Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients
The paper investigates spectral properties of multi-interval Sturm–Liouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of boundary conditions are given. Sufficient conditions for the resolvents...
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nasplib_isofts_kiev_ua-123456789-1730462025-02-09T14:24:48Z Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients Багатоінтервальні дисипативні крайові задачі Штурма-Ліувілля з коефіцієнтами-розподілами Goriunov, A.S. Математика The paper investigates spectral properties of multi-interval Sturm–Liouville operators with distributional coefficients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of boundary conditions are given. Sufficient conditions for the resolvents of these operators to be operators of the trace class and for the systems of root functions to be complete are found. The results are new for one-interval boundary-value problems as well. Досліджено спектральні властивості багатоінтервальних операторів Штурма—Ліувілля з узагальненими функціями в коефіцієнтах. Дано конструктивний опис усіх самоспряжених, максимальних дисипативних/ акумулятивних розширень мінімального оператора в термінах крайових умов. Знайдено достатні умови ядерності резольвент цих операторів та повноти систем їх кореневих функцій. Результати роботи є новими і для одноінтервальних крайових задач. 2020 Article Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients / A.S. Goriunov // Доповіді Національної академії наук України. — 2020. — № 7. — С. 10-16. — Бібліогр.: 15 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2020.07.010 https://nasplib.isofts.kiev.ua/handle/123456789/173046 517.926 en Доповіді НАН України application/pdf Видавничий дім "Академперіодика" НАН України |
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Математика Математика |
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Математика Математика Goriunov, A.S. Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients Доповіді НАН України |
| description |
The paper investigates spectral properties of multi-interval Sturm–Liouville operators with distributional coefficients.
Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of
boundary conditions are given. Sufficient conditions for the resolvents of these operators to be operators of the
trace class and for the systems of root functions to be complete are found. The results are new for one-interval
boundary-value problems as well. |
| format |
Article |
| author |
Goriunov, A.S. |
| author_facet |
Goriunov, A.S. |
| author_sort |
Goriunov, A.S. |
| title |
Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients |
| title_short |
Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients |
| title_full |
Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients |
| title_fullStr |
Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients |
| title_full_unstemmed |
Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients |
| title_sort |
multi-interval dissipative sturm—liouville boundary-value problems with distributional coefficients |
| publisher |
Видавничий дім "Академперіодика" НАН України |
| publishDate |
2020 |
| topic_facet |
Математика |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/173046 |
| citation_txt |
Multi-interval dissipative Sturm—Liouville boundary-value problems with distributional coefficients / A.S. Goriunov // Доповіді Національної академії наук України. — 2020. — № 7. — С. 10-16. — Бібліогр.: 15 назв. — англ. |
| series |
Доповіді НАН України |
| work_keys_str_mv |
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| first_indexed |
2025-11-26T19:22:07Z |
| last_indexed |
2025-11-26T19:22:07Z |
| _version_ |
1849881977319063552 |
| fulltext |
10 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 7: 10—16
Ц и т у в а н н я: Goriunov A.S. Multi-interval dissipative Sturm–Liouville boundary-value problems with dis -
tri butional coefficients. Допов. Нац. акад. наук Укр. 2020. № 7. С. 10—16. https://doi.org/10.15407/dopovidi
2020.07.010
1. Introduction. Differential operators, generated by the Sturm—Liouville expression
( ) ( ) ,l y py qy′ ′= − +
arise in numerous problems of analysis and its applications. The classical assumptions on its coef-
ficients are the following:
1([ , ]; ), 0 ([ , ]; )q C a b p C a b∈ < ∈ .
Principal statements of the theory of such operators remain true under more general assump-
tions
1,1/ ([ , ], )q p L a b∈ .
