Multiinterval Sturm–Liouville boundary-value problems with distributional potentials
We study the multi-interval boundary-value Sturm–Liouville problems with distributional potentials. For the corresponding symmetric operators boundary triplets are found and the constructive descriptions of all self-adjoint, maximal dissipative and maximal accumulative extensions and generalized re...
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| Cite this: | Multiinterval Sturm–Liouville boundary-value problems with distributional potentials / A.S. Goriunov // Доповiдi Нацiональної академiї наук України. — 2014. — № 7. — С. 43-47. — Бібліогр.: 11 назв. — англ. |
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| citation_txt | Multiinterval Sturm–Liouville boundary-value problems with distributional potentials / A.S. Goriunov // Доповiдi Нацiональної академiї наук України. — 2014. — № 7. — С. 43-47. — Бібліогр.: 11 назв. — англ. |
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| description | We study the multi-interval boundary-value Sturm–Liouville problems with distributional potentials. For the corresponding symmetric operators boundary triplets are found and the constructive descriptions of all self-adjoint, maximal dissipative and maximal accumulative extensions
and generalized resolvents in terms of homogeneous boundary conditions are given. It is shown
that all real maximal dissipative and maximal accumulative extensions are self-adjoint and all
such extensions are described.
Вивчаються багатоiнтервальнi крайовi задачi Штурма–Лiувiлля з потенцiалами-розподiлами. Для вiдповiдних симетричних операторiв побудовано простори граничних значень
i дано конструктивнi описи всiх самоспряжених, максимальних дисипативних i максимальних акумулятивних розширень, а також узагальнених резольвент в термiнах однорiдних крайових умов. Показано, що всi дiйснi максимальнi дисипативнi i максимальнi акумулятивнi розширення самоспряженi, i описано всi такi розширення.
Изучены многоинтервальные краевые задачи Штурма–Лиувилля с потенциалами-распределениями. Для соответствующих симметрических операторов построены пространства
граничных значений и даны конструктивные описания всех самосопряженных, максимальных диссипативных и максимальных аккумулятивных расширений, а также обобщенных
резольвент в терминах однородных краевых условий. Показано, что все вещественные максимальные диссипативные и максимальные аккумулятивные расширения самосопряжены, и описаны все такие расширения.
|
| first_indexed | 2025-11-28T03:21:10Z |
| format | Article |
| fulltext |
UDC 517.926,517.927.2
A.S. Goriunov
Multi-interval Sturm–Liouville boundary-value problems
with distributional potentials
(Presented by the Corresponding Member of the NAS of Ukraine M. L. Gorbachuk)
We study the multi-interval boundary-value Sturm–Liouville problems with distributional poten-
tials. For the corresponding symmetric operators boundary triplets are found and the constructi-
ve descriptions of all self-adjoint, maximal dissipative and maximal accumulative extensions
and generalized resolvents in terms of homogeneous boundary conditions are given. It is shown
that all real maximal dissipative and maximal accumulative extensions are self-adjoint and all
such extensions are described.
In recent years, the interest in multi-interval differential and quasi-differential operators has
increased (see [1–4]). The main attention is paid to the case where a (quasi-)differential expres-
sion is formally self-adjoint. From the operator-theoretic point of view this corresponds to the
situation where we investigate extensions of a symmetric (quasi-)differential operator with equal
deficiency indices in the direct sum of Hilbert spaces on the basis of Glazman–Krein–Naimark
theory [5–8]. In the present paper, we develop another approach to such problems based on the
concept of boundary triplets [9, 10].
Let m ∈ N, a = a0 < a1 < · · · < am = b be a partition of a finite interval [a, b] into m parts
and on every interval (ai−1, ai), i ∈ {1, . . . ,m}, let the formal Sturm–Liouville expression
li(y) = −(pi(t)y
′)′ + qi(t)y (1)
be given. Here the measurable finite functions pi and Qi are such that
1
pi
,
Qi
pi
,
Q2
i
pi
∈ L1([ai−1, ai],R), (2)
the potentials qi = Q′
i, and the derivative is understood in the sense of distributions.
For m = 1 the boundary-value problems for the formal differential expression (1) under
assumptions (2) were investigated in [11] on the basis of its regularization by Shin–Zettl quasi-
derivatives. In this paper the most of the results of [11] is extended onto the case of an arbitrary
m ∈ N.
We introduce the quasi-derivatives
D
[0]
i y = y,
D
[1]
i y = piy
′ −Qiy,
D
[2]
i y = (D
[1]
i y)′ +
Qi
pi
D
[1]
i y +
Q2
i
pi
y
on every interval (ai−1, ai), as in [11].
