Dissipative Dirac operator with general boundary conditions on time scales
UDC 517.9 In this paper, we consider the symmetric Dirac operator on bounded time scales. With general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint and the other) of such symmetric operators. We construct a self-adjoint dilation of dissipative...
Saved in:
| Date: | 2020 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2020
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/546 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507040136298496 |
|---|---|
| author | Allahverdiev, B. P. Tuna, H. Allahverdiev, B. P. Tuna, H. Allahverdiev, B. P. Tuna, H. |
| author_facet | Allahverdiev, B. P. Tuna, H. Allahverdiev, B. P. Tuna, H. Allahverdiev, B. P. Tuna, H. |
| author_sort | Allahverdiev, B. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:39Z |
| description |
UDC 517.9
In this paper, we consider the symmetric Dirac operator on bounded time scales. With general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint and the other) of such symmetric operators. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.
|
| doi_str_mv | 10.37863/umzh.v72i5.546 |
| first_indexed | 2026-03-24T02:02:59Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i5.546
UDC 517.9
B. P. Allahverdiev (Süleyman Demirel Univ., Isparta, Turkey),
H. Tuna (Mehmet Akif Ersoy Univ., Burdur, Turkey)
DISSIPATIVE DIRAC OPERATOR
WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES
ДИСИПАТИВНИЙ ОПЕРАТОР ДIРАКА
IЗ ЗАГАЛЬНИМИ ГРАНИЧНИМИ УМОВАМИ НА ЧАСОВИХ ШКАЛАХ
In this paper, we consider the symmetric Dirac operator on bounded time scales. With general boundary conditions, we
describe extensions (dissipative, accumulative, self-adjoint and the other) of such symmetric operators. We construct a
self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a
functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator
are complete.
Розглядається симетричний оператор Дiрака на обмежених часових шкалах. При загальних граничних умовах опи-
сано розширення (дисипативнi, акумулятивнi, самоспряженi та iншi) таких симетричних операторiв. Побудовано
самоспряжене розширення дисипативного оператора та визначено матрицю розсiювання дилатацiї. Також побудо-
вано функцiональну модель цього оператора та визначено його характеристичну функцiю. Насамкiнець доведено,
що всi кореневi вектори цього оператора є повними.
1. Introduction. Continuous systems and discrete systems are two important types of the dynamic
systems. The time scale calculus is one of the main approaches in order to unify continuous and
discrete analysis. It was founded by Stefan Hilger in his Ph. D. thesis [15]. Since then it has received
much attention because it has important applications in some mathematical models of real processes
and phenomena studied in physics, chemical technology, population dynamics, biotechnology and
economics, neural networks and social sciences. For more information, we refer the reader to the
references [16 – 21].
The spectral theory of Sturm – Liouville equations on time scales has been developed extensively
during the past several years. We refer the reader to [26 – 37]. However, in the literature, there exists
a few research about the Dirac system on time scales [25, 38]. In [25], Gulsen and Yılmaz studied
an eigenvalue problem for the Dirac system with separated boundary conditions on an arbitrary time
scale. They improved the results about the spectral theory of the classical Dirac system, such as the
orthogonality of eigenfunctions and the simplicity of the eigenvalues. In [38], the author gave some
sufficient conditions for the disconjugacy of Dirac systems and obtained a formula about the number
of the eigenvalues of the problem. In this paper, we prove some theorems on the completeness of the
system of root functions of the Dirac system on time scales. Hence, our study could fill an important
gap in the spectral theory of the Dirac system on time scales.
The Dirac system is a cornerstone in the history of physics. It provides a natural description of
the electron spin, predicts the existence of antimatter and is able to reproduce accurately the spectrum
c\bigcirc B. P. ALLAHVERDIEV, H. TUNA, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5 583
584 B. P. ALLAHVERDIEV, H. TUNA
of the hydrogen atom. Dirac systems describes particles known as fermions, such as electrons. For
more information, we refer the reader to the books [14, 22, 23].
In the operator theory, dissipative operators is one of the important class of operators. When we
study the spectral analysis of dissipative operators, some of the basic methods are resolvent analysis,
Riesz integrals and the theory of dilations with applications of functional models. Using a functional
model of a dissipative operator, spectral properties of such operators were investigated in [3 – 5, 9 –
11]. In this article, we use this method for the one dimensional dissipative Dirac operator on time
scales. Although the Dirac system on the time scale is not a particular example of the known general
theory because \Delta -differentiation is different from ordinary differentiation, the main results of this
time scale version coincide with the general theory.
The paper is organized as follows. In Section 2, some preliminary concepts related to time
scales are presented for the convenience of the reader. In Section 3, we describe all the maximal
dissipative, maximal accumulative, self-adjoint and other extensions of minimal symmetric Dirac
operator derived from the one dimensional Dirac system
- \Delta y\rho 2 + p(t)y1 = \lambda y1, t \in \BbbT ,
\Delta y1 + r(t)y2 = \lambda y2,
where p(.) and r(.) are real-valued continuous functions defined on bounded time scale \BbbT , y\rho 2(t) =
= y2 (\rho (t)) and p(.), r(.) \in L1
\Delta (\BbbT ) . In Section 4, we construct a self-adjoint dilation of dissipative
operator and its incoming and outgoing spectral representations of dilation. Hence, we determine
the scattering matrix of the dilation according to the Lax and Phillips scheme [1, 2]. We construct a
functional model of the maximal dissipative Dirac operator with general boundary conditions, using
incoming spectral representations. Further, we determine characteristic function of this operator.
Finally, in Section 5, we prove that all root vectors of the maximal dissipative Dirac operator are
complete.
2. Preliminaries. Let \BbbT be a time scale, i.e., a nonempty closed subset of real numbers \BbbR . The
forward jump operator \sigma : \BbbT \rightarrow \BbbT is defined by
\sigma (t) = \mathrm{i}\mathrm{n}\mathrm{f} \{ s \in \BbbT : s > t\} , where t \in \BbbT ,
and the backward jump operator \rho : \BbbT \rightarrow \BbbT is defined by
\rho (t) = \mathrm{s}\mathrm{u}\mathrm{p} \{ s \in \BbbT : s < t\} , where t \in \BbbT .
It is convenient to have graininess operators \mu \sigma : \BbbT \rightarrow [0,\infty ) and \mu \rho : \BbbT \rightarrow ( - \infty , 0] defined by
\mu \sigma (t) = \sigma (t) - t
and
\mu \rho (t) = \rho (t) - t,
respectively.
Definition 1. A point t \in \BbbT is left scattered if \mu \rho (t) \not = 0 and left dense if \mu \rho (t) = 0. A point
t \in \BbbT is right scattered if \mu \sigma (t) \not = 0 and right dense if \mu \sigma (t) = 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
DISSIPATIVE DIRAC OPERATOR WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES 585
Now, we introduce the sets \BbbT k, \BbbT k which are derived form the time scale \BbbT as follows. If \BbbT
has a left scattered maximum t1, then \BbbT k = \BbbT - \{ t1\} , otherwise \BbbT k = \BbbT . If \BbbT has a right scattered
minimum t2, then \BbbT k = \BbbT - \{ t2\} , otherwise \BbbT k = \BbbT .
Definition 2. A function f on \BbbT is said to be \Delta -differentiable at some point t \in \BbbT k if there is
a number f\Delta (t) such that for every \varepsilon > 0 there is a neighborhood U \subset \BbbT of t such that
| f(\sigma (t)) - f(s) - f\Delta (t)(\sigma (t) - s)| \leq \varepsilon | \sigma (t) - s| , where s \in U. (1)
Analogously, one may define the notion of \nabla -differentiability of some function using the backward
jump \rho .
