Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces
Let A be an unbounded self-adjoint operator in a Hilbert separable space \(H_0\) with rigging \(H_ - \sqsupset H_0 \sqsupset H_ +\) such that \(D(A) = H_ +\) in the graph norm (here, \(D(A)\) is the domain of definition of A). Assume that \(H_ +\) is decomposed into the orthogonal sum \(H_ + =...
Gespeichert in:
| Datum: | 2005 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2005
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3628 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509749304360960 |
|---|---|
| author | Bozhok, R. V. Koshmanenko, V. D. Божок, Р. В. Кошманенко, В. Д. |
| author_facet | Bozhok, R. V. Koshmanenko, V. D. Божок, Р. В. Кошманенко, В. Д. |
| author_sort | Bozhok, R. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:00:32Z |
| description | Let A be an unbounded self-adjoint operator in a Hilbert separable space \(H_0\) with rigging \(H_ - \sqsupset H_0 \sqsupset H_ +\) such that \(D(A) = H_ +\) in the graph norm (here, \(D(A)\) is the domain of definition of A). Assume that \(H_ +\) is decomposed into the orthogonal sum \(H_ + = M \oplus N_ +\) so that the subspace \(M_ +\) is dense in \(H_0\). We construct and study a singularly perturbed operator A associated with a new rigging \(H_ - \sqsupset H_0 \sqsupset \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ +\), where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ + = M_ + = D(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} )\), and establish the relationship between the operators A and A. |
| first_indexed | 2026-03-24T02:46:03Z |
| format | Article |
| fulltext |
UDK 517.9
R. V. BoΩok, V. D. Koßmanenko (In-t matematyky NAN Ukra]ny, Ky]v)
SYNHULQRNI ZBURENNQ
SAMOSPRQÛENYX OPERATORIV, ASOCIJOVANI
Z OSNAWENYMY HIL|BERTOVYMY PROSTORAMY
*
Let A be an unbounded self-adjoint operator in a separable Hilbert space H0 which is equipped
H H H− +� �0 in such a way that the domain of definition D H( )A = + in the norm of a graph.
Assume that H + is decomposed into the orthogonal sum H M N+ + += � so that the subspace M +
in dense in H 0 . In the paper, we construct and investigate the singularly perturbed operator
�
A
associated with a new rigging
� �
H H H− +� �0 , where
� �
H M D+ += = ( )A . We establish the relation
between the operators A and
�
A .
Nexaj A [ neobmeΩenym samosprqΩenym operatorom v separabel\nomu hil\bertovomu prostori
H0 , qkyj osnaweno H H H− +� �0 takym çynom, wo oblast\ vyznaçennq D H( )A = + v nor-
mi hrafika. Prypustymo, wo H + rozkladeno v ortohonal\nu sumu H M N+ + += � tak, wo
pidprostir M + [ wil\nym v H0 . U roboti budu[t\sq i vyvça[t\sq synhulqrno zburenyj opera-
tor
�
A , asocijovanyj z novym osnawennqm
� �
H H H− +� �0 , de
� �
H M D+ += = ( )A . Vstanov-
leno zv’qzok miΩ operatoramy A ta
�
A .
1. Vstup. Rozhlqnemo v separabel\nomu prostori Hil\berta H 0 neobmeΩenyj
samosprqΩenyj operator A = A
* ≥ 1 z oblastg vyznaçennq D ( A ) . Z koΩnym
takym operatorom asocig[t\sq osnawenyj prostir Hil\berta [1, 2]
H – � H 0 � H + ,
de � oznaça[ wil\ne neperervne vkladennq, H + = D ( A ) za normog hrafika, a
H – — sprqΩenyj prostir ( cej prostir [ popovnennqm H 0 za normog f − : =
: = A f−1 , f ∈ H 0 ) . Prypustymo, wo synhulqrne zburennq zadano operatorom
T : H + → H – tak, wo mnoΩyna M + : = Ker T [ wil\nog v H 0 . Zhidno z za-
hal\novyznanog v teori] synhulqrnyx zburen\ procedurog (dyv., napryklad, [3 –
20]) synhulqrno zburenyj operator à , qkyj vidpovida[ formal\nij sumi
A T+̃ , vyznaça[t\sq qk odne iz samosprqΩenyx rozßyren\ symetryçnoho opera-
tora Ȧ : = A M+ .