© A. S. Goriunov, 2014
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №7 43
Then the maximal and minimal operators
Li,1 : y → li[y], Dom(Li,1) : = {y ∈ L2 | y,D
[1]
i y ∈ AC([ai−1, ai],C),D
[2]
i y ∈ L2},
Li,0 : y → li[y], Dom(Li,0) := {y ∈ Dom(Li,1) | D
[k]
i y(ai−1) = D
[k]
i y(ai) = 0, k = 0, 1}
are defined in the spaces L2((ai−1, ai),C). According to [11] the operators Li,1, Li,0 are closed and
densely defined in L2([ai−1, ai],C). The operator Li,0 is symmetric with the deficiency indices
(2, 2) and
L∗
i,0 = Li,1, L∗
i,1 = Li,0.
Recall that a boundary triplet of a closed densely defined symmetric operator T with equal
(finite or infinite) deficiency indices is called a triplet (H,Γ1,Γ2) where H is an auxiliary Hilbert
space and Γ1, Γ2 are the linear maps from Dom(T ∗) to H such that
1) for any f , g ∈ Dom(T ∗) there holds
(T ∗f, g)H − (f, T ∗g)H = (Γ1f,Γ2g)H − (Γ2f,Γ1g)H ;
2) for any g1, g2 ∈ H there is a vector f ∈ Dom(T ∗) such that Γ1f = g1 and Γ2f = g2.
It is proved in [11] that for every i = 1, . . . ,m the triplet (C2,Γ1,i,Γ2,i), where Γ1,i, Γ2,i
are linear maps
Γ1,iy := (D
[1]
i y(ai−1+),−D
[1]
i y(ai−)), Γ2,iy := (y(ai−1+), y(ai−)),
from Dom(Li,1) to C
2 is a boundary triplet for the operator Li,0.
We consider the space L2([a, b],C) as a direct sum ⊕m
i=1L2([ai−1, ai],C) which consists of
vector functions f = ⊕m
i=1fi such that fi ∈ L2([ai−1, ai],C). In this space we consider operators
Lmax = ⊕m
i=1Li,1 and Lmin = ⊕m
i=1Li,0.
Then the operators Lmax, Lmin are closed and densely defined in L2([a, b],C). The operator
Lmin is symmetric with the deficiency indices (2m, 2m) and
L∗
min = Lmax, L∗
max = Lmin.
Note that the minimal operator Lmin may be not semi-bounded even in the case of a single-
interval boundary-value problem since the function p may reverse sign.
Theorem 1. The triplet (C2m,Γ1,Γ2) where Γ1, Γ2 are linear maps
Γ1y := (Γ1,1y,Γ1,2y, . . . ,Γ1,my), Γ2y := (Γ2,1y,Γ2,2y, . . . ,Γ2,my)
from Dom(Lmax) onto C
2m is a boundary triplet for Lmin.
Denote by LK the restriction of Lmax onto set of functions y(t) ∈ Dom(Lmax) satisfying the
homogeneous boundary condition
(K − I)Γ1y + i(K + I)Γ2y = 0.
Similarly, denote by LK the restriction of Lmax onto the set of functions y(t) ∈ Dom(Lmax)
satisfying the homogeneous boundary condition
(K − I)Γ1y − i(K + I)Γ2y = 0.
Here K is a bounded operator in C
2m.
44 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2014, №7
The constructive description of the various classes of extensions of the operator Lmin is given
by the following theorem.
Theorem 2. Every LK with K being a contracting operator in C
2m is a maximal dissipative
extension of Lmin. Similarly, every LK with K being a contracting operator in C
2m is a maximal
accumulative extension of the operator Lmin.
Conversely, for any maximal dissipative (respectively, maximal accumulative) extension L̃ of
the operator Lmin there exists the unique contracting operator K such that L̃ = LK (respectively,
L̃ = LK).
The extensions LK and LK are self-adjoint if and only if K is a unitary operator on C
2m.
Recall that a linear operator T acting in L2([a, b],C) is called real if:
1) for every function f from Dom(T ) the complex conjugate function f also lies in Dom(T );
2) the operator T maps complex conjugate functions into complex conjugate functions, that
is T (f) = T (f).
One can see that the maximal and minimal operators are real.
Theorem 3. All real maximal dissipative and maximal accumulative extensions of the mi-
nimal operator Lmin are self-adjoint. The self-adjoint extension LK or LK is real if and only if
the unitary matrix K is symmetric.
