On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis
For the symmetric differential system of the first order that contains a spectral parameter in Nevanlinna's manner the limit of regular boundary value problems with dissipative or accumulative nonseparated boundary conditions is studied when the interval stretches to the semiaxis. When for the...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1065332016-10-01T03:01:46Z On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis Khrabustovskyi, V.I. For the symmetric differential system of the first order that contains a spectral parameter in Nevanlinna's manner the limit of regular boundary value problems with dissipative or accumulative nonseparated boundary conditions is studied when the interval stretches to the semiaxis. When for the considered system the case of the limit point takes place in one of the complex half-planes, we obtain the condition which guarantees the non-self-adjointness of the boundary condition at zero that corresponds to the limit boundary problem. This result is illustrated on the perturbed almost periodic systems. When the boundary condition in the prelimit regular problems is periodic, we show that the limit characteristic matrix is also the characteristic matrix on the whole axis if the coefficients of the system are extended in a certain way on the negative semiaxis. In the general case we find the condition when the convergence of characteristic matrixes implies the convergence of resolvents. 2009 Article On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis / V.I. Khrabustovskyi // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 54-81. — Бібліогр.: 30 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106533 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
For the symmetric differential system of the first order that contains a spectral parameter in Nevanlinna's manner the limit of regular boundary value problems with dissipative or accumulative nonseparated boundary conditions is studied when the interval stretches to the semiaxis. When for the considered system the case of the limit point takes place in one of the complex half-planes, we obtain the condition which guarantees the non-self-adjointness of the boundary condition at zero that corresponds to the limit boundary problem. This result is illustrated on the perturbed almost periodic systems. When the boundary condition in the prelimit regular problems is periodic, we show that the limit characteristic matrix is also the characteristic matrix on the whole axis if the coefficients of the system are extended in a certain way on the negative semiaxis. In the general case we find the condition when the convergence of characteristic matrixes implies the convergence of resolvents. |
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Khrabustovskyi, V.I. |
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Khrabustovskyi, V.I. On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis Журнал математической физики, анализа, геометрии |
| author_facet |
Khrabustovskyi, V.I. |
| author_sort |
Khrabustovskyi, V.I. |
| title |
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis |
| title_short |
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis |
| title_full |
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis |
| title_fullStr |
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis |
| title_full_unstemmed |
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis |
| title_sort |
on the limit of regular dissipative and self-adjoint boundary value problems with nonseparated boundary conditions when an interval stretches to the semiaxis |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/106533 |
| citation_txt |
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems with Nonseparated Boundary Conditions when an Interval Stretches to the Semiaxis / V.I. Khrabustovskyi // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 54-81. — Бібліогр.: 30 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT khrabustovskyivi onthelimitofregulardissipativeandselfadjointboundaryvalueproblemswithnonseparatedboundaryconditionswhenanintervalstretchestothesemiaxis |
| first_indexed |
2025-07-07T18:36:46Z |
| last_indexed |
2025-07-07T18:36:46Z |
| _version_ |
1837014353264508928 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 1, pp. 54�81
On the Limit of Regular Dissipative and Self-Adjoint
Boundary Value Problems with Nonseparated Boundary
Conditions when an Interval Stretches to the Semiaxis
V.I. Khrabustovskyi
Ukrainian State Academy of Railway Transport
7, Feyerbakh Sq., Kharkiv, 61050, Ukraine
E-mail:v_khrabustovskyi@ukr.net
Received April 2, 2007
For the symmetric di�erential system of the �rst order that contains
a spectral parameter in Nevanlinna's manner the limit of regular bound-
ary value problems with dissipative or accumulative nonseparated boundary
conditions is studied when the interval stretches to the semiaxis. When
for the considered system the case of the limit point takes place in one of
the complex half-planes, we obtain the condition which guarantees the non-
self-adjointness of the boundary condition at zero that corresponds to the
limit boundary problem. This result is illustrated on the perturbed almost
periodic systems. When the boundary condition in the prelimit regular
problems is periodic, we show that the limit characteristic matrix is also the
characteristic matrix on the whole axis if the coe�cients of the system are
extended in a certain way on the negative semiaxis. In the general case we
�nd the condition when the convergence of characteristic matrixes implies
the convergence of resolvents.
Key words: characteristic matrix, nonseparated boundary conditions,
resolvent convergence, almost periodic function.
Mathematics Subject Classi�cation 2000: 34B07, 34B20, 34B40, 47E05.
1. Introduction
In a �nite dimensional Hilbert space H we consider the system
i
2
�
(Q (t)x (t))
0
+Q (t) x0 (t)
�
�H� (t) x (t) = w� (t) f (t) ; t 2 [0; 1) ; (1)
c
V.I. Khrabustovskyi, 2009
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
where Q (t) = Q� (t) 2 ACloc;H� (t) = H�
��
(t) are n � n matrixes, detQ (t) 6=
0;H� (t) 2 L1
loc on t and analytically depends on nonreal �, the weight w� (t) =
=H� (t) /=� � 0; (=� 6= 0).
In the paper, the limits of characteristic matrixes of the Weyl�Titchmarsh
type and the limits of resolvents of regular boundary value problems on the in-
terval (0; b) with nonseparated dissipative or accumulative boundary conditions
(in particular, with self-adjoint boundary conditions that include periodic ones)
are studied when (0; b) stretches to the semiaxis (0;1). The limiting transition
from the regular problems to the singular ones was studied and used by many au-
thors [1�13]. However, the question on a type of boundary condition at zero that
corresponds to the limit of the boundary problems with nonseparated boundary
conditions and the resolvent convergence for the system (1) have not been studied
in the general case.
Let w�(t) be the weight of positive type (see (4) below) and let for some non-
real � the number of linearly independent and square integrable on the semiaxis
(0;1) with the weight w�(t) solutions of homogeneous system (1) be minimal?,
and therefore (see Remark 6 below) any characteristic matrix of the system (1)
on the semiaxis depends only on the boundary condition at zero. Under this
assumption we obtain the conditions for which the boundary condition at zero
that corresponds to the limit characteristic matrix depends on � and it is strictly
dissipative or strictly accumulative as =� > 0 or =� < 0 ?? (even if the prelimit
boundary conditions are self-adjoint). It is shown that when these conditions are
not ful�lled, the boundary condition at zero can be self-adjoint (even if the pre-
limit boundary conditions are nonself-adjoint). The classes of systems satisfying
these conditions are given. In the case of periodic boundary conditions in the
prelimit regular boundary value problems that are often met in Physics (see, e.g.,
[15, Ch. 1, � 6]) we show that under some assumptions the limit characteristic
matrix of the system (1) on the semiaxis (0;1) is the characteristic matrix of
this system on the axis (�1;1) if the coe�cients of the system are extended in
a certain way to the negative semiaxis. In some cases the limit characteristic ma-
trix can be calculated due to the mentioned above. The proofs of our results are
based on the possibility to approximate the limit characteristic matrix by means
of characteristic matrixes considered in [13] belonging to special regular bound-
ary value problems with separated and strictly dissipative or strictly accumulative
boundary conditions.
Also, in the paper in the general case (i.e, without requirement that for some
?For constant Q(t) = J = J�1 this means that the rank of one of the radiuses of the Weyl
limit disc for the system (1) is minimal [14].
??And therefore, if in (1) H�(t) = H0(t) + �H(t);H0(t) = H�
0 (t), then the nonorthogonal
generalized resolvent of the corresponding minimal relation corresponds to the limit boundary
problem.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 55
V.I. Khrabustovskyi
nonreal � there should be the minimal number of linearly independent and square
integrable on the semiaxis (0;1) with the weight w�(t) solutions of homogeneous
system (1)) it is shown that the convergence of characteristic matrixes of regular
dissipative or accumulative boundary value problems, in particular, self-adjoint
ones, with separated or nonseparated boundary conditions implies the convergence
of resolvents as � = �0 (as =�=�0 > 0, in the case when H�(t) contains � in the
linear manner) only if the limit characteristic matrix corresponds to the self-
adjoint boundary conditions as � = �0. This fact (in another form) is announced
in [8] for the limits of characteristic matrixes of the self-adjoint regular scalar
di�erential operators of even order with real coe�cients.