We note the following.
If t \in \BbbT \setminus \BbbT k, then f\Delta (t) is not uniquely defined, since for such a point t, small neighborhoods
U of t consist only of t, and besides, we have \sigma (t) = t. Therefore, (1) holds for an arbitrary number
f\Delta (t)(see [29]).
One can show (see [29])
f\Delta (t) = f\nabla (\sigma (t)), f\nabla (t) = f\Delta (\rho (t))
for continuously differentiable functions.
If \BbbT = \BbbR , then
f\Delta (t) = f \prime (t).
If \BbbT = h\BbbZ , h > 0, then
f\Delta (t) =
f (t+ h) - f(t)
h
.
If \BbbT = q\BbbN 0 , q > 1, then
f\Delta (t) =
f (qt) - f(t)
(q - 1) t
.
The product and quotient rules on time scales have the following form: If f, g : \BbbT \rightarrow \BbbR , then
(fg)\Delta (t) = f\Delta (t)g(t) + f(\sigma (t))g\Delta (t),
(fg)\nabla (t) = f\nabla (t)g(t) + f(\rho (t))g\nabla (t),\biggl(
f
g
\biggr) \Delta
(t) =
f\Delta (t)g(t) - f(t)g\Delta (t)
g(t)g (\sigma (t))
,
\biggl(
f
g
\biggr) \nabla
(t) =
f\nabla (t)g(t) - f(t))g\nabla (t)
g(t)g (\rho (t))
.
Let f : \BbbT \rightarrow \BbbR be a function, and a, b \in \BbbT . If there exists a function F : \BbbT \rightarrow \BbbR such that
F\Delta (t) = f(t) for all t \in \BbbT k, then F is a \Delta -antiderivative of f. In this case the integral is given by
the formula
b\int
a
f(t)\Delta t = F (b) - F (a) for a, b \in \BbbT .
Analogously, one may define the notion of \nabla -antiderivative of some function. If \BbbT = \BbbR and f is
continuous, then
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
586 B. P. ALLAHVERDIEV, H. TUNA
b\int
a
f(t)\Delta t =
b\int
a
f(t)dt.
If \BbbT = h\BbbZ , h > 0, and a = hx, b = hy, x < y, then
b\int
a
f(t)\Delta t = h
y - 1\sum
k=x
f(hk).
If \BbbT = q\BbbN 0 , q > 1, and a = qx, b = qy, x < y, then
b\int
a
f(t)\Delta t = (q - 1)
y - 1\sum
k=x
qkf
\Bigl(
qk
\Bigr)
.
Let a < b be fixed points in \BbbT and a \in \BbbT k, b \in \BbbT k. Let L2
\Delta (\BbbT ) be the space of all functions
defined on \BbbT such that
\| f\| :=
\left( b\int
a
| f(t)| 2\Delta t
\right) 1/2
< \infty .
The space L2
\Delta (\BbbT ) is a Hilbert space with the inner product (see [24])
(f, g) :=
b\int
a
f(t)g(t)\Delta t, f, g \in L2
\Delta (\BbbT ).
Now, we introduce convenient Hilbert space H := L2
\Delta (\BbbT ;E) (E := \BbbC 2) of vector-valued functions
using the inner product
(f, g)H :=
b\int
a
(f(x), g(x))E\Delta t.
3. All extensions of the symmetric Dirac operators on time scales. In this section, we
construct a space of boundary value for minimal symmetric Dirac operator on time scales and describe
all extensions (maximal dissipative, accumulative, self-adjoint and other) of such operators.
Let us consider the Dirac systems
My :=
\Biggl\{
- \Delta y\rho 2 + p(t)y1
\Delta y1 + r(t)y2
= \lambda y =
\biggl(
\lambda y1
\lambda y2
\biggr)
, t \in \BbbT , (2)
where \Delta f(t) = f\Delta (t), \lambda is a complex eigenvalue parameter, p(.) and r(.) are real-valued continuous
functions defined on bounded time scale \BbbT and p(.), r(.) \in L1
\Delta (\BbbT ).
Let us consider the set Dmax consisting of all vector-valued functions y =
\biggl(
y1
y2
\biggr)
\in H in which
y1 and y2 are \Delta -absolutely continuous functions on \BbbT and My \in H. We define the maximal operator
\Lambda max on the set Dmax by the equality \Lambda maxy = My.
Let Dmin denote the linear set of all vectors y \in Dmax satisfying the conditions
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
DISSIPATIVE DIRAC OPERATOR WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES 587
y1(a) = y\rho 2(a) = y1(b) = y\rho 2(b) = 0.
If we restrict the operator \Lambda max to the set Dmin, then we obtain the minimal operator \Lambda min. It is
clear that \Lambda \ast
min = \Lambda max, and \Lambda min is a closed symmetric operator with deficiency indices (2, 2). In
fact since \BbbT is a bounded time scale, the two linearly independent solutions of the Dirac systems
defined by (2) both lie in L2
\Delta (\BbbT ). Therefore, the deficiency indices of the symmetric operator \Lambda min
are (2, 2) (see [6]). Now we recall the following definitions.
Definition 3. A linear operator S (with dense domain D (S)) acting on some Hilbert space
H is called dissipative (accumulative) if \mathrm{I}\mathrm{m} (Sf, f) \geq 0 (\mathrm{I}\mathrm{m} (Sf, f) \leq 0) for all f \in D (S) and
maximal dissipative (maximal accumulative) if it does not have a proper dissipative (accumulative)
extension (see [3 – 5]).
Definition 4 (space of boundary values). A triplet (\BbbH , T1, T2) is called a space of boundary
values of a closed symmetric operator S with equal deficiency numbers on a Hilbert space H if
T1 and T2 are linear maps from D (S\ast ) to \BbbH , and such that:
i) for every f, g \in D (S\ast ) we have
(S\ast f, g)H - (f, S\ast g)H = (T1f, T2g)\BbbH - (T2f, T1g)\BbbH ;
ii) for any F1, F2 \in \BbbH there is a vector f \in D (A\ast ) such that T1f = F1 and T2f = F2 (see [7]).
Now we have the following lemma.
Lemma 1 (Green’s formula). Let y =
\biggl(
y1
y2
\biggr)
, z =
\biggl(
z1
z2
\biggr)
\in Dmax. Then we have
(My, z) - (y,Mz) = [y, z]b - [y, z]a ,
where [y, z]t := y1(t)z
\rho
2(t) - z1(t)y
\rho
2(t).
Proof. Let y, z \in Dmax. Then we obtain
(My, z)H - (y,Mz)H =
b\int
a
( - \Delta y\rho 2 + p(t)y1) z1\Delta t+
b\int
a
(\Delta y1 + r(t)y2) z2\Delta t -
-
b\int
a
y1( - \Delta z\rho 2 + p(t)z1)\Delta t -
b\int
a
y2(\Delta z1 + r(t)z2)\Delta t =
= -
b\int
a
\Bigl[
(\Delta y\rho 2) z1 + y2(\Delta z1)
\Bigr]
\Delta t+
b\int
a
\Bigl[
(\Delta y1) z2 + y1(\Delta z\rho 2)
\Bigr]
\Delta t.
Since
\Delta
\Bigl(
z1(t)y
\rho
2(t)
\Bigr)
= z1(t) (\Delta y\rho 2(t)) + (y\rho 2(t))
\sigma
(\Delta z1(t)) =
= \Delta y\rho 2(t)z1(t) + y2(t)(\Delta z1(t))
and
\Delta
\Bigl(
z\rho 2(t)y1(t)
\Bigr)
= (\Delta z\rho 2(t))y1(t) + (z\rho 2(t))
\sigma
(\Delta y1(t)) =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
588 B. P. ALLAHVERDIEV, H. TUNA
= (\Delta z\rho 2(t))y1(t) + z2(t)\Delta y1(t).
