developed in
parallel in recent years. Actually, Sturm-Liouville equations and their
discrete counterparts, Jacobi matrices are analysed using similar and re-
lated methods. Therefore, there is no doubt that the theory of Jacobi
matrices is far much developed. This shows that the theory of difference
equations have surely grown.
In this study, we have investigated the absolutely continuous spectrum of
a fourth order self-adjoint extension operator of minimal operator gener-
ated by difference equation;
Ly(t) = w
-1
(t)
4
y(t - 2) - i{ (q(t)
2
y(t - 2)) +
2
(q(t) y(t - 1)} -
(p(t) y(t - 1)) +
i{r(t) y(t - 1) +
(r(t)y(t)} + m(t)y(t),
(1.1)
1
CHAPTER 1. INTRODUCTION
2
defined on a weighted Hilbert space
2
w
(
N) with the weight function w(t) >
0, t N where p(t), q(t), r(t) and m(t) are real-valued functions.Here the
equation is in the form that makes it symmetric and also of order 4. In
this case the coefficients are allowed to be unbounded. is a forward dif-
ference operator such that f (t) = f (t + 1) - f (t); for t N.The method
applied is asymptotic summation as outlined in Levinson-Benzaid -Lutz
theorem [8] and whose spectral parameter uniform version is given in
[1,4,5]. For simplicity in computation and analysis ,we have assumed
that w(t) = 1 unless otherwise stated. For the spectral analysis we
have solved the equation Ly(t) = zy(t) where L is the difference oper-
ator generated by (1.1) and z is the spectral parameter,z C. We have
applied the M-matrix theory as developed in Hinton and Shaw [14] in
order to compute the spectral multiplicity and the location of the ab-
solutely continuous spectrum of self-adjoint extension operator. These
results has been an extension of some known spectral results of fourth
order differential operators to difference setting. Similarly,they have ex-
tended results found in Jacobi matrices [10] .In this thesis,chapter 1 is
about introduction and some preliminary results including literature re-
view,objectives,methodology and basic definitions.In chapter 2,we have
given the results on the computation of the eigenvalues,dichotomy condi-
tions and some results on singular continuous spectrum.Chapter 3 con-
tains the main results in deficiency indices,absolutely continuous spectrum
and the spectral multiplicity.Finally,we have summarized our results in
chapter 4 and also highlighted areas of further research.
CHAPTER 1. INTRODUCTION
3
1.2
Basic Concepts
Definition 1.2.1
An operator T defined on Hilbert space H is said to be symmetric if it is
densely defined and D(T ) D(T
) where T
is a Hilbert adjoint of T
Definition 1.2.2
An operator T on a Hilbert space H is said to be self-adjoint if T = T
so that self adjoint operators are symmetric by definition.
Definition 1.2.3
Let T be an operator defined on Hilbert space H . A number is said
to be in the spectrum of T if the operator T - I is not invertible. The
spectrum of T is denoted by (T ) and is defined by
(T ) = { C : T - I is not invertible}
. In addition, the complement of the spectrum,
C\(T ) is called the re-
solvent operator T and is denoted by (T ) that is,
(T ) = { C : T - I is invertible}
. In this, one says R
(T ) = (T - I)
-1
is the resolvent operator of T .
Here
(T ) (T ) =
We note that an operator T - I fails to be invertible if it is neither
one-to-one nor onto.
Definition 1.2.4
If the operator is not one-to -one, it implies that is an eigenvalue of the
CHAPTER 1. INTRODUCTION
4
operator T . Thus the set of such C which makes T - I not one-to-
one forms the components of the spectrum known as the point(discrete)
spectrum denoted by
(T )
Definition 1.2.5
If T - I is not invertible (does not have a bounded inverse) because
T -I is not onto then the spectral values in this case form a continuous
spectrum. The set of all such is denoted by
c
(T ) = { C : T - I
does not have a bounded inverse, T - I is not onto}.
Definition 1.2.6
Let T be the maximal multiplication operator defined by Tu(x) = xu(x)
on a Hilbert space H,then the spectrum of T is absolutely continuous
with the D(T) consisting of all u H with xu(x) H.
Definition 1.2.7
A mapping
is known as forward difference operator if for any function
f (t), t N then
f (t) = f (t + 1) - f (t).
Similarly
or
is backward operator if
f(t) = f(t) - f(t - 1).