In the general case also a criterion of the separating of boundary conditions
that correspond to the limit of regular boundary value problems with nonsepa-
rated boundary conditions is obtained.
We notice that due to the fact that in the considered system (1) the weight
w�(t) can be degenarated, the obtained results should be applied to the matrix
di�erential operators and relations of arbitrary order (dissipative and, in partic-
ular, symmetric).
2. Auxiliary Results
In this section, some results of [13] are given in a convenient but not most
general form. These results will be used later on. By ( ; ) and k � k the scalar
product and the norm in various spaces with special indexes, if necessary, are
denoted.
Let X� (t) be the matrix solution of the homogeneous system (1) satisfying
the initial condition X� (0) = kÆjkk
n
j; k=1
def
= I . Since H� (t) = H�
��
(t), then
X�
��
(t)Q (t)X� (t) = Q (0)
def
= G; =� 6= 0: (2)
Under the condition 0 � � � � <1, we denote ��(�; �) =
Z �
�
X�
�(t)w�(t)
X�(t)dt. One has
X�
� (�)Q (�)X� (�)�X�
� (�)Q (�)X� (�) = 2=��� (�; �) =� 6= 0: (3)
Further it is supposed that?
9�0
�
=�0 6= 0
�
; � 2 (0; 1) : ��0 (0; �) > 0 (4)
(by [13] the condition (4) is valid if �0 is changed by an arbitrary nonreal �).
For x (t) 2 H or x (t) 2 B (H) we denote U [x (t)] = x� (t)Q (t)x (t).
?The condition =�0 6= 0 can be weakened in the same way as in [13].
56 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
De�nition 1 [13]. An analytic n � n-matrix function M (�) = M�
�
��
�
on
nonreal � is called a characteristic matrix (c.m.) of the system (1) on I =
(0; b) ; (b � 1) (or simply, c.m.) if for =� 6= 0 and for any vector function
f (t) 2 L2
w�
(I) with compact support the corresponding solution x� (t) of the sys-
tem (1) of the form
x� (t) = R�f =
Z
I
X� (t)
�
M (�)�
1
2
sgn (s� t) (iG)�1
�
X�
��
(s)w� (s) f (s) ds
(5)
satis�es the condition
(=�) lim
�"b
(U [x� (�)]� U [x� (0)]) � 0; =� 6= 0: (6)
The following remark establishes a relation between the c.m. and the reg-
ular boundary value problems for the system (1) with the boundary condition
depending on spectral parameter.
Remark 1 [13]. Let the interval I = (0; b) be �nite. Then:
10. If the matrix-functions M�; N� analytically depend on nonreal � and
=� (N �
�Q (b)N� �M
�
�Q (0)M�) � 0; =� 6= 0; (7)
kM�hk+ kN�hk > 0; 0 6= h 2 H; =� 6= 0; (8)
M�
��
Q (0)M� = N �
��
Q (b)N�; =� 6= 0; (9)
then the boundary value problem, obtained by connecting to the system (1) the
boundary condition
9h = h (�; f) 2 H : x (0) =M�h; x (b) = N�h; (10)
has the unique solution as =� 6= 0. The solution is equal to x� (t) (5), where
M (�) = �
1
2
�
M� +X�1
� (b)N�
� �
M� �X�1
� (b)N�
��1
(iG)
�1
; (11)
where
det
�
M� �X�1
� (b)N�
�
6= 0; =� 6= 0:
The matrix-function M (�) (11) is a c.m. of the system (1) on I.
20. If M (�) is the c.m. of the system (1) on I, then x� (t) (5) is the solution
of some boundary value problem from n010.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 57
V.I. Khrabustovskyi
De�nition 2 [13]. Let M (�) be the c.m. of the system (1) on I. We say that
the corresponding condition (6) is separated for nonreal � = �0 if for any vector
function f (t) 2 L2
w� 0
(I) with compact support the following two inequalities hold
simultaneously for the solution x�0 (t) (5) of the system (1):
=�0U [x�0 (0)] � 0; lim
�"b
=�0U [x�0 (�)] � 0: (12)
Theorem 1 [13]. Let M (�) be the c.m. of the system (1). We represent
M (�) in the form
M (�) =
�
P (�)�
1
2
I
�
(iG)�1 : (13)
Then the condition (6) corresponding to M (�) is separated for � = �0 if and
only if the operator P (�0) is a projection, i.e.,
P (�0) = P2 (�0) : (14)
The following remark establishes a relation between the c.m. with separated
boundary condition (6) and the boundary value problems with separated bound-
ary conditions depending on spectral parameter.
Remark 2 [13]. Let the interval I = (0; b) be �nite. Then:
10. If the operator-functions M�;N� from n010 of Remark 1 are such that
=�M�
�Q (0)M� � 0; =�N �
�Q (b)N� � 0 (=� 6= 0) (i.e., the boundary condition
(7)�(10) is separated), then the solution of the boundary value problem (1),(7)�
(10) for any vector function f (t) 2 L2
w�
(I) with compact support is equal to
x� (t) (5), where M (�) is some c.m. of the system (1) on (0; b) with separated
condition (6). And therefore M (�) admits the representation (13), where P (�)
is a projection that is equal to
P (�) = �X�1
� (b)N�
�
M� �X�1
� (b)N�
��1
: (15)
20. If M (�) (13) is the c.m. of the system (1) on I and, moreover, P (�) =
P2 (�), then x� (t) (5) is a solution of some boundary value problem from n010.
De�nition 3 [13]. If the matrix-function M (�) of the form (13) is the
c.m. of the system (1) on I and, moreover, P (�) = P2 (�), then P (�) is called
a characteristic projection (c.p.) of the system (1) on I (or simly c.p.).
3. Main Results
In the lemma below, the matrixes M� and N� from Remark 1 can depend
on b. We denote the corresponding matrix (15) by P (�; b).
58 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
It follows from [13] (and [9] as Q (t) = J = J�1 ):
Lemma 1. Any sequence bn " 1 contains a subsequence bnk such that for
any nonreal � there exists
limP (�; bnk) = P (�) ; (16)
and M (�) (13), (16) are the c.m.'s of the system (1) on (0;1).
Any c.m. of the system (1) on (0;1) can be obtained in a similar way.
Remark 3. As it is seen from Example 2 (see below), even with the constant
Q (t), with the periodic H� (t) and with periodic boundary condition (10), the limit
(16) can depend on the choice of subsequence bnk .
By R� (�) we denote the operator (5), (13) corresponding to P (�) = � .
The following theorem shows when the convergence of c.m.'s implies the con-
vergence of corresponding resolvents.
Theorem 2.10. Let for the c.m. M (�) (13), (16) of the system (1) on (0;1)
for some nonreal � = �0 there be an equality in the condition (6)?.
Then, for � = �0 and for any vector function f (t) 2 L2
w�0
(0; 1) with compact
support:
lim
bnk!1
k[R� (P (�))�R� (P (�; bnk))] fkL2w�(0; bnk)
! 0: (17)
If in (1)
H�(t) = H0(t) + �H(t); H0(t) = H�
0 (t); (18)
then (17) holds for =�=�0 > 0.
20. If for the prelimit c.m.'s M (�; bnk) (13), (16) there is the equality in the
condition (6) for some nonreal � = �0 (and therefore by [9, 13] it takes place for
=� 6= 0 ) and (17) is valid for � = �0, then for � = �0 (for =�=�0 > 0 in the
case (18)) there is the equality in (6) for the limit c.m. M (�) (13), (16).