Hence, we get
(My, z)H - (y,Mz)H = -
b\int
a
\Delta
\Bigl(
z1(t)y
\rho
2(t)
\Bigr)
\Delta t+
b\int
a
\Delta
\Bigl(
y1(t)z
\rho
2(t)
\Bigr)
\Delta t =
=
b\int
a
\Delta
\Bigl[
y1(t)z
\rho
2(t) - z1(t)y
\rho
2(t)
\Bigr]
\Delta t = [y, z]b - [y, z]a .
Lemma 1 is proved.
Let us define by T1, T2 the linear maps from Dmax to \BbbC 2 by the formula
T1y =
\biggl(
- y1(a)
y1(b)
\biggr)
, T2y =
\biggl(
y\rho 2(a)
y\rho 2(b)
\biggr)
. (3)
Now we will prove the following theorem.
Theorem 1. The triplet
\bigl(
\BbbC 2, T1, T2
\bigr)
defined by (3) is a boundary spaces of the operator \Lambda min.
Proof. Let y, z \in Dmax. Then we obtain
(T1y, T2z)\BbbC 2 - (T2y, T1z)\BbbC 2 = - y1(a)z
\rho
2 (a) + z1(a)y
\rho
2(a)+
+y1(b)z
\rho
2(b) - z1(b)y
\rho
2(b).
By using Green’s formula, we get
(T1y, T2z)\BbbC 2 - (T2y, T1z)\BbbC 2 = [y, z]b - [y, z]a .
Hence,
(\Lambda maxy, z)H - (y,\Lambda maxz)H = (T1y, T2z)\BbbC 2 - (T2y, T1z)\BbbC 2 .
Thus, we prove the first condition of the definition of a space of boundary value.
Now, we will prove the second condition. Let
u =
\biggl(
u1
u2
\biggr)
, v =
\biggl(
v1
v2
\biggr)
\in \BbbC 2.
Then the vector-valued function
y(t) =
\biggl(
y1(t)
y2(t)
\biggr)
= \alpha 1(t)u1(t) + \alpha 2(t)v1(t) + \beta 1(t)u2(t) + \beta 2(t)v2(t),
where
\alpha 1(t) =
\biggl(
\alpha 11(t)
\alpha 12(t)
\biggr)
, \alpha 2(t) =
\biggl(
\alpha 21(t)
\alpha 22(t)
\biggr)
,
\beta 1(t) =
\biggl(
\beta 11(t)
\beta 12(t)
\biggr)
, \beta 2(t) =
\biggl(
\beta 21(t)
\beta 22(t)
\biggr)
\in Dmax
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
DISSIPATIVE DIRAC OPERATOR WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES 589
satisfy the conditions
\alpha 11(a) = - 1, \alpha \rho
12(a) = \alpha 11(b) = \alpha \rho
12(b) = 0,
\alpha \rho
22(a) = 1, \alpha 21(a) = \alpha 21(b) = \alpha \rho
22(b) = 0,
\beta 11(b) = 1, \beta 11(a) = \beta \rho
12(a) = \beta \rho
12(b) = 0,
\beta \rho
22(b) = 1, \beta 21(a) = \beta 21(b) = \beta \rho
22(a) = 0,
belong to the set Dmax and T1y = u, T2y = v.
Theorem 1 is proved.
Corollary 1. For any contraction K in \BbbC 2 the restriction of the operator \Lambda max to the set of
functions y \in Dmax satisfying either
(K - I)T1y + i (K + I)T2y = 0 (4)
or
(K - I)T1y - i (K + I)T2y = 0 (5)
is, respectively, the maximal dissipative and accumulative extension of the operator \Lambda min. Conversely,
every maximal dissipative (accumulative) extension of the operator \Lambda min is the restriction of \Lambda max
to the set of functions y \in Dmax satisfying (4) ((5)), and the extension uniquely determines the
contraction K. Conditions (4) ((5)), in which K is an isometry describe the maximal symmetric
extensions of \Lambda min in H. If K is unitary, these conditions define self-adjoint extensions. In the latter
case, (4) and (5) are equivalent to the condition
(\mathrm{c}\mathrm{o}\mathrm{s}S)T1y - (\mathrm{s}\mathrm{i}\mathrm{n}S)T2y = 0,
where S is a self-adjoint operator in \BbbC 2. The general form of dissipative and accumulative extension
of the operator \Lambda min is given by the conditions
K (T1y + iT2y) = T1y - iT2y, T1y + iT2y \in D (K), (6)
K (T1y - iT2y) = T1y + iT2y, T1y - iT2y \in D (K), (7)
where K is a linear operator with
\| Kf\| \leq \| f\| , f \in D (K).
The general form of symmetric extensions is given by formulae (6) and (7), where K is an isometric
operator. In particular, the boundary conditions
y\rho 2(a) - \sigma 1y1(a) = 0, (8)
y\rho 2(b) + \sigma 2y1(b) = 0 (9)
with \mathrm{I}\mathrm{m}\sigma 1 \geq 0 or \sigma 1 = \infty , \mathrm{I}\mathrm{m}\sigma 2 \geq 0 or \sigma 2 = \infty (\mathrm{I}\mathrm{m}\sigma 1 = 0 or \sigma 1 = \infty , \mathrm{I}\mathrm{m}\sigma 2 = 0 or
\sigma 2 = \infty ) describe the maximal dissipative (self-adjoint) extensions of \Lambda min with separated boundary
conditions. Note that if \sigma 1 = \infty (\sigma 2 = \infty ), then the boundary condition (8) ((9)) should be replaced
by y1(a) = 0 (y1(b) = 0).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
590 B. P. ALLAHVERDIEV, H. TUNA
Now, we study the maximal dissipative operator \Lambda K , where K is the strict contraction in \BbbC 2
generated by the expression M and boundary condition (4). Since K is a strict contraction, the
operator K + I must be invertible, and the boundary condition (4) is equivalent to the condition
T2y +GT1y = 0, (10)
where G = - i (K + I) - 1 (K - I), \mathrm{I}\mathrm{m}G > 0, and - K is the Cayley transform of the dissipative
operator G. We denote \Lambda G (= \Lambda K) the dissipative operator generated by the expression M and
boundary condition (10).
Let
G =
\biggl(
\sigma 1 0
0 \sigma 2
\biggr)
,
where \mathrm{I}\mathrm{m}\sigma 1 > 0, \mathrm{I}\mathrm{m}\sigma 2 > 0 and C2 = 2 \mathrm{I}\mathrm{m}G, C > 0. Then the boundary condition (10) coincides
with the separated boundary conditions (8) and (9).
4. Self-adjoint dilation, incoming and outgoing spectral representations. In this section,
we set up a self-adjoint dilation of the maximal dissipative Dirac operator on time scales and its
incoming and outgoing spectral representations. Later, we determine the scattering matrix of the
dilation according to the Lax and Phillips scheme [1, 2]. Using incoming spectral representations,
we establish a functional model of this operator. Finally, we determine characteristic function of this
operator.
Now, we consider the spaces L2(( - \infty , 0);E) and L2((0,\infty );E). The orthogonal sum \scrH =
= L2(( - \infty , 0);E)\oplus H \oplus L2((0,\infty );E) is called main Hilbert space of the dilation.