Definition 1.2.8
Let H be a separable Hilbert space and let T be a densely defined sym-
metric linear operator on H. The operator T is closed if its graph
{x T x H H : x D(T )}
CHAPTER 1. INTRODUCTION
5
is closed. If T is a symmetric operator on H with D(T ) D(T ) and
T - \D(T ) = T we call T' a symmetric extension of T. symmetric op-
erators have maximal symmetric extensions and the maximal symmetric
extensions are closed only if not self-adjoint. In order to convert equa-
tion (1.1), into a first order system, we define the vector valued functions
x(t),u(t) and y(t) by,
x(t) = (x
1
(t), x
2
(t))
tr
, u(t) = (u
1
(t), u
2
(t))
tr
, y(t) = (x(t), u(t))
tr
where the superscript tr denotes transpose and
x
1
(t) = y(t - 1)
x
2
(t) =
y(t - 2)
u
1
(t) = p(t) y(t - 1) -
3
y(t - 2) + i{ (q(t) y(t - 1)) +
q(t)
2
y(t - 2)} - ir(t)y(t)
u
2
(t) =
2
y(t - 2) - iq(t) y(t - 1).
Now we let
x(t) =
x
1
(t)
x
2
(t)
and
u(t) =
u
1
(t)
u
2
(t)
Therefore the discrete linear Hamiltonian system as outlined by Hinton
and Shaw[14] for differential operators and discritised by Shi[20] is of the
CHAPTER 1. INTRODUCTION
6
form
J Y (t) = [zW (t) + P (t)]R(Y )(t)
(1.2)
where t N, W (t) and P (t) are 4 x 4 complex Hamiltonian matrices.
W (t) = diag(w(t), 0..., 0), w(t) is a weighted function, x(t), u(t) C
2
, J
is a symplectic matrix, that is
J =
0
-I
2
I
2
0
and
P(t) =
-C
(t) A
(t)
A(t)
B(t)
.
For non-zero elements of 2 x 2 matrices A,B, and C are given by
A
1,2
= 1, A
2,2
= iq, B
2,2
= 1, C
1,1
= m, C
1,2
=
-C
2,1
= ir, and C
2,2
= p
Definition 1.2.9
Let
2
w
[(0, )] be a Hilbert space with weight function w(t) and define
this Hilbert space using the vector valued function x(t), u(t) and Y (t) by
2
w
[(0, )] = {y; y = y(t)
t=0
C and
t=0
(RY
)(t)W (t)(RY )(t) < }
where RY (t) is a partial shift operator
Ry(t) =
x
(t + 1)
u(t)
Like in differential operators,a regularity condition is needed for spectral
analysis of higher order difference operators,that is, there exists an n
0
such
that non-trivial solutions Y(t,z) of (1.1) viz (1.2) and all z C,Shi[20]
CHAPTER 1. INTRODUCTION
7
n
t=0
(RY (t, z)
W (t)(RY (t, z)) > 0, n 0.
The scalar product for the vector valued functions system is,
t=0
y
1
(t + 1)w(t)y(t + 1) = y
1
, y w, y, y
1
2
w
([o, ) see [20].
In this case, one defines maximal difference operator L
on
2
w
(
N) by
D(L
) =
{y(t)
2
w
(
N):there exists f(t)
2
w
(
N) such that
JY (t) - P (t)RY (t) = W (t)f (t), t N.
This implies that for y(t),f (t)
2
w
(
N),then
L
y(t) = w(t)f (t).
The restriction of L
by boundary conditions at 0 and all t n + 1 for
some n N results into a pre-minimal difference operator defined by
D(L ) = {y(t) D(L
): there exists n N such that y(0) = y(t) = 0, for
all t n + 1}.Thus for y(t) = D(L ) then
L
y(t) = L y(t).
The closure of pre-minimal operator L , ¯
L is defined as the minimal dif-
ference operator. This means that a minimal operator is a restriction of
maximal operator L
. We shall denote by L and L
minimal and maximal
operators respectively. It follows that L and L
are symmetric, L L
and L = L
as required.
In order to compute deficiency indices of L,we have used a > 2 as the
left-end point in order for L
to be densely defined .The result can be ex-
trapolated to the set
N using Remlings results [19] since deficiency indices
of an operator are invariant of left-end points
Definition 1.2.10
The deficiency indices of the operator L is the pair (N
-
, N
+
) defined
CHAPTER 1. INTRODUCTION
8
by, dim N(L
± iI) and denoted by N
-
and N
+
, for dim N(L
- i) and
dim N(L
+ i) respectively. Here N(L
± i) is the null space of L
± i.
Thus
2
N
-
, N
+
4
and if N
-
= N
+
then, there exists a symmetric self-adjoint extension H
of an operator L.
Definition 1.2.11
Let
Y
(., z) =
U
(., z)
V
(., z)
be the fundamental matrix of
J
x
(t)
u(t)
=
0
-
0
x
(t)
u(t)
=
-C
(t) + zW A
(t)
A(t)
B(t)
x
(t)
u(t)
(1.3)
with initial values of
Y
(a, z) =
1
-
2
2
1
,
where
1
,
2
satisfy
1
1
+
2
2
= I
2
,
1
2
-
2
1
= 0
2
and
1
2
Y (a) = 0
(1.4)
CHAPTER 1. INTRODUCTION
9
1
and
2
are 2 x 2 matrices, that is = (
1
,
2
)
C
2
x
2
.