P r o o f. 10. In (1) substitute
x (t) = T (t) ~x (t) ; T (t) = � (t)S; (19)
where matrix � (t) is the solution of the Cauchy problem
i
2
�
(Q (t)� (t))
0
+Q (t)�0 (t)
�
= 0; � (0) = I;
?Therefore there will also be an equality for � = ��0 if the number of linearly independent and
square-integrable on the semiaxis (0;1) with the weight w� (t) solutions of the homogeneous
system (1) for some nonreal � = �+; � = ��; =�+=�� < 0 coincides. This statement can be
obtained from [14], [16�18] and from Lem. 3.2 from [13].
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 59
V.I. Khrabustovskyi
and the matrix S is such that S�GS = J = J�1.
We get the system
iJ ~x0 (t)� ~H� (t) ~x� (t) = ~w� (t) ~f (t) ; (20)
where ~H� (t) = T �(t)H�(t)T (t), ~w� (t) = T � (t)w� (t)T (t), ~f (t) = T�1 (t) f (t).
The c.m.'s of the system (1) and (20) are in a one-to-one correspondence
M (�) = S ~M (�)S�; P (�) = S ~P (�)S�1;
and, moreover, for the corresponding resolvents we have
k[R� (P (�))�R� (P (�; bnk))] f (t)kL2w�(0; bnk)
=
h ~R�
�
~P (�)
�
� ~R�
�
~P (�; bnk)
�i
~f (t)
L2
~w�
(0; bnk)
;
where the analog of R� (�) for the system (20) is denoted by ~R�
�
~�
�
.
Since (4) is valid for the system (20), then the instruments of the matrix discs
from [14] can be applied to (20).
Due to [9, 14] any c.m. of the system of the type of (20) on (0; b) can be
represented in the form
~M (�; b) =
1
2
i
�
R2
1 (�; b) +R2 (�; b) + 2R1 (�; b) v� (b)R (�; b)
�
; =� > 0; (21)
where
R2
1(�; b) = [2=���(0; b)]
�1; R2 (�; b) = [2=����(0; b)]
�1 ;
v�(b) 2 B(H); v�� (b) v� (b) � I;
and any c.m. of the system of the type of (20) on (0; 1) can be represented in
the form
~M (�) =
1
2
i
�
R2
1 (�) +R2 (�) + 2R1 (�) v�R (�)
�
; =� > 0; (22)
where
R1(�) = lim
b!1
R1(�; b); R (�) = lim
b!1
R (�; b) ; v� 2 B(H); v��v� � I:
It can be proved (compare with [9], where v�1� = v��) that the fact that the
ful�lment of the condition (6) with b =1 for ~M (�) (22) as =� > 0 or =� < 0 is
equivalent to the following inequality conditions, respectively:
C�1 (�) (I � P1 (�))C1 (�) +R (�) (I � v��v�)R (�) � 0; =� > 0; (23)
C(�) (I � P (�))C� (�) +R1 (�) (I � v�v
�
�)R1 (�) � 0; =� > 0; (24)
60 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
where C1 (�) = R1 (�) + v�R (�) ; C (�) = R (�) +R1 (�) v�, P1 (�) and P (�) are
orthoprojections on R1 (�)H and R (�)H, respectively. The equality in (6) for
~M (�) (22) for =� > 0 (=� < 0) is equivalent to the equality in (23) ((24)). But
the equality in (23) or (24) is equivalent to the simultaneous ful�lment of two
inequalities, respectively:
(I � P1 (�)) v�P (�) = 0; P (�) = v��v�P (�) ; =� > 0; (25)
(I � P (�)) v��P1 (�) = 0; P1 (�) = v�v
�
�P1 (�) ; =� > 0: (26)
Let
~M (�) = lim
bnk!1
~M (�; bnk) : (27)
For de�niteness, set =�0 < 0.
Since ~M (�) is a c.m. of the system (20) on (0; bnk), then ~M (�) can be
represented in the form (21) with v� (bnk) being replaced by some ~v� (bnk).
Then
h ~R�0
�
~P (�0)
�
� ~R�0
�
~P0 (�; bnk)
�i
~f
2
L2
~w�0
(0; bnk)
(28)
=
1
2=��0
h~v���0 (bnk)� v���0 (bnk)
i
R1
�
��0; bnk
�
~h
2 ; (29)
where ~h = ~h(�0; ~f) =
1R
0
~X�
��0
(s) ~w��0
(s) ~f (s) ds .
We denote by P1 (�; bnk) and P (�; bnk) the Riesz projections corresponding
to those parts of � (R1 (�; bnk)) and � (R (�; bnk)) that are separated from zero
as bnk !1. Then, the using of (21),(22),(26),(27) makes it possible to show that
there exist equal limits
lim ~v���0
(bnk)P1
�
��0; bnk
�
= lim v���0 (bnk)P1
�
��0; bnk
�
= P
�
��0
�
v���0P1
�
��0
�
;
and therefore the right-hand side in (28) converges to 0 as � = �0.
If for the c.m. M(�) of the system (1), (18) on (0;1) there is the equality in
(6) for � = �0, then there is the equality in (6) for =�=�0 > 0 in view of Lem. 3.2
[14]. Thus n010 is proved.
20 follows from the formula (1.70) [13] and from the concluding arguments of
the proof of (17) from nÆ1. The theorem is proved.
Notice that n020 of Th. 2 or Remark 8 shows that in n010 it is impossible
to omit the condition of the equality in (6). Also notice that, as it is seen from
the example in the proof of Remark 10, the equality in (6) is possible for the
c.m. (13) of the system (1) on (0; 1), although there exists the sequence of c.m.
M (�; bnk) for which there is no equality in (6), but (16) is valid.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 61
V.I. Khrabustovskyi
The remark below follows from (25), (26).
Remark 4. Let the matrix Q (t) in the system (1) be constant, moreover
Q (t) = J = J�1. Then:
10. Let us choose a basis such that P (�0) = diag (Im; On�m) if =�0 > 0
and P1(��0) = diag(Im1
; On�m1
) if =�0 < 0. Then the statement that for the
c.m. ~M (�) of the system (1) on (0; 1) with nonreal � = �0 there is the equality
in the condition (6) is equivalent to the statement that all contractions v� in the
representation ~M (�) (22) with � = �0 are described by the formulae:
v�0 =
�
~v11 ; : : : ; ~v
1
m; ~x
1
1; : : : ; ~x
1
n�m
�
; m = rgP (�0); =�0 > 0; (30)
v ��0
= (~v1; : : : ; ~vm1
; ~x1; : : : ; ~xn�m1
)� ; m1 = rgP1(��0); =�0 < 0; (31)
where ~v1j (~vj) are some �xed orthonormal eigenvectors columns of P1 (�0) (P
�
��0
�
)
that correspond to the eigenvalue 1, ~x1k(~xk) are arbitrary vector columns such that
v�0 is the contraction.
20. Let for the c.m. ~M (�) (22) of the system (1) on (0; 1) there be the
equality in the condition (6) for � = �0, =�0 > 0, and � = ��0
?.
This is equivalent to the simultaneous ful�lment of the following three equali-
ties:
v�0P (�0) = P1 (�0) v�0 ; P (�0) = v� (�0)P1 (�0) v (�0) ;
P1 (�0) = v (�0)P (�0) v
� (�0) ;
that is, in its turn, equivalent to the fact that all contractions v� in the represen-
tation ~M (�) (22) with � = �0 are described by the formula
v�0 = U�1 (�0) diag (U; V )U (�0) ; (32)
where U is some �xed unitary m�m-matrix, V is an arbitrary (n�m)�(n�m)-
contraction; U1(�); U (�) are such unitary matrixces that
P1 (�) = U�1 (�) diag (Im; On�m)U1 (�) ; P (�) = U� (�) diag (Im; On�m)U (�) :
For V = In�m, (32) passes into the formula obtained in [9] under the condition
that v�0 is unitary.