In the space \scrH , we define the operator \Upsilon on the set D (\Upsilon ) , its elements consisting of vectors
w = \langle \eta - , y, \eta +\rangle , generated by the expression
\Upsilon \langle \eta - , y, \eta +\rangle =
\biggl\langle
i
d\eta -
d\xi
,My, i
d\eta +
d\varsigma
\biggr\rangle
, (11)
\eta - \in W 1
2 (( - \infty , 0);E) , \eta + \in W 1
2 ((0,\infty );E) , y \in H,
T2y +GT1y = C\eta - (0) , T2y +G\ast T1y = C\eta + (0) , C2 := 2 \mathrm{I}\mathrm{m}G, C > 0, (12)
where W 1
2 is the Sobolev space.
Theorem 2. The operator \Upsilon is self-adjoint in \scrH .
Proof. We first prove that \Upsilon is symmetric in \scrH . Let f, g \in D (\Upsilon ) , f = \langle \eta - , y, \eta +\rangle and
g = \langle \zeta - , z, \zeta +\rangle . Then we have
(\Upsilon f, g)\scrH - (f,\Upsilon g)\scrH = i (\eta - (0) , \zeta - (0))E - i (\eta + (0) , \zeta + (0))E + [y, z]b - [y, z]a. (13)
By direct computation, we get
i (\eta - (0) , \zeta - (0))E - i (\eta + (0) , \zeta + (0))E + [y, z]b - [y, z]a = 0.
Thus, \Upsilon \subseteq \Upsilon \ast , i.e., \Upsilon is a symmetric operator.
It is easy to check that \Upsilon and \Upsilon \ast are generated by the same expression (11). Let us describe
the domain of \Upsilon \ast . We shall compute the terms outside the integral sign, which are obtained by
integration by parts in bilinear form (\Upsilon f, g)\scrH , f \in D (\Upsilon ), g \in D (\Upsilon \ast ) . Their sum is equal to zero,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
DISSIPATIVE DIRAC OPERATOR WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES 591
[y, z]b - [y, z]a + i (\eta - (0) , \zeta - (0))E - i (\eta + (0) , \zeta + (0))E = 0.
Further, solving the boundary conditions (12) for T1y and T2y, we find
T1y = - iC - 1 (\eta - (0) - \eta + (0)) , T2y = C\eta - (0) + iTC - 1 (\eta - (0) - \eta + (0)) .
Therefore, using (3), we find that (13) is equivalent to the equality
i (\eta + (0) , \zeta + (0))E - i (\eta - (0) , \zeta - (0))E =
= [y, z]b - [y, z]a = (T1y, T2z)E - (T2y, T1z)E =
= - i
\bigl(
C - 1 (\eta - (0) - \eta + (0)) , T2z
\bigr)
E
- (C\eta - (0) , T1z)E -
- i
\bigl(
TC - 1 (\eta - (0) - \eta + (0)) , T1z
\bigr)
E
.
Since the values \eta \mp (0) can be arbitrary vectors, a comparison of the coefficients of \eta i\mp (0), i = 1, 2,
on the left and right of the last equality proves that the vector g = \langle \zeta - , z, \zeta +\rangle satisfies the boundary
conditions (12), T2z + GT1z = C\zeta - (0), T2z + G\ast T1z = C\zeta + (0). Therefore, D (\Upsilon \ast ) \subseteq D (\Upsilon ),
and, hence, \Upsilon = \Upsilon \ast .
Theorem 2 is proved.
Note that the self-adjoint operator \Upsilon generates a unitary group Ut = \mathrm{e}\mathrm{x}\mathrm{p} (i\Upsilon t) , t \in \BbbR , on \scrH .
Let denote by \scrP : \scrH \rightarrow H and \scrP 1 : H \rightarrow \scrH the mapping acting according to the formulae \scrP :
\langle \eta - , y, \eta +\rangle \rightarrow y and \scrP 1 : y \rightarrow \langle 0, y, 0\rangle . Let Zt := \scrP Ut\scrP 1, t \geq 0, by using Ut. The family \{ Zt\} ,
t \geq 0 of operators is a strongly continuous semigroup of completely nonunitary contraction on H.
Let us denote byB the generator of this semigroup: By = \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow +0
\biggl(
Zty - y
it
\biggr)
. The domain of B
consists of all the vectors for which the limit exists. The operator B is dissipative. The operator \Upsilon
is called the self-adjoint dilation of B. Then we have the following theorem.
Theorem 3. The operator \Upsilon is a self-adjoint dilation of the operator \Lambda G (= \Lambda K).
Proof. We will show that B = \Lambda G, hence \Upsilon is self-adjoint dilation of B. If we prove that the
equality
\scrP (\Upsilon - \lambda I) - 1 \scrP 1y = (\Lambda G - \lambda I) - 1 y, y \in H, \mathrm{I}\mathrm{m}\lambda < 0,
the assertion follows (see [8]). For this purpose, we set (\Upsilon - \lambda I) - 1 \scrP 1y = g = \langle \zeta - , z, \zeta +\rangle implies
that (\Upsilon - \lambda I) g = \scrP 1y, and, hence, Mz - \lambda z = y, \zeta - (\xi ) = \zeta - (0) e - i\lambda \xi and \zeta + (\xi ) = \zeta + (0) e - i\lambda \xi .
Since g \in D (\Upsilon ) , then \zeta - \in W 1
2 (( - \infty , 0);E), it follows that \zeta - (0) = 0, and consequently z
satisfies the boundary condition T2z + GT1z = 0. Therefore, z \in D (\Lambda G), and since point \lambda with
\mathrm{I}\mathrm{m}\lambda < 0 cannot be an eigenvalue of dissipative operator, then z = (\Lambda G - \lambda I) - 1 y. Thus,
(\Upsilon - \lambda I) - 1 \scrP 1y = \langle 0, (\Lambda G - \lambda I) - 1 y, C - 1 (T2y +G\ast T1y) e
- i\lambda \xi \rangle
for y \in H and \mathrm{I}\mathrm{m}\lambda < 0. On applying the mapping \scrP , we get
(\Lambda G - \lambda I) - 1 = \scrP (\Upsilon - \lambda I) - 1 \scrP k = - i\scrP
\infty \int
0
Ute
- i\lambda tdt\scrP k =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
592 B. P. ALLAHVERDIEV, H. TUNA
= - i
\infty \int
0
Zte
- i\lambda tdt = (B - \lambda I) - 1 , \mathrm{I}\mathrm{m}\lambda < 0,
i.e., \Lambda G = B.
Theorem 3 is proved.
On the other hand, the unitary group \{ Ut\} has an important property which makes it possible
to apply it to the Lax – Phillips theory (see [1]). It has orthogonal incoming and outgoing subspaces
D - = \langle L2(( - \infty , 0);E), 0, 0\rangle and D+ = \langle 0, 0, L2((0,\infty );E)\rangle and they have following properties.
Lemma 2. UtD - \subset D - , t \leq 0, and UtD+ \subset D+, t \geq 0.
Proof. We will just prove for D+ since the proof for D - is similar. Set \scrR \lambda = (\Upsilon - \lambda I) - 1 .
Then, for all \lambda , with \mathrm{I}\mathrm{m}\lambda < 0, we have
\scrR \lambda f =
\Biggl\langle
0, 0, - ie - i\lambda \xi
\xi \int
0
ei\lambda s\eta + (s) ds
\Biggr\rangle
, f = \langle 0, 0, \eta +\rangle \in D+.
Hence, we have \scrR \lambda f \in D+. If g \bot D+, then we get
0 = (\scrR \lambda f, g)\scrH = - i
\infty \int
0
e - i\lambda t (Utf, g)\scrH dt, \mathrm{I}\mathrm{m}\lambda < 0.
Thus, we obtain (Utf, g)\scrH = 0 for all t \geq 0, i.e., UtD+ \subset D+ for t \geq 0.