Let us recall that a generalized resolvent of a closed symmetric operator T in a Hilbert space
H is an operator-valued function λ 7→ Rλ defined on C \ R, which can be represented as
Rλf = P+(T+ − λI+)−1f, f ∈ H,
where T+ is a self-adjoint extension of T which acts in a certain Hilbert space H+ ⊃ H, I+ is the
identity operator on H+, and P+ is the orthogonal projection operator from H+ onto H. It is
known that an operator-valued function Rλ (Im λ 6= 0) is a generalized resolvent of a symmetric
operator T if and only if it can be represented as
(Rλf, g)H =
+∞∫
−∞
d(Fµf, g)
µ− λ
, f, g ∈ H,
where Fµ is a generalized spectral function of the operator T , i. e. µ 7→ Fµ is an operator-valued
function Fµ defined on R and taking values in the space of continuous linear operators in H
with the following properties:
1) for µ2 > µ1, the difference Fµ2
− Fµ1
is a bounded non-negative operator;
2) Fµ+ = Fµ for any real µ;
3) for any x ∈ H,
lim
µ→−∞
‖Fµx‖H = 0, lim
µ→+∞
‖Fµx− x‖H = 0.
The following theorem provides a description of all generalized resolvents of the operator Lmin.
Theorem 4. 1. Every generalized resolvent Rλ of the operator Lmin in the half-plane Imλ < 0
acts by the rule Rλh = y, where y is a solution of the boundary-value problem
l(y) = λy + h,
(K(λ)− I)Γ1f + i(K(λ) + I)Γ2f = 0.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №7 45
Here h(x) ∈ L2([a, b],C) and K(λ) is a 2m × 2m matrix-valued function which is holomorphic
in the lower half-plane and satisfies ‖K(λ)‖ 6 1.
2. In the half-plane Imλ > 0, every generalized resolvent of Lmin acts by the rule Rλh = y,
where y is a solution of the boundary-value problem
l(y) = λy + h,
(K(λ)− I)Γ1f − i(K(λ) + I)Γ2f = 0.
Here h(x) ∈ L2([a, b],C) and K(λ) is a 2m × 2m matrix-valued function which is holomorphic
in the lower half-plane and satisfies ‖K(λ)‖ 6 1.
The parametrization of the generalized resolvents by the matrix-valued functions K is bijective.
This research is supported by the grant no. 03–01–12 of the National Academy of Sciences of Ukraine
(under the joint Ukrainian-Russian project of the NAS of Ukraine and the Siberian Branch of RAS).
1. Everitt W.N., Zettl A. Sturm–Liouville differential operators in direct sum spaces // Rocky Mountain
J. Math. – 1986. – 16, No 3. – P. 497–516.
2. Everitt W.N., Zettl A. Quasi-differential operators generated by a countable number of expressions on the
real line // Proc. London Math. Soc. – 1992. – 64, No 3. – P. 524–544.
3. Sokolov M. S. An abstract approach to some spectral problems of direct sum differential operators //
Electron. J. Differential Equations. – 2003. – 2003, No 75. – P. 1–6.
4. Sokolov M. S. Representation results for operators generated by a quasi-differential multi-interval system
in a Hilbert direct sum space // Rocky Mountain J. Math. – 2006. – 36, No 2. – P. 721–739.
5. Zettl A. Formally self-adjoint quasi-differential operators // Ibid. – 1975. – 5, No 3. – P. 453–474.
6. Everitt W.N., Markus L. Boundary value problems and symplectic algebra for ordinary differential and
quasi-differential operators. – Providence, RI: Amer. Math. Soc., 1999. – 187 p.
7. Zettl A. Sturm–Liouville theory. – Providence, RI: Amer. Math. Soc., 2005. – 328 p.
8. Naimark M.A. Linear differential operators. Part 2. – New York: F. Ungar, 1968. – 352 p. (Rus. ed.: Nauka,
Moscow, 1969).
9. Gorbachuk V. I., Gorbachuk M. L. Boundary value problems for operator differential equations. – Dordrecht:
Kluwer, 1991. – 347 p. (Rus. ed.: Naukova Dumka, Kiev, 1984).
10. Kochubei A.N. Symmetric operators and nonclassical spectral problems // Mat. Zametki. – 1979. – 25,
No 3. – P. 425–434.
11. Goriunov A. S., Mikhailets V.A. Regularization of singular Sturm–Liouville equations // Meth. Funct.
Anal. Topol. – 2010. – 16, No 2. – P. 120–130.