Theorem 3. If in the system (1) H� (t) =(18), and for the c.m. M (�) of
this system on (0; 1) in the condition (6) there is the equality for some nonreal
� = �0, then there should be the sequence of c.m.'s M (�; bnk) for which there is
the equality in (6) for =� 6= 0, and (16) is valid.
?As it can be obtained from [9, 14] and Lem. 3.2 from [13], rgR1(�) = rgR(�) = m, =� > 0.
62 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
P r o o f. Taking into account the beginning of the proof of Th. 2, one can
see that it is su�cient to prove Th. 3 for the system (20). For de�niteness, set
=�0 > 0.
If for the c.m. ~M (�) of the system (20) on (0; 1) in (6) there is the equality
for � = �0, then in view of Remark 4 the matrix v�0 in the representation (22) has
the form (30) as � = �0 if the basis is chosen such that P (�0) = diag (Im; On�m).
Choose the vectors ~x11; : : : ; ~x
1
n�m so that the matrix v�0 (30) becomes unitary.
Consider the matrix ~M (�0; b) = ~M (�0; b; v�0) (21) with the unitary matrix
v�0 and the boundary value problem (7)�(10) for the system (20) with
M� = ~P (�0; b)� I; N� = ~X� (b) ~P (�0; b) ; ~P(�0; b) = ~P(�0; b; v�0); (33)
where ~P(�0; b) and ~M(�0; b) are joined with G = J by (13).
From [14] it is known:
1) The c.m. ~M (�; b) of this problem as � = �0 coincides with ~M (�0; b; v�0)
and, therefore, ~M (�0; b) ���!
b!1
~M (�0).
2) Since v�0 is unitary, then in the condition (6)for the c.m. ~M(�; b) there is
the equality as � = �0 (and therefore as =� 6= 0 by [9, 13]).
By virtue of n01 of Remark 4 in (6) for the c.m.
~~M (�) = lim ~M (�; bnk) there
is the equality as � = �0 and, consequently, in (6) (see the proof of n01 Th. 2)
there is the equality for it as =�=�0 > 0.
By virtue of Lem. 3.2 from [13], the operators R�
�
~P (�; b)
�
and R�
�
~P (�)
�
,
R�
�
~~P�
�
corresponding to ~M (�; b) and ~M (�) ;
~~M (�), respectively, are the re-
solvents of maximal symmetric relations as =�=�0 > 0 generated by the system
(20) in L2
~H
(0; b) and L2
~H
(0; 1), respectively. Therefore ~M (�) =
~~M (�), and
from the strong convergence of the resolvents R�0
�
~P (�0; bnk)
�
! R�0
�
~~P (�0)
�
in L2
~H
(0; 1)? taking place due to Th. 2, there follows their strong convergence
as =�=�0 > 0. In view of (4) we obtain ~M (�; bnk)!
~M (�) as =�=�0 > 0, and
it is valid for =� 6= 0 since for any c.m. M (�) =M�
�
��
�
. The theorem is proved.
The remark below follows from the proofs of Ths. 2, 3 and from Remark 4.
Remark 5. Let for the c.m. ~M(�) (22) of the system (20) on (0;1) the
contraction v�0 , =�0 > 0 or v��0 , =�0 < 0 be unitary, but not having the repre-
sentation of the form (30) or (31) ?? (such c.m. exists in view of [14]). Then for
the c.m. corresponding to the boundary value problem of the type (20), (10), (33),
?Here we set R�
�
~P (�; b)
�
~f = 0 if ~f 2 L2~H (b; 1)
??Then, if the system (1) corresponds to the Sturm�Liouville scalar equation or if for this
system the condition of Lem. 2 holds as =�0=�0 < 0, then the boundary condition of the type
(33) for the corresponding system (20) with this unitary matrix v�0 is not separated.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 63
V.I. Khrabustovskyi
there is the equality in (6) as =� 6= 0. However, for the corresponding limit c.m.
this is not true as =�=�0 > 0.
Theorems 2 (in another form), 3 and Remark 5 are announced in [8] for the
limits of the c.m. of the self-adjoint regular scalar di�erential operators of even
order with real coe�cients.
Lemma 2. Let for some nonreal � = �0 the number of linearly independent
solutions of the homogeneous system (1) that belong to L2
w�0
(0; 1) is equal to the
number of negative eigenvalues of the matrix =�0G (i.e., it is minimal in view of
[16]).
Then for any nonreal �, for any c.m. M (�) (13) of the system (1) on the
semi-axis (0; 1) in the representation (13) the operator P (�) is the projection.
P r o o f. In view of [14, 16] the conditions of the lemma hold for any �
such that =�=�0 > 0. For de�niteness, set =�0 > 0. From the de�nition of c.p.
it follows (see [13]) that for =� > 0 the initial condition x(0) for the solution of
the homogeneous system (1) that belongs to L2
w�
(0;1) satis�es x (0) 2 P+ (�)H,
where P+ (�) is the c.p. (4.17) from [13] of this system on (0; 1) .
We extend the coe�cients of the system (1) to the left semi-axis (�1; 0) in
such a way that
Q (t) = G; H� (t) = �I; t < 0: (34)
If for =� > 0 this system with f (t) = 0 has the solution from L2
w�
�
R1
�
, then,
in view of (34), its initial condition satis�es x (0) 2 P+H
T
P+ (�)H, where P+ is
the Riesz projection of the operator G that corresponds to its positive spectrum.
But the subspace P+H is G-positive, and P+ (�)H is G-negative [13]. Therefore
x (0) = 0.
Hence in view of n06 of Th. 1.1 [13] the c.m. ~M (�) of the system (1) extended
by means of (34) is unique and in view of [13] equal to (13), with P (�) being
substituted by
~P (�) = P+ (�) (P+ (�) +P�)
�1 ; P� = I �P+;
det (P+ (�) +P�) 6= 0;
�=� > 0: (35)
In L2
= ~H�0
�
R1
�
, where =�0 > 0, consider the minimal closed relation L0 that
corresponds to the di�erential expression
i
��
~Q (t) y (t)
�0
+ ~Q (t) y0 (t)
�
�Re ~H�0 (t) y (t) ;
where ~Q (t) ; ~H� (t) are the coe�cients of the system (1) extended by means
of (34).
64 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
The analog of X� (t) for the system (1), (34) on the axis we denote by ~X� (t).
In view of [17] (see also [18, 19]) and [13] the operator in L2
~H� 0
�
R1
�
~x�� 0 (t) =
~R�� 0
~f
=
1Z
�1
~X�� 0
(t)
�
~M
�
��0
�
�
1
2
sgn (s� t) (iG)
�1
�
~X�
� 0
(s)= ~H� 0 (s)
~f (s) ds
is equal to (L0 + i)�1. In view of Lagrange's formula 8 ~f (s) 2 L2
~w�0
�
R1
�
lim
(s; t)"R1
�
U
�
~x�� 0 (t)
�
� U
�
~x�� 0 (s)
��
= 0:
It is obvious that for any ~f (s) 2 L2
~w�0
�
R1
�
with compact support
lim
s!�1
~x�� 0 (s) = 0;
and therefore for any f (s) 2 L2
w� 0
(0; 1) with compact support
lim
t!1
U
�
~x�� 0 (t)
�
= 0;
and hence in view of (35), we have
lim
t!1
P�+
�
��0
�
X�
�� 0
(t)Q (t)X�� 0
(t)P+
�
��0
�
= 0: (36)
Let M (�) be an arbitrary c.m. of the system (1) on (0; 1). In view of
Theorem 4.4 from [13] for the matrix P (�) corresponding to M(�) by (13) the
following holds:
P
�
��0
�
H � P+
�
��0
�
H; (37)
whence
P
�
��0
�
= P+
�
��0
�
P
�
��0
�
; (38)
and therefore for the matrix P (�) that corresponds to M (�) there is the equality
of the type (36). Thus for the c.m. M (�) the condition (6) is separated for
� = ��0, and in view of Th. 1 P
�
��0
�
= P2
�
��0
�
. Consequently, P (�) = P2 (�) as
=� 6= 0 because �0 is arbitrary and in view of (3.6) from [13]. Lemma 2 is proved.