U cij roboti proponu[t\sq novyj metod pobudovy synhulqrno zburenoho ope-
ratora. Sut\ c\oho metodu polqha[ v nastupnomu. Vyxodqçy z ortohonal\noho
rozkladu H + =
M N+ +� , de, nahada[mo, M + = Ker T [ pidprostorom, wil\-
nym u H 0 , vvodymo novyj osnawenyj prostir:
�
H − � H 0 �
�
H + , pokladagçy
�
H + ≡ M + . Pislq c\oho vyznaça[mo synhulqrno zburenyj operator (poznaça[-
mo joho
�
A ) qk [dyno vyznaçenyj operator, asocijovanyj iz novym osnawennqm
prostoru H 0 . Takyj operator
�
A fiksu[t\sq umovog D( )
�
A = M + .
Takym çynom, my rozßyrg[mo zvyçajnyj klas synhulqrno zburenyx opera-
toriv. Okrim usi[] sim’] samosprqΩenyx rozßyren\ symetryçnoho operatora Ȧ
vklgça[mo v klas synhulqrno zburenyx operatoriv we j operator
�
A . Vyqvlq-
[t\sq, wo spektral\ni vlastyvosti operatoriv à ta
�
A [ istotno riznymy. Na
dumku avtoriv, vybir operatora
�
A v qkosti synhulqrno zburenoho operatora
* Çastkovo pidtrymano proektamy DFG 436 UKR 113/67, 113/78 ta INTAS 00-257.
© R. V. BOÛOK, V. D. KOÍMANENKO, 2005
622 ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
SYNHULQRNI ZBURENNQ SAMOSPRQÛENYX OPERATORIV … 623
bil\ß adekvatno vraxovu[ fizyçnu ideg ideal\no tverdoho qdra (abo absolgtno
neprozoroho ekrana) v teori] synhulqrnyx zburen\.
Nastupni dva rozdily statti [ dopomiΩnymy. Osnovnym rezul\tatom roboty [
teoremaH4. Zokrema, v cij teoremi opysano konstrukcig operatora
�
A , a takoΩ
vstanovleno zv’qzok miΩ operatoramy A ta
�
A .
2. Osnaweni prostory Hil\berta. Nahada[mo deqki zahal\ni fakty z teori]
osnawenyx hil\bertovyx prostoriv (dokladniße dyv. [1, 2]).
Za oznaçennqm trijka hil\bertovyx prostoriv
H – � H 0 � H + (2.1)
utvorg[ osnawenyj prostir Hil\berta, qkwo vykonugt\sq taky umovy:
a) obydva vkladennq [ neperervnymy i wil\nymy, wo poznaça[t\sq symvo-
lomHH� ;
b) normy u prostorax H – , H 0 ta H + zadovol\nqgt\ nerivnosti
⋅ − ≤ ⋅ 0 ≤ ⋅ + ; (2.2)
v) prostory H – ta H + [ vza[mno sprqΩenymy vidnosno H 0 .
Ostannq umova oznaça[, wo dlq koΩnoho vektora ϕ ∈ H + linijnyj funkcio-
nal l f fϕ ϕ( ) : ( , )= 0 , f ∈ H 0 , ma[ prodovΩennq za neperervnistg na uves\ pros-
tir H – . I tomu tak zvanu pozytyvnu normu ϕ + moΩna obçyslyty za formulog
ϕ + = sup ( , )
f
f
− =1
0ϕ , f ∈ H 0 .