Lemma 2 is proved.
Lemma 3.
\bigcap
t\leq 0
UtD - =
\bigcap
t\geq 0
UtD+ = \{ 0\} .
Proof. Let us define the mappings \scrP + : \scrH \rightarrow L2 ((0,\infty );E) and \scrP +
1 : L2 ((0,\infty );E) \rightarrow D+
as follows \scrP + : \langle \eta - , \widehat y, \eta +\rangle \rightarrow \eta + and \scrP +
1 : \eta \rightarrow \langle 0, 0, \eta \rangle , respectively. We take into consider that
the semigroup of isometries U+
t := \scrP +Ut\scrP +
1 , t \geq 0, is a one-sided shift in L2 ((0,\infty );E) . Indeed,
the generator of the semigroup of the one-sided shift Vt in L2 ((0,\infty );E) is the differential operator
i
d
d\xi
with the boundary condition \eta (0) = 0. On the other hand, the generator S of the semigroup of
isometries U+
t , t \geq 0, is the operator
S\eta = \scrP +\Upsilon \scrP +
1 \eta = \scrP +\Upsilon \langle 0, 0, \eta \rangle = \scrP +
\biggl\langle
0, 0, i
d\eta
d\xi
\biggr\rangle
= i
d\eta
d\xi
,
where \eta \in W 1
2 ((0,\infty );E) and \eta (0) = 0. Since a semigroup is uniquely determined by its generator,
it follows that U+
t = Vt, and, therefore, we obtain\bigcap
t\geq 0
UtD+ = \langle 0, 0,
\bigcap
t\leq 0
VtL
2 ((0,\infty );E)\rangle = \{ 0\} .
Lemma 3 is proved.
Definition 5. The linear operator T acting in the Hilbert space H is called simple if there is no
subspace N \not = \{ 0\} invariant for T such that N \cap D(T ) = N and the restriction of T to N \cap D(T )
is self-adjoint on N.
Lemma 4. The operator \Lambda G is simple.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
DISSIPATIVE DIRAC OPERATOR WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES 593
Proof. Let H \prime \subset H be a nontrivial subspace in which \Lambda G induces a self-adjoint operator \Lambda \prime
G
with domain D (\Lambda \prime
G) = H \prime \cap D (\Lambda G) . If f \in D (\Lambda \prime
G), then f \in D (\Lambda \ast
G) and
0 =
d
dt
\| ei\Lambda Gtf\| 2H =
d
dt
\bigl(
ei\Lambda Gtf, ei\Lambda Gtf
\bigr)
H
=
= - 2
\bigl(
\mathrm{I}\mathrm{m}GT1e
i\Lambda Gtf, T1e
i\Lambda Gtf
\bigr)
E
.
Consequently, we obtain T1e
i\Lambda Gtf = 0. For eigenvectors y \in H \prime of the operator \Lambda G we get
T1y = 0. By using this result with boundary condition T2y+GT1y = 0, we have T2y = 0, i.e., y =
= 0. Since all solutions of My = \lambda y belong to L2
q ((0, a);E) , from this it can be concluded that the
resolvent R\lambda (\Lambda G) is a compact operator, and the spectrum of \Lambda G is purely discrete. Consequently, by
the theorem on expansion in the eigenvectors of the self-adjoint operator \Lambda \prime
G, we obtain H \prime = \{ 0\} .
Hence, the operator \Lambda G is simple.
Lemma 4 is proved.
Now, we set H - =
\bigcup
t\geq 0
UtD - , H+ =
\bigcup
t\leq 0
UtD+. Then we have the following lemma.
Lemma 5. The equality H - +H+ = \scrH holds.
Proof. From Lemma 2, it is easy to show that the subspace \scrH \prime = \scrH \circleddash (H - +H+) is invariant
relative to the group \{ Ut\} and has the form \scrH \prime = \langle 0, H \prime , 0\rangle , where H \prime is a subspace in H. Therefore,
if the subspace \scrH \prime (and also H \prime ) were nontrivial, then the unitary group \{ U \prime
t\} restricted to this
subspace would be a unitary part of the group \{ Ut\} , and, hence, the restriction \Lambda \prime
G of \Lambda G to H \prime
would be a self-adjoint operator in H \prime . Since the operator \Lambda G is simple, it follows that H \prime = \{ 0\} .
Lemma 5 is proved.
Assume that \chi (\lambda ) and \omega (\lambda ) are solutions of My = \lambda y satisfying the conditions
\chi 1 (a, \lambda ) = 0, \chi \rho
2 (a, \lambda ) = - 1, \omega 1 (a, \lambda ) = 1, \omega \rho
2 (a, \lambda ) = 0.
We denote by m(\lambda ) the matrix-valued function satisfying the conditions
m(\lambda )T1\chi = T2\chi , m(\lambda )T1\omega = T2\omega ;
m(\lambda ) is a meromorphic function on the complex plane \BbbC with a countable number of poles on the
real axis. Further, it is possible to show that the function m(\lambda ) possesses the following properties:
\mathrm{I}\mathrm{m} m(\lambda ) \leq 0 for all \mathrm{I}\mathrm{m}\lambda \not = 0, and m\ast (\lambda ) = m
\bigl(
\lambda
\bigr)
for all \lambda \in \BbbC , except the real poles m(\lambda ).
We denote by \varsigma j (x, \lambda ) and \tau j (x, \lambda ), j = 1, 2, the solutions of system My = \lambda y which satisfy
the conditions
T1\varsigma j = (m(\lambda ) +G) - 1Cej , T1\tau j = (m(\lambda ) +G\ast ) - 1Cej , j = 1, 2,
where e1 and e2 are an orthonormal basis for E.
We set
U -
\lambda j (x, \xi , \rho ) = \langle e - i\lambda \xi ej , \varsigma j(x, \lambda ), C
- 1 (m+G\ast ) (m+G) - 1Ce - i\lambda \rho ej\rangle , j = 1, 2.
We note that the vectors U -
\lambda j (x, \xi , \rho ), j = 1, 2, for real \lambda do not belong to the space \scrH . However,
U -
\lambda j (x, \xi , \rho ) , j = 1, 2, satisfies the equation \Upsilon U = \lambda U and the corresponding boundary conditions
for the operator \Upsilon .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
594 B. P. ALLAHVERDIEV, H. TUNA
By means of vector U -
\lambda j (x, \xi , \rho ), j = 1, 2, we define the transformation \scrF - : f \rightarrow \widetilde f - (\lambda ) by
(\scrF - f) (\lambda ) := \widetilde f - (\lambda ) := 1\surd
2\pi
2\sum
j=1
(f, U\lambda j)\scrH ej
on the vectors f = \langle \eta - , y, \eta +\rangle in which \eta - , \eta +, y are smooth, compactly supported functions.
Lemma 6. The transformation \scrF - isometrically maps H - onto L2 (\BbbR ;E). For all vectors
f, g \in H - the Parseval equality and the inversion formulae hold:
(f, g)\scrH = ( \widetilde f - , \widetilde g - )L2 =
\infty \int
- \infty
2\sum
j=1
\widetilde fj - (\lambda )\widetilde gj - (\lambda )d\lambda ,
f =
1\surd
2\pi
\infty \int
- \infty
2\sum
j=1
\widetilde fj - (\lambda )U\lambda jd\lambda ,
where \widetilde f - (\lambda ) = (\scrF - f) (\lambda ) and \widetilde g - (\lambda ) = (\scrF - g) (\lambda ).