Let the condition of Lemma 2 hold. Then, in view of [13] and (38), for
the system (1) on the semi-axis (0; 1) the Weyl function K+ (�) = K�
+
�
��
�
2
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 65
V.I. Khrabustovskyi
B (P�H; P�H), �=� > 0 is unique, where P� are such complementary ortho-
projections that ��G� = P+�P�. In view of Theorem 4.4 from [13] the following
remark is valid.
Remark 6. Let the condition of Lemma 2 hold. Then the formula
P (�) = � (P� +K+ (�)P�) (I� � U� (�) K+ (�))�1 (P� � U� (�)P�) �
�1;
�=� > 0; (39)
where I� are identical operators in P�H, establishes a one-to-one correspondence
between the c.p.'s P (�) of the system (1) on semi-axis (0; 1) and the contractions
U� (�) = U��
�
��
�
2 B (P�H; P�H), �=� > 0 that analytically depend on non-
real �. Moreover, x� (t) (5), (13), (39) for any vector function f (t) 2 L2
w�
(0; 1)
with compact support is the unique solution of the system (1) that belongs to
L2
w�
(0; 1) and satis�es the boundary condition?
9h = h (�; f) 2 H : x (0) = � (P� + U� (�)P�)h; �=� > 0: (40)
Now let us consider the boundary problem (1), (10) as b � � (see (4)) with
M� = I; N� = � (b)U� (b) ; (41)
where � (b) is an arbitrary matrix such that
�� (b)Q (b) � (b) = G; (42)
the matrix U� (b), analytically depending on the nonreal �, is such that??
(=�) (U�� (b)GU� (b)�G) � 0; U��� (b)GU� (b) = G (43)
In view of Theorem 2.6 [13] this problem satis�es the conditions of Remark 1
and therefore its c.m. is equal to (13), where???
P (�) = (I � Z� (b))
�1 def
= P (�; b) (44)
hence
I �P (�; b) = (Z� (b)� I)�1 Z� (b) ; (45)
?This condition can be self-adjoint as � = �0 only if the signature of the matrix =�0G is
nonpositive. Otherwise it is accumulative or dissipative, but not necessarily strictly.
??If for =� 6= 0 (43) holds and for some nonreal � and b = b0 there is the equality in (43),
then U� (b0) does not depend on � [8] (see also [13] as dimH =1 ).
???Below we denote P (�; b) (in contrast to Lem. 1 and Ths. 2, 3) only (44).
66 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
where
Z� (b) = U�1� (b) ��1 (b)X� (b) : (46)
In view of (3), (42), (43)
1
=�
[Z�� (b)GZ� (b)�G] � 2�� (0; b) ; (47)
moreover, if there is the equality in (43), then it is in (47), too.
It follows from (4) and (47) that the matrix Z� (b) (46) is unitary dichotomic
if the matrix G is inde�nite?. We denote by P (Z� (b)) the Riesz projection of
the matrix Z� (b) that corresponds to its spectrum lying inside the unit circle.
Since in view of [20, p. 64] the subspaces P (Z�(b))H and (I �P (Z�(b)))H are
=�G-negative and =�G-positive, respectively, then in view of (2), (43) P (Z� (b))
is the c.p. of the system (1) on (0; b).
From Lemma 1 there follows
Lemma 3. Any sequence bn " 1 contains such subsequences
�
bnk
;
�
bn0
k
that for any nonreal � the following limits exist:
limP (�; bnk) = P (�) ; (48)
limP
�
Z�
�
bn0
k
��
= Q (�) = Q2 (�) : (49)
The matrix functions M (�) (13) with P (�) (48) and P (�) = Q (�) (49) are
the c.m.'s of the system (1) on (0; 1).
The following proposition follows from Th. 1 and (44).
Proposition. Boundary condition (6), that corresponds to the limit c.m.
M(�) (13), (48) is separated for nonreal � = �0 if and only if
80 6= f 2 H : k(Z�0 (bnk) + Z�1�0 (bnk))fk ! 1: (50)
Example 1. Let in the system (1) Q(t) = G;H�0(t) 2 L1(0;1);=�0 6= 0.
Then in view of [20, p. 166] 9 lim
t!1
X�0(t) = X; detX 6= 0, and (50) holds
if in Lemma 3 bn = f(n); U�0(bn) = X�n, where f(n) is such a function that
(kXkn + kX�1kn)kX�0(f(n)) � Xk ! 0. Here P(�)(48)= P(X). P(X) is the
analog of P(Z�(b)) for X. In this example the condition of Lem. 2 does not hold
as � = �0 because 8h 2 H : X�0(t)h 2 L2
w�0
(0;1).
?Further for simplicity it is considered to be true. In the case of de�nite G the statements
and proofs are modi�ed in a obvious way.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 67
V.I. Khrabustovskyi
Lemma 4. Let the condition of Lemma 2 hold and in Lemma 3 let one of the
subsequences
�
bnk
,
�
bn0
k
be a subsequence of another one. Then in (48), (49)
P (�) = Q (�) : (51)
P r o o f. We denote the subsequence that is a subsequence of another one by
f�ng. Since in view of Lem. 2 P (�) = P2 (�) ; then
kP (�; �n) (I �P (�; �n))k ! 0: (52)
Thus in view of (44), (45)
(I � Z� (�n))
�2 Z� (�n)
! 0; (53)
and therefore the spectrum
�
�
(I � Z� (�n))
�2 Z� (�n)
�
! 0: (54)
From the theorem on a mapping of spectrum and with the unitary dichotomy
of Z� (b) being taking into account, we obtain
max fj�j j� 2 �i (Z� (�n))g ! 0; min fj�j j� 2 �e (Z� (�n))g ! 1; (55)
where �i (Z� (�n)) and �e (Z� (�n)) are the parts of spectrum Z� (�n) lying inside
and outside the unit circle, respectively.
One has
P (�; �n)�P (Z� (�n)) = An +Bn; (56)
where
An = (P (�; �n)� I)P (Z� (�n)) ; Bn = P (�; �n) (I �P (Z� (bn))) : (57)
In view of (52) it follows from (57) that
kP (�; �n)Ank ! 0:
Since in view of (55) �
�
P (�; �n)
��
P(Z�(�n))H
�
! 1, then (for example, in view
of [21, p. 42]) there could be found a constant Æ = Æ (�) > 0 not depending on n
and such that
8f 2 H : kP (�; �n)Anfk � Æ kAnfk ;
68 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
from which An ! 0. In the same way one can prove that Bn ! 0. The lemma is
proved in view of (56).
The �rst statement in (55) is strengthened by
Remark 7. There exists such not depending on b constant K = K (�), that
kZ� (b)P (Z� (b))k � K: (58)
Besides if the conditions of Lem. 2 hold, then
Z� (�n)P (Z� (�n))! 0: (59)
P r o o f. In view of (4),(47) and =�G-nonnegativity of P (Z� (b))H one has
9Æ = Æ(�) > 0 8f 2 H:
1
=�
f�P� (Z� (b))X
�
� (b)Q (b)X� (b)P (Z� (b)) f � �Æ kZ� (b)P (Z� (b)) fk
2 :
(60)
The statement (58) is based on the fact that the left-hand side of the inequality
(60) is bounded as kfk = 1 in view of Lem. 1.8, formula (1.77) and Th. 4.2 [13].