U
, V
are 4 x 2 complex-valued matrices whose every column solves
Ly = zy and that V
(, ..z) satisfy self-adjoint boundary conditions at
a. Thus,columns of Y
(, ...z) span the 4-dimensional vector space of solu-
tions of (1.3). Therefore in the limit point case with Imz > 0 one has a
matrix M C
2
x
2
such that
X
(t, z) = Y
(a, z) =
I
2
M(z)
= U
(t, z) + V
(t, z)M(z)
where
X
(t, z) satisfy the boundary conditions of (1.4).It has been shown
in [20] that if L is limit point as t ,then one can construct the
M-matrix M(z) for the Hamiltonian restriction to [a, ) with Dirichlet
boundary conditions.To do this ,let
W
1
(a, z)
W
2
(a, z)
be a system of 2 square summable solutions for Imz > 0.
Then from the theory of Hinton and Shaw [14], it follows that this solu-
tions also arise from Y
a
(t, z)
I
n
M(z)
,where Y
a
(t, z) is the fundamental
solution of the system satisfying the appropriate boundary conditions at
a.
Definition 1.2.12
If L is limit point, then L has self-adjoint extensions.With a=0,the self-
adjoint extension H of L are precisely defined by
D(H) = y D(L
) : (
1
,
2
)y(0) = 0, L
y = Hy
and L H = H
L
.
CHAPTER 1. INTRODUCTION
10
1.3
Literature Review
The spectral analysis and deficiency indices of Sturm-Liouville operat-
ors have been generating a lot of interest in the field of mathematical
research. Sturm-Liouville equations and their discrete counterparts, Jac-
obi matrices are analysed in related and almost in a similar way. It is
a known fact that the spectral theory of Sturm- Lioville operators and
Jacobi matrices are developed in parallel.
The accelerated growth of the theory of difference equations has played an
important role in the applicable analysis and in mathematical research as
a whole. The difference equations appeared earlier than differential equa-
tions and played an important role in the development of the latter.
The qualitative study of the solutions of difference systems is periodic
and one can easily include method of variation of constants, the concept
of exact and adjoint equations and Lagranges' and Green's identities into
this analysis. The method of generating functions, a very important tech-
nique for obtaining the closed form solutions of higher order difference
equations will follow immediately.
Currently,there are research papers that have developed and expanded the
M-function theory for difference systems. These papers include Fischer
and Remling[12], Clark and Gesztesy [9], Behncke and Nyamwala [4,5],
Behncke [7], Shi [20].For example,Remling made an attempt to establish
asymptotic integration as a valuable tool in spectral analysis in conjuc-
tion with the theory of the M-matrix.Remling could prove some results on
the spectral theory of fourth order operators,though unbounded middle
CHAPTER 1. INTRODUCTION
11
terms formed an obstacle.Behncke,Hinton and Remling finally developed
the spectral theory for higher even order operators with bounded coeffi-
cients satisfying some regularity conditions.Due to this and other results
on asymptotic integration by Behncke and Hinton,it was clear that one
obstacle to analysis of the absolutely continuous spectrum of operators
with unbounded coefficients, is the understanding of the zero's of polyno-
mials, here the Fourier polynomials. This is experienced especially when
proving some results on the spectral theory of fourth order operators,
where the middle terms form an obstacle even though unbounded. The
theory of M-functions as developed in these papers are equivalent but the
approach in [20] has been relevant in this study because the results are
closer to the traditional approach of Hinton and Shaw [14]. Actually, the
analysis has been parallel to that of Shi [20].
Even though attempts have been made to compute deficiency indices
and the location of absolutely continuous spectrum of unbounded operat-
ors, much has not been done for discrete operators except for papers by
Behncke and Nyamwala [4,5] and that of Agure, Ambogo and Nyamwala
[1] where the coefficients that were taken to be unbounded were the even
order coefficients.
We have investigated the absolutely continuous spectrum of fourth or-
der difference operator generated by (1.1) when odd order coefficients are
unbounded. This has been done using asymptotic summation. Asymp-
totic summation is based on the discretized version of Levinson's theorem
which appeared in the Benzaid-Lutz paper [8] and the result which is z-
uniform is stated here below,since the assumptions in the Theorem have
Excerpt out of 54 pages
- Quote paper
- Evans Mogoi (Author), 2015, Absolutely continuous spectrum of fourth order difference operators with unbounded coefficients on a Hilbert space, Munich, GRIN Verlag, https://www.grin.com/document/375444
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