Proof. By Paley – Wiener theorem, we get
\widetilde fj - (\lambda ) = 1\surd
2\pi
(f, U\lambda j)\scrH =
=
1
2\pi
0\int
- \infty
\Bigl(
\eta - (\xi ) , e - i\lambda \xi ej
\Bigr)
E
d\xi \in H2
- (E) , j = 1, 2,
where f = \langle \eta - , 0, 0\rangle , g = \langle \zeta +, 0, 0\rangle \in D - . If we will use the Parseval equality for Fourier integrals,
then we obtain
(f, g)\scrH =
\infty \int
- \infty
(\eta - (\xi ) , \zeta - (\xi ))E d\xi =
\infty \int
- \infty
( \widetilde f - (\lambda ), \widetilde g - (\lambda ))Ed\lambda = (\scrF - f,\scrF - g)L2 ,
where H2
\pm denote the Hardy classes in L2 (\BbbR ;E) consisting of the functions analytically extendible
to the upper and lower half-planes, respectively. We now extend to the Parseval equality to the
whole of H - . We consider in H - the dense set of H \prime
- of the vectors obtained as follows from
the smooth, compactly supported functions in D - : f \in H \prime
- if f = UT f0, f0 = \langle \eta - , 0, 0\rangle ,
\eta - \in C\infty
0 (( - \infty , 0);E) , where T = Tf is a nonnegative number depending on f. If f, g \in H \prime
- ,
then, for T > Tf and T > Tg, we have U - T f, U - T g \in D - , moreover, the first components of
these vectors belong to C\infty
0 (( - \infty , 0);E) . Therefore, since the operators Ut, t \in \BbbR , are unitary, by
the equality
\scrF - Utf =
1\surd
2\pi
2\sum
j=1
(Utf, U\lambda j)\scrH ej = ei\lambda t\scrF - f,
we have
(f, g)\scrH = (U - T f, U - T g)\scrH = (\scrF - U - T f,\scrF - U - T g)L2 =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
DISSIPATIVE DIRAC OPERATOR WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES 595
= (e - i\lambda T\scrF - f, e
- i\lambda T\scrF - g)L2 = ( \widetilde f, \widetilde g)L2 .
By taking the closure, we obtain the Parseval equality for the space H - . The inversion formula is
obtained from the Parseval equality if all integrals in it are considered as limits in the of integrals
over finite intervals. Finally, we obtain desired result
\scrF - H - =
\bigcup
t\geq 0
\scrF - UtD - =
\bigcup
t\geq 0
ei\lambda tH2
- = L2 (\BbbR ;E) .
Lemma 6 is proved.
Now, we set
U+
\lambda j (x, \xi , \rho ) = \langle SG (\lambda ) e - i\lambda \xi ej , \tau j (x, \lambda ) , e
- i\lambda \rho ej\rangle , j = 1, 2,
where
SG(\lambda ) = C - 1 (m(\lambda ) +G) (m(\lambda ) +G\ast ) - 1C. (14)
We note that the vectors U+
\lambda j (x, \xi , \rho ) for real \lambda do not belong to the space \scrH . However, U+
\lambda j (x, \xi , \rho )
satisfies the equation \Upsilon U = \lambda U and the corresponding boundary conditions for the operator \Upsilon . With
the help of vector U+
\lambda j (x, \xi , \rho ) , we define the transformation \scrF + : f \rightarrow \widetilde f+(\lambda ) by
(\scrF +f) (\lambda ) := \widetilde f+(\lambda ) := 2\sum
j=1
\widetilde fj+(\lambda )ej := 1\surd
2\pi
2\sum
j=1
\Bigl(
f, U+
\lambda j
\Bigr)
\scrH
ej
on the vectors f = \langle \eta - , y, \eta +\rangle in which \eta - , \eta + and y are smooth, compactly supported functions.
Lemma 7. The transformation \scrF + isometrically maps H+ onto L2 (\BbbR ;E). For all vectors
f, g \in H+ the Parseval equality and the inversion formula hold:
(f, g)\scrH = ( \widetilde f+, \widetilde g+)L2 =
\infty \int
- \infty
2\sum
j=1
\widetilde fj+(\lambda )\widetilde gj+(\lambda )d\lambda ,
f =
1\surd
2\pi
\infty \int
- \infty
2\sum
j=1
\widetilde fj+(\lambda )U+
\lambda jd\lambda ,
where \widetilde f+(\lambda ) = (\scrF +f) (\lambda ) and \widetilde g+(\lambda ) = (\scrF +g) (\lambda ).
Proof. The proof is similar to Lemma 4.
It is clear that the matrix-valued function SG(\lambda ) is meromorphic in \BbbC and all poles are in the
lower half-plane. From (14), we obtain \| SG(\lambda )\| \leq 1 for \mathrm{I}\mathrm{m}\lambda > 0; and SG (\lambda ) is the unitary matrix
for all \lambda \in \BbbR . Therefore, we have
U+
\lambda j =
2\sum
k=1
Sjk(\lambda )U
-
\lambda k, j = 1, 2, (15)
where Sjk, j, k = 1, 2, are elements of the matrix SG(\lambda ). From Lemmas 4 and 5, we get H - = H+.
With Lemma 5 this shows that H - = H+ = \scrH , therefore we obtain been proved the following
lemma for the incoming and outgoing subspaces (for D - and D+).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
596 B. P. ALLAHVERDIEV, H. TUNA
Lemma 8.
\bigcup
t\geq 0
UtD - =
\bigcup
t\leq 0
UtD+ = \scrH .
Lemma 9. D - \bot D+.
Proof. It is clear.
Thus, the transformation \scrF - isometrically maps H - onto L2 (\BbbR ) with the subspace D - mapped
onto H2
- and the operators Ut are transformed into the operators of multiplication by ei\lambda t. This
means that \scrF - is the incoming spectral representation for the group \{ Ut\} . Similarly, \scrF + is the
outgoing spectral representation for the group \{ Ut\} . It follows from (15) that the passage from the
\scrF - representation of an element f \in \scrH to its \scrF + representation is accomplished as \widetilde f+ (\lambda ) =
= S - 1
G (\lambda ) \widetilde f - (\lambda ). Consequently, according to [1], we have proved the following theorem.
Theorem 4. The function S - 1
G (\lambda ) is the scattering matrix of the group \{ Ut\} (of the self-adjoint
operator \Upsilon ).
Now, we recall the following definition.
Definition 6 [2]. The analytic matrix-valued function S(\lambda ) on the upper half-plane \BbbC + is called
inner function on \BbbC + if \| S(\lambda )\| \leq 1 for all \lambda \in \BbbC + and S(\lambda ) is a unitary matrix for almost all
\lambda \in \BbbR .
Let S(\lambda ) be an arbitrary nonconstant inner function on the upper half-plane. Let us define K by
the formula K = H2
+ \circleddash SH2
+. It is clear that K \not = \{ 0\} is a subspace of the Hilbert space H2
+. We
consider the semigroup of operators Zt (t \geq 0) acting in K according to the formula
Zt\varphi = \scrP [ei\lambda t\varphi ], \varphi = \varphi (\lambda ) \in K,
where \scrP is the orthogonal projection from H2
+ onto K. The generator of the semigroup \{ Zt\} is
denoted by
T\varphi = \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow +0
\biggl(
Zt\varphi - \varphi
it
\biggr)
which T is a maximal dissipative operator acting in K and with the domain D(T ) consisting of
all functions \varphi \in K, such that the limit exists. The operator T is called a model dissipative
operator. This model dissipative operator is a special case of a more general model dissipative
operator constructed by Nagy and Foiaş [2], which is associated with the names of Lax – Phillips [1].
Here, the basic assertion is that S(\lambda ) is the characteristic function of the operator T.