The statement (59) follows from (53), (58) and Lem. 1.8 [13] since
Z� (�n)P (Z� (�n)) = ((I � Z� (�n))P (Z� (�n)))
2 (I � Z� (�n))
�2 Z� (�n) :
(61)
The remark is proved.
From Lemma 4, Remark 7, formula (47) and the fact that Z� (b)P (�; b) =
P (�; b)� I there follows
Corollary 1. Let the condition of Lemma 2 hold and let f�ng be the subse-
quence from the proof of Lemma 4. Then for any f (t) 2 L2
w�
(0; 1) with compact
support
lim
�n!1
k[R� (P (Z� (�n)))�R� (P (�; �n))] f (t)k
2
L2w
�
(0; �n)
�
1
2=�
(G (I �P (�))h; (I �P (�))h) ; (62)
where h = (iG)�1
1R
0
X�
��
(s)w� (s) f (s) ds. If there is the equality in (43), then
there is also the equality in (62) with lim instead of lim.
Remark 8. For the system (1) with 1-periodic coe�cients and with periodic
condition (43) the equality (17) does not hold with bnk = n for any nonreal �.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 69
V.I. Khrabustovskyi
P r o o f. Let Q (t+ 1) = Q (t), H� (t+ 1) = H� (t) in the system (1) and
let � � 1 in the condition (4). Then, as shown in [22�24] (see also Th. 5 below),
the system (1) satis�es the condition of Lem. 2. Let � (n) = U� (n) = I, n =
1; 2 : : : in the boundary conditions (41)�(43). Then (48) holds with bnk = n and
P (Z� (n)) = P (Z� (1)) in view of Floke theorem. Therefore P (�) = P (Z� (1)) by
Lem. 4 with �n = n and, consequently, kR� (P (Z� (n)))�R� (P (�))k
B
�
L2w�
(0;n)
�
= 0.
Therefore in view of Cor. 1 for any f (t) 2 L2
w�
(0; 1) with compact support
lim
n!1
k[R� (P (�))�R� (P (�; n))] f (t)k2L2w�(0; n)
=
1
2=�
(G [I �P (Z� (1))]h; [I �P (Z� (1))]h) e
� Æ (�) kI �P (Z� (1)) hk
2 ;
where Æ (�) > 0 as =� 6= 0 in view of (4), (47), [20, p. 64]. The corollary is
proved.
It follows from [9, 14, 27] that =M (�) /=� > 0 for limit c.m. (13), (48) which
is obtained if in the boundary condition (41)�(43) we set �(b) = �(b); U� (b) =
��1 (b) ~T (b)S. Here ~T (b) = T (b)U(b); T (t); S � see (19), U(t) 2 ACloc is the
J-unitary matrix such that J-module (see, e.g., [24]) of matrix S�1��1(t)Xi(t)S
is equal to U�1(t)S�1��1Xi(t)S. But
=M (�) = G�1 ((I �P� (�))G (I �P (�))�P� (�)GP (�))G�1:
Let the condition of Lem. 2 hold. Then P2 (�) = P (�). Since the subspace
P (�)H is maximal =�G-negative [13], then rg(I �P(�)) is equal to the number
of positive eigenvalues of the matrix =�G. Therefore rg (I �P� (�))G (I �P (�))
is equal to the number of these eigenvalues because =M(�)==� > 0. Therefore
the following remark is valid.
Remark 9. Let the condition of Lemma 2 hold. Then there exists such
a sequence �n " 1 and such independent from � matrixes U� (�n) satisfying
(41)�(43) (with the equality in (43)) that the strictly dissipative or accumulative
boundary condition (40) at zero corresponds to the limit c.m. M (�) (13), (48),
(bnk = �n).
Theorem 4 below gives the conditions under which any limit c.m. corresponds
to strictly dissipative or accumulative boundary condition at zero (if P (�) 6= I ).
To prove it the following lemma is necessary.
Lemma 5. Let for some nonreal � = �0 the sequences f�ng ; f
ng exist and
the constants Æ > 0; M; � > 0 exist such that � (�n) = I, and the following
70 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
conditions hold:
��0 (0; �n) � ÆX�
�0
(
n)X�0 (
n) ; (63)������
�nZ
n
Q�1 (t)
�
H�0 (t)�
i
2
Q0 (t)
�
dt
������ �M; (64)
�
�
U��0 (�n)U�0 (�n)
�
� �; (65)
where � (A) is the smallest eigenvalue of the matrix A = A�.
Then for �n � � (see. (4))
1
=�0
(I �P� (Z�0 (�n)))G (I �P (Z�0 (�n)))
> 2�e�M (I �P� (Z�0 (�n))) (I �P (Z�0 (�n))) :
P r o o f. We extend the coe�cients Q (t) and H� (t) of the system (1)
periodically on [��n; 0) from [0; �n) if the period is equal to �n.
In view of (42) Q (t) 2 ACloc and therefore
X� (t) = X� (s)X
�1
� (�n) ;
where t 2 [��; 0] ; s = t+ �n. Whence in view of (63), (64) 8f 2 H
f���0 (��n; 0) f = f�X��1
�0
(�n)��0 (0; �n)X
�1
�0
(�n) f
� Æf�X�1�
�0
(�n)X
�
�0
(
n)X�0 (
n)X
�1
�0
(�n) f �
Æ kfk2
X�0 (�n)X
�1
�0
(
n)
2
� Æ exp
8<
:�
������
�kZ
k
Q�1 (t)
�
H�0 (t)�
i
2
Q0 (t)
�
dt
������
9=
; kfk2 � Æe�M kfk2 ;
where the inequality before the last one holds true in view of [20, p. 162].
As a result, in view of (47), (65):
1
=�0
h
G� Z��1�0
(�n)GZ
�1
�0
(�n)
i
� 2U��0(�n)� (��n; 0)U�0(�n) � 2�e�MI;
whence
1
=�0
[(I �P� (Z�0 (�n)))G (I �P (Z�0 (�n)))
�Z��1�0
(��n) (I �P
� (Z�0 (�n)))G (I �P (Z�0 (�n)))Z
�1
�0
(��n)
i
� 2�e�M (I �P� (Z�0 (�n))) (I �P (Z�0 (�n))) :
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 71
V.I. Khrabustovskyi
But for �n � � due to (4), (47) [20, p. 61] we have
9Æn > 0 :
1
=�0
(I �P� (Z�0 (�n)))G (I �P (Z�0 (�n)))
� Æn (I �P
� (Z�0 (�n))) (I �P (Z�0 (�n))) :
Lemma is proved.
Theorem 4. Let the condition of Lemma 2 hold and for the sequence bn " 1
let the following limits? exist for all nonreal �
limP (�; bn) = limP(Z�(bn)) = P (�) ; (66)
that is the c.p. by Lem. 3. Let the sequence fbng contain the subsequence f�ng,
for which the following conditions hold with some nonreal � = �0:
10. For U�0 (b) (43) a constant � > 0 exists such that (65) holds.
20. There exists the sequence f
ng and the constants Æ > 0 and M such that
either
a) � (�n) = I and (63),(64) hold for su�ciently large n
or
b) in (42) � (t) 2 ACloc; � (0) = I, and for su�ciently large n
��0 (0; �n) � ÆX�
�0
(
n) �
��1 (
n) �
�1 (
n)X�0 (
n) ; (67)������
�nZ
n
i2
�
��
0
(t) ���1 (t)G�G��1 (t) �0 (t)
�
+ �� (t)H�0 (t) � (t)
dt
������ �M: (68)
Then for =�=�0 > 0:
9Æ0 = Æ0 (�) > 0 :
1
=�
(I �P� (�))G (I �P (�)) � Æ0 (I �P
� (�)) (I �P (�)) :
(69)
P r o o f. The proof of (69) with � = �0 and when 10, 20a) hold, follows from
Lems. 4, 5. The proof of (69) with � = �0 and when 10, 20b) hold, follows from
from Lems. 4, 5 and the fact that the c.m.'s of the problems (1), (10), (41)�(43)
and
iGy0 (t)�
�
i
2
�
��
0
(t) ���1 (t)G�G��1 (t) �0 (t)
�
+ �� (t)H� (t) � (t)
�
y (t)
= �� (t)w�� (t) g (t) 9h = h (�; g) : y (0) = h; y (b) = U� (b)h
?That are equal by Lem. 4.