Received 31.03.2014Institute of Mathematics of the NAS of Ukraine, Kiev
А.С. Горюнов
Багатоiнтервальнi крайовi задачi Штурма–Лiувiлля
з потенцiалами-розподiлами
Вивчаються багатоiнтервальнi крайовi задачi Штурма–Лiувiлля з потенцiалами-розподi-
лами. Для вiдповiдних симетричних операторiв побудовано простори граничних значень
i дано конструктивнi описи всiх самоспряжених, максимальних дисипативних i макси-
мальних акумулятивних розширень, а також узагальнених резольвент в термiнах одно-
рiдних крайових умов. Показано, що всi дiйснi максимальнi дисипативнi i максимальнi аку-
мулятивнi розширення самоспряженi, i описано всi такi розширення.
46 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2014, №7
А.С. Горюнов
Многоинтервальные краевые задачи Штурма–Лиувилля
с потенциалами-распределениями
Изучены многоинтервальные краевые задачи Штурма–Лиувилля с потенциалами-распре-
делениями. Для соответствующих симметрических операторов построены пространства
граничных значений и даны конструктивные описания всех самосопряженных, максималь-
ных диссипативных и максимальных аккумулятивных расширений, а также обобщенных
резольвент в терминах однородных краевых условий. Показано, что все вещественные мак-
симальные диссипативные и максимальные аккумулятивные расширения самосопряжены,
и описаны все такие расширения.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2014, №7 47
|
| id | nasplib_isofts_kiev_ua-123456789-87950 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-11-28T03:21:10Z |
| publishDate | 2014 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Goriunov, A.S. 2015-11-01T18:44:58Z 2015-11-01T18:44:58Z 2014 Multiinterval Sturm–Liouville boundary-value problems with distributional potentials / A.S. Goriunov // Доповiдi Нацiональної академiї наук України. — 2014. — № 7. — С. 43-47. — Бібліогр.: 11 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/87950 517.926,517.927.2 We study the multi-interval boundary-value Sturm–Liouville problems with distributional potentials. For the corresponding symmetric operators boundary triplets are found and the constructive descriptions of all self-adjoint, maximal dissipative and maximal accumulative extensions and generalized resolvents in terms of homogeneous boundary conditions are given. It is shown that all real maximal dissipative and maximal accumulative extensions are self-adjoint and all such extensions are described. Вивчаються багатоiнтервальнi крайовi задачi Штурма–Лiувiлля з потенцiалами-розподiлами. Для вiдповiдних симетричних операторiв побудовано простори граничних значень i дано конструктивнi описи всiх самоспряжених, максимальних дисипативних i максимальних акумулятивних розширень, а також узагальнених резольвент в термiнах однорiдних крайових умов. Показано, що всi дiйснi максимальнi дисипативнi i максимальнi акумулятивнi розширення самоспряженi, i описано всi такi розширення. Изучены многоинтервальные краевые задачи Штурма–Лиувилля с потенциалами-распределениями. Для соответствующих симметрических операторов построены пространства граничных значений и даны конструктивные описания всех самосопряженных, максимальных диссипативных и максимальных аккумулятивных расширений, а также обобщенных резольвент в терминах однородных краевых условий. Показано, что все вещественные максимальные диссипативные и максимальные аккумулятивные расширения самосопряжены, и описаны все такие расширения. This research is supported by the grant no. 03–01–12 of the National Academy of Sciences of Ukraine (under the joint Ukrainian-Russian project of the NAS of Ukraine and the Siberian Branch of RAS). en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Математика Multiinterval Sturm–Liouville boundary-value problems with distributional potentials Багатоiнтервальнi крайовi задачi Штурма–Лiувiлля з потенцiалами-розподiлами Многоинтервальные краевые задачи Штурма–Лиувилля с потенциалами-распределениями Article published earlier |
| spellingShingle | Multiinterval Sturm–Liouville boundary-value problems with distributional potentials Goriunov, A.S. Математика |
| title | Multiinterval Sturm–Liouville boundary-value problems with distributional potentials |
| title_alt | Багатоiнтервальнi крайовi задачi Штурма–Лiувiлля з потенцiалами-розподiлами Многоинтервальные краевые задачи Штурма–Лиувилля с потенциалами-распределениями |
| title_full | Multiinterval Sturm–Liouville boundary-value problems with distributional potentials |
| title_fullStr | Multiinterval Sturm–Liouville boundary-value problems with distributional potentials |
| title_full_unstemmed | Multiinterval Sturm–Liouville boundary-value problems with distributional potentials |
| title_short | Multiinterval Sturm–Liouville boundary-value problems with distributional potentials |
| title_sort | multiinterval sturm–liouville boundary-value problems with distributional potentials |
| topic | Математика |
| topic_facet | Математика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/87950 |
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