However, many problems of mathematical physics require the study of differential operators with
complex coefficients which are Radon measures or even more singular distributions. In papers
https://doi.org/10.15407/dopovidi2020.07.010
UDC 517.926
A.S. Goriunov
Institute of Mathematics of the NAS of Ukraine, Kyiv
E-mail: goriunov@imath.kiev.ua
Multi-interval dissipative Sturm–Liouville
boundary-value problems
with distributional coefficients
Presented by the Corresponding Member of the NAS of Ukraine A.N. Kochubei
The paper investigates spectral properties of multi-interval Sturm–Liouville operators with distributional coeffi-
cients. Constructive descriptions of all self-adjoint and maximal dissipative/accumulative extensions in terms of
boundary conditions are given. Sufficient conditions for the resolvents of these operators to be operators of the
trace class and for the systems of root functions to be complete are found. The results are new for one-interval
boundary-value problems as well.
Keywords: Sturm–Liouville operator; multi-interval boundary value problems; distributional coefficients; maximal
dissipative extension; completeness of root functions.
11ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 7
Multi-interval dissipative Sturm–Liouville boundary-value problems with distributional coefficients
[1—4], a new approach to the investigation of such operators was proposed based on the defini-
tion of these operators as quasi-differential, which also allows one to consider differential ope-
rators of higher order [3, 5].
The purpose of this paper is to develop a spectral theory of not self-adjoint Sturm—Liou-
ville operators given on a finite system of bounded intervals under minimal conditions for the
regularity of the coefficients.
Multi-interval differential and quasi-differential operators were investigated, particularly,
in papers [7—9].
2. Preliminary results. Let [ , ]a b be a compact interval, m∈, and let 0 1 ma a a a b= < <…< =
be a partition of the interval [ , ]a b into m parts. Let us consider the space 2([ , ], )L a b as a
direct sum 1 2 1([ , ], )m
k k kL a a= −⊕ which consists of vector functions 1
m
k kf f== ⊕ such that
2 1([ , ], )k k kf L a a−∈ .
Let, on each interval 1( , )k ka a− , {1, , }k m∈ … , the formal Sturm-Liouville differential ex-
pression
( ) ( ( ) ) ( ) (( ( ) ) ( ) )k k k k kl y p t y q t y i r t y r t y′ ′ ′ ′= − + + + , (1)
be given with coefficients kp , kq , and kr which satisfy the conditions:
2 1
1
, , , ([ , ], )
| | | | | |
k k
k k k k
k k k
Q r
q Q L a a
p p p
−
′= ∈ , (2)
where the derivatives kQ ′ are understood in the sense of distributions.
Similarly to [3] (see also [1, 4]), we introduce the quasi-derivatives by the coefficients of
expression (1) on each interval 1[ , ]k ka a− in the following way:
[0] :kD y y= ;
[1] : ( )k k kkD y p y Q ir y′= − + ;
2 2
[2] [1] [1]: ( ) k k k k
k k k
k k
Q ir Q r
D y D y D y y
p p
− +′= + + .
We also denote, for all 1[ , ]k kt a a−∈ : [0] [1] 2ˆ ( ) ( ( ), ( ))k k ky t D y t D y t= ∈ .
Under assumptions (2), these expressions are Shin—Zettl quasi-derivatives (see [11, 10]).
One can easily verify that, for the smooth coefficients kp , kq , and kr , the equality ( ) kl y = [2] kD y= − holds.
Therefore, one may correctly define expressions (1) under assumptions (2) as Shin—Zettl
quasi-differential expressions:
[2][ ] :k kl y D y= − .
The corresponding Shin—Zettl matrices (see [10, 11]) have the form
12 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 7
A.S. Goriunov
2 2
12 2
1
([ , ]; ).k
Q ir
p p
A L a b
Q r Q ir
p p
×
+⎛ ⎞
⎜ ⎟
⎜ ⎟= ∈⎜ ⎟+ −− −⎜ ⎟
⎝ ⎠
(3)
Then, on the Hilbert spaces 2 1(( , ), )k kL a a− , the minimal and maximal differential opera-
tors are defined, which are generated by the quasi-differential expressions [ ]kl y (see [10, 11]):
[1] [2]
, 1 , 1 2 1 2: [ ], Dom( ) : { , ([ , ], ), }k k k k kk kL y l y L y L y D y AC a a D y L−→ = ∈ ∈ ∈ ,
, 0 , 0 , 1 1ˆ ˆ: [ ], Dom( ) : { Dom( ) ( ) ( ) 0.}k k k k k k k kL y l y L y L y a y a−→ = ∈ = = .