72 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
coincide. It follows from (69), ( � = �0 ) that the contraction U� (�) in (40) is
strict as � = �0. Therefore, in view of [26, p. 210], this contraction is strict for
any � such that =�=�0 > 0. Thus (69) is valid for any of these � in view of (40).
The theorem is proved.
Remark 10. If such sequences f�ng; f
ng that the conditions (63) and (64)
(or (67) and (68)) hold simultaneously do not not exist, then it may happen that
(I �P� (�))G (I �P (�)) = 0 for self-adjoint or even nonself-adjoint boundary
condition (10), (41) for a sequence of regular boundary problems not depending
on b.
P r o o f. Let in the system (1)
Q (t) =
�
0 i
�i 0
�
; H� (t) = �H (t) ;
0 � H (t) =
8>><
>>:
�
B 0
0 0
�
; 0 � t < 1;�
0 0
0 An
�
; 1 � n � t < n+ 1;
where B > 0;
1P
n=1
An > 0. It can be directly veri�ed (it also follows from [7])
that the condition of Lem. 2 holds for this equation if and only if
1P
n=1
An = 1.
We have for N� =
�
1 ��A0
0 1
�
, where A0 � 0:
I �P (�) = lim
n!1
h
I + (Z� (n+ 1)� I)�1
i
=
0
@ 0 �B
�
1
�
1P
j=0
Aj
0
1
A.
Thus with bnk = n+1 for the limit P (�) (48) we have (I�P�(�))G(I�P(�))
= 0 if and only if
1P
n=1
An =1 (for self-adjoint boundary condition (41) (A0 = 0)
or for nonself-adjoint boundary condition (41) (as A0 > 0 ) ). Thereby for the
considered equation the simultaneous ful�lment of the conditions (63), (64) is
impossible as
1P
n=1
An = 1, although (63) holds when for example
n = 0, while
(64) holds when for example An and j�n �
nj are bounded simultaneously. The
remark is proved.
We will illustrate Th. 4 by using the perturbed almost periodic systems as an
example.
Theorem 5. Let in the system (1) the matrix-function Q (t) be uniformly
almost periodic, inf
t
jdetQ(t)j > 0.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 73
V.I. Khrabustovskyi
Let for some nonreal �0
H�0 (t) = A (t) +B (t) ; where =A(t)==�0 � 0: (70)
Suppose the matrix function A (t) to be uniformly almost periodic, and the
normalized at zero on I fundamental matrix of the system
i
2
(Q (t) y (t))
0
+Q (t) y0 (t) = A (t) y (t) (71)
to have a representation of the Floke type
Y (t) = Z (t) e�t; (72)
where the matrix function Z (t) is uniformly almost periodic, inf
t
jdetZ(t)j > 0,
and
9�0 > 0 :
�0Z
0
Y � (t)
�
=A (t)
=�0
�
Y (t) dt > 0: (73)
Let
1Z
0
t2q�2 kB (t)k dt <1; (74)
where q is equal to the maximal order of the Jordan cells of matrix � in represen-
tation (72)?.
Then for the system (1):
10. The condition of Lem. 2 holds for any �0 such that =�0=�0 > 0.
20. The condition (63) in n020 of Th. 4 holds for any sequence f�ng " 1
if f
ng " 1 is such a sequence of positive common "-almost periods of matrixes
A (t) and Z (t), that �n�
n � �0 (see (73)), " is su�ciently small. For these �n
and
n the condition (67) in n020 of Th. 4 also holds if the norms k�(
n)k are
bounded.
Besides, if the di�erence �n�
n is bounded, then all other conditions in n020
of Th. 4 also hold
in the case a) if we additionally require in (42) � (�n) = I and the norms
kQ0 (t)k to be bounded;
in the case b) if we additionally require in (42) � (t) 2 ACloc, �(0) = I and
the norms k� (t)k ; k�0 (t)k to be bounded on the semi-axis (0;1).
?As it is seen from (80), obtained below in the proof of Th. 5, the ful�lment of the condition
(4) for the equation (1) follows from (72)�(74).
74 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
P r o o f. It follows from (71),(72),(74) and Yakubovich's Theorem [28, p.
383] (see also [20, p. 275]) that
X� 0 (t) = Z (t) (I + S (t)) e�tC; (75)
where the matrixes S (t)! 0 as t!1.
For de�niteness, set =�0 > 0.
Let us prove 10. In view of the equality of the type (3) for the equation (71)
and the conditions (72), (73) we have
e�tZ� (t)Q (t)Z (t) e�t �G > 0; t � �0: (76)
In view of (76): �(e�� \ f� 2 C : j�j = 1g = ?, where �0 � � is the
common "-almost period of Q (t) and Z (t) for su�ciently small ". Therefore
�(e� \ f� 2 C : j�j = 1g = ?. Let us denote the invariant subspaces of e� that
correspond to the parts of its spectrum lying inside and outside the unit circle
by Hi and He, respectively. We denote the corresponding Riesz projections of
the matrix e� by Pi and Pe. If f 2 PiH, then e�tf ! 0 as t ! 1. Therefore
(Gf; f) < 0 for f 6= 0 in view of (76). Therefore dimPiH does not exceed the
number of negative eigenvalues of the matrix G.
On the other hand, in view of (70), (72), (75) 8� > 0:
��0(�; � + �) = C�e�
��
�Z
0
e�
�t(I + S�(t+ �))Z�(t+ �)[=A(t+ �)==�0
+=B(t+ �)==�0]Z(t+ �)(I + S(t+ �))e�tdtee
��
C: (77)
Now let 0 < � be a �xed and su�ciently large "-almost period common for Z(t)
and A(t), where " is su�ciently small. In view of boundness of A(t); Z(t); S(t)
and in view of (74) there exists a constant Æ1, and also in view of (73) there exists
a constant Æ2 > 0 such that due to (77) we have
Æ1
1X
n=0
ke�n�Cfk2 � f���0(0;1)f � Æ2
1X
n=0
ke�n�Cfk2: (78)
It follows from (78) that
f���0(0;1)f =1; if 0 6= f 2 C�1PeH;
f���0(0;1)f <1; if 0 6= f 2 C�1PiH;
Therefore the number of linearly independent solutions of the homogeneous
equations (1), (71)�(74), belonging to L2
W�0
(0;1), is equal to dimPiH, i.e., it does
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 75
V.I. Khrabustovskyi
not exceed the number of negative eigenvalues of the matrix G, but in view of [16]
the number of these solutions is not less than the number of these eigenvalues.
The statement 10 is proved in view of [16].
Let us prove 20. We check, for example, the ful�lment of condition (67) of
Theorem 4. Using (70), (74), (77) we have
��0(0; �n) �
�nZ
n
X�
�0
(t)w�0(t)X�0(t)dt
� C�e�
�
n
( �0Z
0
e�
�t(I + S�(t+
n))Z
�(t+
n)
�
=A(t+
n)
=�0
�
Z(t+
n)(I + S(t+
n))e
�tdt+
(
n)
)
e�
nC; (79)
where the expressions under integrals in (73) di�ers a little from those in (79) for
large
n uniformly on t 2 [0; �0],
(
n)! 0. Therefore in view of (73), (79) there
exists such constant Æ3 > 0 not depending on n that
��0(0; �n) � Æ3C
�e�
�
ne�
nC (80)
for su�ciently large n. Since
X�
�(
n)�
��1(
n)�
�1(
n)X�(
n)
�
n
max
n
k��1(
n)Z(
n)(I + S(
n))k
2
o
C�e�
�
ne�
nC;
then (67) holds in view of (42).