Let K = \langle 0, H, 0\rangle , so that \scrH =D - \oplus K \oplus D+. From the explicit form of the unitary transfor-
mation \scrF - under the mapping \scrF - , we obtain
\scrH \rightarrow L2 (\BbbR ;E) , f \rightarrow \widetilde f - (\lambda ) = (\scrF - f) (\lambda ),
D - \rightarrow H2
- (E) , D+ \rightarrow SGH
2
+ (E) ,
K \rightarrow H2
+ (E)\circleddash SGH
2
+ (E) ,
Ut \rightarrow (\scrF - Ut\scrF - 1
-
\widetilde f - )(\lambda ) = ei\lambda t \widetilde f - (\lambda ).
(16)
Formulae (16) show that operator \Lambda G (\Lambda K) is a unitarily equivalent to the model dissipative ope-
rator with the characteristic function SG(\lambda ). Since the characteristic functions of unitary equivalent
dissipative operator coincide (see [2]), we have thus proved the following theorem.
Theorem 5. The function SG(\lambda ) defined (14) coincides with the characteristic function of the
maximal dissipative operator \Lambda G (\Lambda K).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
DISSIPATIVE DIRAC OPERATOR WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES 597
5. Completeness of root vectors of the maximal dissipative Dirac operator on time scales. In
this section, we prove that all root vectors of the maximal dissipative Dirac operator on time scales are
complete. we will prove the characteristic function SG(\lambda ) is a Blaschke – Potapov product. Because
we know that the absence of the singular factor in the factorization of the characteristic function is
guarantee the completeness of the system of root vectors of maximal dissipative operators [2, 12].
Lemma 10 [4]. The characteristic function \widetilde SK (\lambda )of the operator \Lambda K has the form
\widetilde SK(\lambda ) := SG(\lambda ) = X1 (I - K1K
\ast
1 )
1/2 (\Theta (\xi ) - K1) (I - K\ast
1\Theta (\xi )) - 1 (I - K1K
\ast
1 )
1/2X2,
where K1 = - K is the Cayley transformation of the dissipative operator G, and \Theta (\xi ) is the Cayley
transformation of the matrix-valued function m(\lambda ),
\xi = (\lambda - i) (\lambda + i) - 1
and
X1 := (\mathrm{I}\mathrm{m}G) - 1/2 (I - K1)
- 1 (I - K1K
\ast
1 )
1/2 ,
X2 := (I - K\ast
1K1)
- 1/2 (I - K\ast
1 )
- 1 (\mathrm{I}\mathrm{m}G)1/2 ,
| \mathrm{d}\mathrm{e}\mathrm{t}X1| | \mathrm{d}\mathrm{e}\mathrm{t}X2| = 1.
Recall that the inner matrix-valued function \widetilde SG (\lambda ) is a Blaschke – Potapov product if and only
if \mathrm{d}\mathrm{e}\mathrm{t} \widetilde SG(\lambda ) is a Blaschke product (see [2, 12]). By Lemma 10, the characteristic function \widetilde SG(\lambda ) is
a Blaschke – Potapov product if and only if the matrix-valued function
XK (\xi ) = (I - K1K
\ast
1 )
1/2 (\Theta (\xi ) - K1) (I - K\ast
1\Theta (\xi )) - 1 (I - K1K
\ast
1 )
1/2
is a Blaschke – Potapov product in the unit disk.
Definition 7 [13]. Let \widetilde E be an n-dimensional (n < \infty ). In \widetilde E we fix an orthonormal basis
e1, e2, . . . , en and denote by Ek, k = 1, 2, . . . , n, the linear span of vectors e1, e2, . . . , ek. If L \subset Ek,
then the population of x \in Ek - 1 with the property
\mathrm{C}\mathrm{a}\mathrm{p} \{ \lambda : \lambda \in \BbbC , (x+ \lambda ek) \subset L\} > 0
will be shown by \Gamma k - 1L (\mathrm{C}\mathrm{a}\mathrm{p}G is the inner logarithmic capacity of a set G \subset \BbbC ). The \Gamma -capacity
of a set L \subset \widetilde E is a number
\Gamma - \mathrm{C}\mathrm{a}\mathrm{p}L := \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{C}\mathrm{a}\mathrm{p} \{ \lambda : \lambda e1 \subset \Gamma 1\Gamma 2 . . .\Gamma n - 1L\} ,
where supremum is taken with respect to all orthonormal basis in \widetilde E.
It is known that every set L \subset \widetilde E of zero \Gamma -capacity has zero 2n2-dimensional Lebesque measure,
however the converse is not true.
Denote by [E] the set of all linear operators in E. To convert [E] into an 4-dimensional Hilbert
space, we give the inner product \langle T, S\rangle = \mathrm{t}\mathrm{r}S\ast T for T, S \in [E] (\mathrm{t}\mathrm{r}S\ast T is the trace of the operators
S\ast T ). Hence we may give the \Gamma -capacity of a set in E.
We use the following result of [12].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
598 B. P. ALLAHVERDIEV, H. TUNA
Lemma 11. Let X (\xi ), | \xi | < 1, be a analytic function with the values to be contractive opera-
tors in [E] (\| X (\xi )\| \leq 1). Then for \Gamma -quasievery strictly contractive operators (i.e., for all strictly
contractive K \in [E] possible with the exception of a set of \Gamma of zero capacity) the inner part of the
contractive function
XK (\xi ) = (I - K1K
\ast
1 )
1/2 (X (\xi ) - K1) (I - K\ast
1X (\xi )) - 1 (I - K1K
\ast
1 )
1/2
is a Blaschke – Potapov product.
By summing all obtained result for the dissipative operator \Lambda K (\Lambda G) , we have proved the fol-
lowing theorem.
Theorem 6. For \Gamma -quasievery strictly contractive K \in [E] the characteristic function \widetilde SK(\lambda ) of
the dissipative operator \Lambda K is a Blaschke – Potapov product and spectrum of \Lambda K is purely discrete
and belongs to the open upper half-plane. For \Gamma -quasievery strictly contractive K \in [E] the
operator \Lambda K has an countable number of isolated eigenvalues with finite multiplicity and limit
points at infinity, and the system of eigenvectors and associated vectors (or root vectors) of this
operators is complete in H.
References
1. P. D. Lax, R. S. Phillips, Scattering theory, Acad. Press, New York (1967).
2. B. Sz. Nagy, C. Foiaş, Analyse Harmonique des Operateurs de L’espace de Hilbert, Masson, Akad. Kiado, Paris,
Budapest (1967).
3. B. P. Allahverdiev, Spectral problems of nonself-adjoint 1D singular Hamiltonian systems, Taiwanese J. Math., 17,
№ 5, 1487 – 1502 (2013).
4. B. P. Allahverdiev, Extensions, dilations and functional models of Dirac operators, Integral Equat. and Oper. Theory,
51, 459 – 475 (2005).
5. B. P. Allahverdiev, Spectral analysis of dissipative Dirac operators with general boundary conditions, J. Math. Anal.
and Appl., 283, 287 – 303 (2003).
6. M. A. Naimark, Linear differential operators, 2nd ed., Nauka, Moscow (1969).
7. M. L. Gorbachuk, V. I. Gorbachuk, Boundary value problems for operator differential equations, Naukova Dumka,
Kiev (1984).
8. A. Kuzhel, Characteristic functions and models of nonself-adjoint operators, Kluwer Acad., Dordrecht (1996).
9. B. S. Pavlov, Self-adjoint dilation of a dissipative Schrödinger operator and eigenfunction expansion, Funct. Anal.
and Appl., 98, 172 – 173 (1975).
10. B. S. Pavlov, Self-adjoint dilation of a dissipative Schrödinger operator and its resolution in terms of eigenfunctions,
Math. USSR Sbornik, 31, № 4, 457 – 478 (1977).