Results of [10, 11] for general Shin—Zettl quasi-differential operators together with for-
mula (3) imply that the operators , 1kL , , 0kL are closed and densely defined on the space
2 1([ , ], )k kL a a− .
In the case where kp , kq , and kr are real-valued, the operator , 0kL is symmetric with the
deficiency index (2, 2), *
, 0 , 1,k kL L= and *
, 1 , 0k kL L= .
3. Dissipative boundary-value problems. We consider the space 2([ , ], )L a b as a direct sum
1 2 1([ , ], )m
k k kL a a= −⊕ which consists of vector functions 1
m
i if f== ⊕ such that 2 1([ , ], )i i if L a a−∈ .
In this space 2([ , ], )L a b , we consider the maximal and minimal operators max 1 , 1
m
i iL L== ⊕ and
min 1 , 0
m
i iL L== ⊕ .
It is easy to see that the operators maxL , minL are closed and densely defined on the space
2([ , ], )L a b .
Throughout the rest of the paper, we assume that (the) functions kp , kq , and kr are real-valued
for all k, and, therefore, the operators , 0kL are symmetric with the deficiency indices (2, 2). Then
the operator minL is symmetric with the deficiency index (2 , 2 )m m and * *
min max max min,L L L L= = .
Then the problem of describing all its self-adjoint, maximal dissipative and maximal accu-
mulative extensions in terms of homogeneous boundary conditions of the canonical form na-
turally arises. For this purpose, it is convenient to apply the approach based on the concept of
boundary triplets. It was developed in the papers by Kochubei [12], see also book [13] and re-
ferences therein.
Note that the minimal operator minL may be not semi-bounded even in the case of a single-
interval boundary-value problem since the function p may reverse sign.
Recall that a boundary triplet of a closed densely defined symmetric operator T with equal
(finite or infinite) deficiency indices is called a triplet 1 2( , , )H Γ Γ , where H is an auxiliary Hil-
bert space, and 1 2,Γ Γ are the linear maps from *Dom( )T into H such that:
1. for any *, Dom( )f g T∈ , there holds
* *
1 2 2 1( , ) ( , ) ( , ) ( , )H HT f g f T g f g f g− = Γ Γ − Γ Γ ,
2. for any 1 2,g g H∈ , there is a vector *Dom( )f T∈ such that 1 1f gΓ = and 2 2f gΓ = .
The definition of the boundary triplet implies that Dom( )f T∈ , iff 1 2 0f fΓ = Γ = . A boun-
dary triplet 1 2( , , )H Γ Γ with dim H n= exists for any symmetric operator T with equal non-zero
deficiency indices ( , )n n ( )n ∞� , but it is not unique.
13ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 7
Multi-interval dissipative Sturm–Liouville boundary-value problems with distributional coefficients
For the minimal quasi-differential operators , 0kL , the boundary triplet is explicitly given
by the following theorem which follows from the results of [2].
Theorem 1. For every 1, ,k m= … , the triplet 2
1, 2,( , , )k kΓ Γ , where 1, 2,,k kΓ Γ are linear maps
[1] [1]
1, 1 2, 1: ( ( ), ( )), : ( ( ), ( ))k k k k k kk ky D y a D y a y y a y a− −Γ = + − − Γ = + − ,
from , 1Dom( )kL onto 2 is a boundary triplet for the operator , 0kL .
For the minimal operator minL in the space 2([ , ], )L a b , the boundary triplet is explicitly
given by the following theorem.