Under the assumptions of Th. 5 all other conditions of n020 of Th. 4 evidently
hold and Th. 5 is proved.
Remark 11. We can set in (42):
�(t) = [Q2(t)]�1=4V (t)[Q2(0)]1=4 2 ACloc;
where the unitary matrix V (t) is a solution to Cauchy problem
V 0 = P 0(t)(2P (t) � I)V (t); V (0) = I;
P (t) is a Riesz projection of the matrix Q(t) that corresponds to its negative
spectrum. Moreover, the norms k�(t)k; k�0(t)k are bounded on axis if the matrix
function Q(t) is uniformly almost periodic, inf
t
jdetQ(t)j > 0, Q0(t) is bounded.
76 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
P r o o f is the corollary of [7].
The following theorem shows that under some assumptions in the case of
periodic boundary condition (10),(41) the limit c.m. of the system (1) on the
semiaxis (0;1) is a c.m. of this system on the axis (�1;1) if its coe�cients are
extended in a certain way to the negative semiaxis.
Theorem 6. Let the condition of Lem. 2 hold and let for the sequence f�ng
from the proof of Lemma 4 there be the subsequence f�njg such that in (41)
�(�nj ) = U�(�nj ) = I and Q0(t+ �nj) are locally uniformly bounded for t < 0.
Let the following limits exist for:
1) any t � 0 : limQ(t+ �nj ) = Q�(t); where detQ�(t) 6= 0;
2) almost all t < 0: limQ0(t+ �nj );
3) any nonreal � and any a < 0: H�(t + �nj ) ! H�
� (t) in L1(a; 0), where
H�
� (t) depends on the nonreal � in Nevanlinna's manner.
Then P(�) (48) (fbnkg = f�ng) is a c.p. of the system (1) on the axis (see
[13]) with the coe�cients?
Q(t) =
(
Q�(t)
Q(t)
; H�(t) =
(
H�
� (t); t � 0
H�(t); t > 0:
(81)
P r o o f. We extend the coe�cients Q(t) and H�(t) of the system (1) period-
ically on [��nj ; 0) if the period equals �nj . The corresponding extension of X�(t)
we denote X
nj
� (t). Let us �x t = t0 < 0. Then for ��nj < t0 and due to (3) we
have
1
=�
(I �P�(X�(�nj )))X
n�
j
� (t0)Q(t0)X
nj
� (t0)(I �P(X�(�nj )))
�
1
=�
(I �P�(X�(�nj )))X
n�
j
� (��nj )Q(��nj )X
nj
� (��nj )(I �P(X�(�nj )))
=
1
=�
X�
�
�1(�nj )(I �P
�(X�(�nj )))G(I �P(X�(�nj )))X
�1
� (�nj ) � 0:
Passing to the limit in the inequality : : : Q(t0) : : : � 0 as �nj !1 and using
the fact that for any nonreal � X
nj
� (t0) ! X�(t0) due to 1)�3) and [20, p. 160],
we obtain
1
=�
(I �P(�))X�
�(t0)Q
�(t0)X�(t0)(I �P(�)) � 0;
where X�(t) is the matrix solution of homogeneous system (1), (81) such that
X�(0) = I. The theorem is proved in view of [13, p. 450] and since t0 is arbitrary.
?Whose properties on t and on � are similar to those of the system (1) on the semiaxis (see
introduction) due to 1)�3).
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1 77
V.I. Khrabustovskyi
We notice that for any sequence f�ng " 1 there is a subsequence f�njg for
which the conditions 1)�3) of Th. 6 hold if the matrix coe�cients of the system (1)
satisfy the following conditions: the coe�cient Q(t) is uniformly almost periodic
matrix function having uniformly almost periodic derivative and inf
t
jdetQ(t)j > 0;
the coe�cient H�(t) =(18) and for some nonreal �0: H�0(t) = h(t) + �(t), where
h(t) is almost periodic Stepanov's matrix function,
x+1R
x
k�(t)kdt! 0 as x! +1.
Remark 12. If for the system (1), (81) from Th. 6 the analog of the condition
(4) on the left semi-axis (�1; 0) holds,? then (69) for c.p. P(�) from this theorem
holds for any nonreal � in view of [13, p. 459]. If additionally this c.p. is the
unique c.p. of the system (1),(81) on the axis, then P(�) can be found explicitly
with the help of Th. 4.1 from [13] by taking any pair of c.p.'s of the system (1),
(81) on the left and right semiaxes.
Example 2. In the system (1) let Q(t) = G; H�(t) 2 Cloc on t, H�(t+ T ) =
H�(t); T > 0, and therefore the condition of Lem. 2 holds. A T -periodic extension
of H�(t) on the whole axis is again denoted by H�(t). Then the sequence f�ng
from Theorem 6 contains the subsequence f�njg such that H�(t+�nj )! H�(t+�)
uniformly on t 2 (�1;1), where � 2 [0; T ) and � = lim�nj (modT ).
Therefore in the considered case the conditions 1)�3) of Th. 6 hold with f�njg
mentioned above, and H�
� (t) = H�(t+ �). In view of [29, p. 225] the c.m. of the
system (1), (81) on axis is unique, and if �(�nj ) = U�(�nj ) = I, then the limit
c.p. P(�) from Th. 6 of the system (1) on (0;1) equals
P(�) = P(X�(T ))[P(X�(T )) +X�(�)(I �P(X�(T )))X
�1
� (�)]�1:
The corresponding spectral matrix that generates Parseval's equality (when
H�(t) =(18)) is equal [29] to �(�) = �ac(�)+�d(�), where �
0
ac(�) can be found by
the formula from [29] for the spectral matrix of the system (1) on axis in the case
H�(t) = H�(t � T�); T� > 0; t 2 R�. In particular, it follows from this formula
and [22]�[24]?? that the absolutely continuous spectrum of the limit problem and
its multiplicity are similar to those in the system (1) on axis if its coe�cients are
extended periodically on the left semi-axis. The component �d can be nonzero if,
for instance, H�
� (t) = H�(�t) (t < 0) and H�(t) (81) is not T -periodic function
on the axis (it follows, e.g., from [29, p. 218]).
?For example, it takes place if the system (1) corresponds to the symmetric matrix di�erential
operator of arbitrary order.
??We notice that for the system (1), (18) with periodic coe�cients the c.m. on the axis and
its asymptotic in singular points have been obtained in [22] (when H(t) is the weight of the
positive type) and in [23, 24] (in the general case). Later an equivalent formulae for the c.m.
and its asymptotic in the mentioned points for the canonical periodic systems with the weight
of the positive type have been obtained by L.A. Sakhnovitch (see [12]) apparently in a more
complicated way.
78 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 1
On the Limit of Regular Dissipative and Self-Adjoint Boundary Value Problems
Example 3. All conditions of Th. 6 and Remark 12 hold if: a) the system
(1) corresponds to the Sturm�Liouville equation on [0;1) with matrix potential
P(t) 2 Cloc for which there exists such sequence of intervals (an; bn) � (0;1)
that: an+1 > bn " 1, bn � an " 1, �(bn) = U�(bn) = I, P(t) 2 C1[an; bn],
9M : jjP(t)jj + jjP0(t)jj �M , t 2 [an; bn]; b) the subsequence f�ng is chosen from
fbng. (Here the uniqueness of c.p. for the system (1), (81) follows, for example,
from Cor. 3 from [30].)
Aknowlegments. The author is grateful to Prof. F.S. Rofe-Beketov for
the attention he paid to this work.
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