11. B. S. Pavlov, Dilation theory and spectral analysis of nonself-adjoint differential operators, Proc. 7th Winter School,
Drogobych (1974), 3-69 (1976) (in Russian); English transl: Transl. II. Ser., Amer. Math. Soc., 115, 103 – 142 (1981).
12. Yu. P. Ginzburg, N. A. Talyush, Exceptional sets of analytical matrix-functions, contracting and dissipative operators,
Izv. Vyssh. Uchebn. Zaved. Math., 267, 9 – 14 (1984).
13. L. I. Ronkin, Introduction to the theory of entire functions of several variables, Nauka, Moscow (1971).
14. J. Weidmann, Spectral theory of ordinary differential operators, Lect. Notes Math., 1258 (1987).
15. S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten (Ph. D. Thesis), Univ. Würzburg
(1988).
16. D. R. Anderson, G. Sh. Guseinov, J. Hoffacker, Higher-order self-adjoint boundary-value problems on time scales,
J. Comput. and Appl. Math., 194, № 2, 309 – 342 (2006).
17. F. Atici Merdivenci, G. Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on
time scales, J. Comput. and Appl. Math., 141, № 1-2, 75 – 99 (2002).
18. M. Bohner, A. Peterson, Dynamic equations on time scales, Birkhäuser, Boston (2001).
19. M. Bohner, A. Peterson (Eds.), Advances in dynamic equations on time scales, Birkhäuser, Boston (2003).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
DISSIPATIVE DIRAC OPERATOR WITH GENERAL BOUNDARY CONDITIONS ON TIME SCALES 599
20. G. Sh. Guseinov, Self-adjoint boundary value problems on time scales and symmetric Green’s functions, Turkish J.
Math., 29, № 4, 365 – 380 (2005).
21. V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic systems on measure chains, Kluwer Acad. Publ.,
Dordrecht (1996).
22. B. M. Levitan, I. S. Sargsjan, Sturm – Liouville and Dirac operators, Math. and Appl. (Soviet Series), Kluwer Acad.
Publ. Group, Dordrecht (1991).
23. B. Thaller, The Dirac equation, Springer (1992).
24. B. P. Rynne, L2 spaces and boundary value problems on time scales, J. Math. Anal. and Appl., 328, 1217 – 1236
(2007).
25. T. Gulsen, E. Yilmaz, Spectral theory of Dirac system on time scales, Appl. Anal., 96, № 16, 2684 – 2694 (2017).
26. G. Sh. Guseinov, An expansion theorem for a Sturm – Liouville operator on semi-unbounded time scales, Adv. Dyn.
Syst. and Appl., 3, 147 – 160 (2008).
27. G. Sh. Guseinov, Eigenfunction expansions for a Sturm – Liouville problem on time scales, Int. J. Different. Equat.,
2, 93 – 104 (2007).
28. A. Huseynov, E. Bairamov, On expansions in eigenfunctions for second order dynamic equations on time scales,
Nonlinear Dyn. Syst. Theory, 9, 77 – 88 (2009).
29. B. P. Allahverdiev, A. Eryilmaz, H. Tuna, Dissipative Sturm – Liouville operators with a spectral parameter in the
boundary condition on bounded time scales, Electron. J. Different. Equat., 95, 1 – 13 (2017).
30. B. P. Allahverdiev, Extensions of symmetric singular second-order dynamic operators on time scales, Filomat, 30,
№ 6, 1475 – 1484 (2016).
31. B. P. Allahverdiev, Non-self-adjoint singular second-order dynamic operators on time scale, Math. Meth. Appl. Sci.,
42, 229 – 236 (2019).
32. B. P. Allakhverdiev, H. Tuna, Spectral analysis of singular Sturm – Liouville operators on time scales, Ann. Univ.
Mariae Curie-Sklodowska, Sect. A, 72, № 1, 1 – 11 (2018).
33. H. Tuna, Dissipative Sturm – Liouville operators on bounded time scales, Mathematica, 56, 80 – 92 (2014).
34. H. Tuna, Completeness of the root vectors of a dissipative Sturm – Liouville operators in time scales, Appl. Math.
and Comput., 228, 108 – 115 (2014).
35. H. Tuna, Completeness theorem for the dissipative Sturm – Liouville operators on bounded time scales, Indian J. Pure
and Appl. Math., 47, № 3, 535 – 544 (2016).
36. H. Tuna, M. A. Özek, The one-dimensional Schrödinger operator on bounded time scales, Math. Meth. Appl. Sci.,
40, № 1, 78 – 83 (2017).
37. A. Huseynov, Limit point and limit circle cases for dynamic equations on time scales, Hacet. J. Math. Stat., 39,
379 – 392 (2010).
38. A. S. Özkan, Parameter-dependent Dirac systems on time scales, Cumhuriyet Sci. J., 39, № 4, 864 – 870 (2018).
Received 04.04.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
|
| id | umjimathkievua-article-546 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:59Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6f/cd94ab0b12c650cc55a8e10baa07e16f.pdf |
| spelling | umjimathkievua-article-5462022-03-26T11:01:39Z Dissipative Dirac operator with general boundary conditions on time scales Dissipative Dirac operator with general boundary conditions on time scales Dissipative Dirac operator with general boundary conditions on time scales Allahverdiev, B. P. Tuna, H. Allahverdiev, B. P. Tuna, H. Allahverdiev, B. P. Tuna, H. UDC 517.9 In this paper, we consider the symmetric Dirac operator on bounded time&nbsp;scales. With general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint and the other) of such symmetric operators. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete. Розглядається симетричний оператор Дірака на обмежених часових шкалах.&nbsp;При загальних граничних умовах описано розширення (дисипативні, акумулятивні, самоспряжені та інші) таких симетричних операторів.&nbsp;Побудовано самоспряжене розширення дисипативного оператора та визначено матрицю розсіювання дилатації. Також побудовано функціональну модель цього оператора та визначено його характеристичну функцію. Насамкінець доведено, що всі кореневі вектори цього оператора є повними. Institute of Mathematics, NAS of Ukraine 2020-04-29 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/546 10.37863/umzh.v72i5.546 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 5 (2020); 583–599 Український математичний журнал; Том 72 № 5 (2020); 583–599 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/546/8689 |
| spellingShingle | Allahverdiev, B. P. Tuna, H. Allahverdiev, B. P. Tuna, H. Allahverdiev, B. P. Tuna, H. Dissipative Dirac operator with general boundary conditions on time scales |
| title | Dissipative Dirac operator with general boundary conditions on time scales |
| title_alt | Dissipative Dirac operator with general boundary conditions on time scales Dissipative Dirac operator with general boundary conditions on time scales |
| title_full | Dissipative Dirac operator with general boundary conditions on time scales |
| title_fullStr | Dissipative Dirac operator with general boundary conditions on time scales |
| title_full_unstemmed | Dissipative Dirac operator with general boundary conditions on time scales |
| title_short | Dissipative Dirac operator with general boundary conditions on time scales |
| title_sort | dissipative dirac operator with general boundary conditions on time scales |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/546 |
| work_keys_str_mv | AT allahverdievbp dissipativediracoperatorwithgeneralboundaryconditionsontimescales AT tunah dissipativediracoperatorwithgeneralboundaryconditionsontimescales AT allahverdievbp dissipativediracoperatorwithgeneralboundaryconditionsontimescales AT tunah dissipativediracoperatorwithgeneralboundaryconditionsontimescales AT allahverdievbp dissipativediracoperatorwithgeneralboundaryconditionsontimescales AT tunah dissipativediracoperatorwithgeneralboundaryconditionsontimescales |