Theorem 2. The triplet 2
1 2( , , )m Γ Γ , where 1 2,Γ Γ are linear maps
1 1, 1 1, 2 1, 2 2, 1 2, 2 2,: ( , , , ), : ( , , , )m my y y y y y y yΓ = Γ Γ … Γ Γ = Γ Γ … Γ , (4)
from maxDom( )L onto 2m is a boundary triplet for the operator minL .
Denote, by KL , the restriction of operator maxL onto the set of functions maxy Dom( )L∈
satisfying the homogeneous boundary condition
1 2( ) ( ) 0K I y i K I y− Γ + + Γ = , (5)
where K is an arbitrary bounded operator on the space 2m .
Similarly, denote by KL , the restriction of maxL onto the set of functions maxy Dom( )L∈
satisfying the homogeneous boundary condition
1 2( ) ( ) 0K I y i K I y− Γ − + Γ = , (6)
where K is an arbitrary bounded operator on the space 2m .
Theorem 1 and [13, Th. 1.6] lead to the following description of all self-adjoint, maximal
dissipative and maximal accumulative extensions of operator maxL .
Theorem 3. Every KL with K being a contracting operator in the space 2m , is a maximal
dissipative extension of the operator minL . Similarly, every KL with K being a contracting operator
in 2m is a maximal accumulative extension of the operator minL . Conversely, for any maximal
dissipative (respectively, maximal accumulative) extension L of the operator minL , there exists
the unique contracting operator K such that KL L= (respectively, KL L= ).
The extensions KL and KL are self-adjoint, iff K is a unitary operator on 2m .
The mappings KK L→ and KK L→ are injective.
All functions from maxDom( )L together with their first quasi-derivatives belong to
1 1([ , ], )m
k k kAC a a= −⊕ . This implies that the following definition is correct.
Denote, by −f(t ) , the left germ and, by +f(t ) , the right germ of the continuous function f at
a point t . Similarly to paper [2], we say that the boundary conditions which define the operator
maxL L⊂ are called local, if, for any functions y Dom( )L∈ and for any 1 max, , Dom( )my y L… ∈ ,
the equalities − = −j j jy (a ) y(a ) , + = +j j jy (a ) y(a ) and 0− = + =j k j ky (a ) y (a ) , k j≠ imply
that y Dom( )j L∈ . For 0j = and j m= , we consider only the right and left germs, respectively.
The following statement gives a description of the extensions KL and KL which are given by
local boundary conditions.
14 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2020. № 7
A.S. Goriunov
Theorem 4. The boundary conditions (5) and (6) defining the extensions KL and KL , res-
pectively, are local, iff the matrix K has the block form
0
1
0 0
0 0
0 0
n
a
a
a
K
K K
K
⎛ ⎞…
⎜ ⎟
⎜ ⎟= …
⎜ ⎟
⎜ ⎟…⎝ ⎠
, (7)
where
1aK and
naK ∈ and other 2 2
jaK ×∈ .
4. Generalized resolvents. Let us recall that a generalized resolvent of a closed symmet-
ric operator L in a Hilbert space is an operator-valued function Rλλ defined on \
which can be represented as
1( ) ,R f P L I f f+ + + −
λ = −λ ∈ ,
where L+ is a self-adjoint extension of the operator L which acts in a certain Hilbert space
+ ⊃ , I + is the identity operator on + , and P+ is the orthogonal projection operator from
+ onto . It is known that an operator-valued function Rλ is a generalized resolvent of a sym-
metric operator L , iff it can be represented as
( , )
( , ) , ,
d F f g
R f g f g
+∞ μ
λ −∞
= ∈
μ−λ∫ ,
where Fμ is a generalized spectral function of the operator L . This implies that the operator-
va lued function Fμ has the following properties:
1. For 2 1μ > μ , the difference
2 1
F Fμ μ− is a bounded non-negative operator.
2. F Fμ+ μ= for any real μ .
3. For any x∈ , the following equalities hold:
|| || 0, || || 0lim limF x F x xμ μ
μ→−∞ μ→+∞
= − = .
The following theorem provides a parametric description of all generalized resolvents of
the symmetric operator minL (see also [14]).
Theorem 5. 1) Every generalized resolvent Rλ of the operator minL in the half-plane Im 0λ <
acts by the rule R h yλ = , where y is a solution of the boundary-value problem
( )l y y h= λ + ,
1 2( ( ) ) ( ( ) ) 0K I f i K I fλ − Γ + λ + Γ = .
Here, 2( ) ([ , ], )h x L a b∈ and ( )K λ is a 2 2m m× matrix-valued function which is holo-
morphic in the lower half-plane and such that || ( ) || 1K λ � .
2) In the half-plane Im 0λ > , every generalized resolvent of operator minL acts by R h yλ = ,
where y is a solution of the boundary-value problem
( )l y y h= λ + ,
15ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2020. № 7
Multi-interval dissipative Sturm–Liouville boundary-value problems with distributional coefficients
1 2( ( ) ) ( ( ) ) 0K I f i K I fλ − Γ − λ + Γ = .
Here, 2( ) ([ , ], )h x L a b∈ , and ( )K λ is a 2 2m m× matrix-valued function which is holo mor-
phic in the upper half-plane and satisfies || ( ) || 1K λ � . This parametrization of the generalized resol-
vents by the matrix-valued functions ( )K λ is bijective.
5. Completeness of the system of root vectors. Results of paper [15] imply that, in the sin-
gle-interval case under the assumptions made and additionally for 0kr r= ≡ , the resolvents of
the operators KL and KL are Hilbert—Schmidt operators. This result is amplified and refined
by the following theorem.
Theorem 6. 1) The resolvents of the maximal dissipative (maximal accumulative) operators KL
and KL are Hilbert—Schmidt operators.
2) Let 0δ > exist such that, for any {1, 2, , }k m∈ … ,
2 1
1
, ([ , ], )k k
k k
k k
Q ir
W a a
p p
δ
−
⎧ ⎫+
⊂⎨ ⎬
⎩ ⎭
.
Then the resolvent of the maximal dissipative (maximal accumulative) operator KL ( KL ) is an
operator from the trace class, and its system of root functions is complete in the Hilbert space
2([ , ], )L a b .
REFERENCES
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and boundary-value problems. Electron. J. Diff. Equ., No. 101, pp. 1-16.
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cients. Math. Notes, 99, No. 5, pp. 779-784.
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operators with distributional coefficients. Opusc. Math., 33, No. 3, pp. 467-563.
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A.S. Goriunov
14. Bruk, V. M. (1976). A certain class of boundary value problems with a spectral parameter in the boundary
condition. Mat. Sb. (N.S.), 100, No. 2, pp. 210-216 (in Russian).
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distributions. Ukr. Math. J., 67, No. 5, pp. 680-689.
Received 30.03.2020
А.С. Горюнов
Інститут математики НАН України, Київ
E-mail: goriunov@imath.kiev.ua
БАГАТОІНТЕРВАЛЬНІ ДИСИПАТИВНІ
КРАЙОВІ ЗАДАЧІ ШТУРМА—ЛІУВІЛЛЯ
З КОЕФІЦІЄНТАМИ-РОЗПОДІЛАМИ
Досліджено спектральні властивості багатоінтервальних операторів Штурма—Ліувілля з узагальненими
функціями в коефіцієнтах. Дано конструктивний опис усіх самоспряжених, максимальних дисипативних/
акумулятивних розширень мінімального оператора в термінах крайових умов. Знайдено достатні умови
ядерності резольвент цих операторів та повноти систем їх кореневих функцій. Результати роботи є нови-
ми і для одноінтервальних крайових задач.
Ключові слова: оператор Штурма—Ліувілля, багатоінтервальна крайова задача, сингулярні коефіцієнти,
максимальне дисипативне розширення, повнота крайових функцій.
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