Zhidno z teoremog Risa l fϕ( ) = ( ),f ϕ∗
− z deqkym ϕ* ∈ H – . OtΩe, ϕ + =
= ϕ∗
−
i tomu vidobraΩennq
D− +, : H + � ϕ → ϕ* ∈ H –
[ unitarnym. Z inßoho boku, prostir H – zbiha[t\sq z popovnennqm H 0 vidnos-
no tak zvano] nehatyvno] normy
f − : = sup ( , )
ϕ
ϕ
+ =1
0f , ϕ ∈ H + .
Na pidstavi (2.2) skalqrnyj dobutok ( , )⋅ ⋅ 0 v H 0 moΩna prodovΩyty do dual\-
noho dobutku miΩ H + ta H – , qkyj my poznaça[mo qk 〈 〉− +ω ϕ, , = 〈 〉+ −ϕ ω, , ,
ω ∈ H – , ϕ ∈ H + . Operatory
D− +, : H + → H – , I+ −, = D− +
−
,
1 : H – → H +
nazyvagt\sq kanoniçnymy unitarnymy izomorfizmamy miΩ H – ta H + . Vony za-
dovol\nqgt\ spivvidnoßennq
( , )f ϕ 0 = 〈 〉− +f , ,ϕ = ( , ),f D− + −ϕ = ( , ),I f+ − +ϕ , f ∈ H 0 , ϕ ∈ H + .
Isnu[ zv’qzok miΩ trijkamy prostoriv vyhlqdu (2.1) ta samosprqΩenymy ope-
ratoramy A v H 0 . Cej zv’qzok fiksu[t\sq vidobraΩennqm D− +, ta umovog
D ( A ) = H + . Spravdi, rozhlqnemo operator
LA : = D− + ++, H , H++ : = D ( LA ) = { },ϕ ϕ∈ ∈+ − +H HD 0 .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
624 R. V. BOÛOK, V. D. KOÍMANENKO
Oçevydno, LA [ symetryçnym u H 0 , oskil\ky dlq usix ϕ, ψ ∈ D ( LA )
( , )LAϕ ψ 0 = ( , ),D− +ϕ ψ 0 =
= 〈 〉∗
− +ϕ ψ, , = ( , )ϕ ψ + = 〈 〉∗
+ −ϕ ψ, , = ( , ),ϕ ψD− + 0 = ( , )ϕ ψLA 0 ,
de element ϕ*
buv oznaçenyj vywe. Naspravdi LA samosprqΩenyj v H 0 ,
oskil\ky zhidno z pobudovog joho oblast\ znaçen\ zbiha[t\sq z usim prostorom
H 0 . Vyznaçymo A LA: /= 1 2 . Zrozumilo, wo D ( A ) = H + zavdqky tomu, wo
( , )LAϕ ψ 0 = ( , )/ /L LA A
1 2 1 2
0ϕ ψ = ( , )ϕ ψ + . Oçevydno takoΩ, wo A ≥ 1, oskil\ky
⋅ + ≥ ⋅ 0.
Navpaky, nexaj A = A
* ≥ 1 [ samosprqΩenym neobmeΩenym operatorom z
oblastg vyznaçennq D ( A ) u prostori H 0 . Vyxodqçy z A , moΩna lehko
pobuduvaty osnawenyj prostir Hil\berta H – � H 0 � H + . Nahada[mo cg po-
budovu. Prostir H + ototoΩng[mo z D ( A ) v skalqrnomu dobutku ( , )ϕ ψ + : =
: = ( , )A Aϕ ψ 0 , ϕ, ψ ∈ D ( A ) . Dali, vyxodqçy z nepovnoho lancgΩka H 0 � H + ,
prodovΩu[mo joho do osnawenoho prostoru (2.1) zvyçajnym çynom (qk bulo opy-
sano pry analizi umovy v) ). OtΩe, spravedlyvog [ nastupna teorema.
Teorema31. KoΩen osnawenyj prostir Hil\berta vydu (2.1) vza[mno odno-
znaçno pov’qzanyj (asocijovanyj) z samosprqΩenym operatorom A = A
* ≥ 1 v
H 0 . Pry c\omu D ( A ) = H + i pozytyvnyj skalqrnyj dobutok ( , )ϕ ψ + =
= ( , )A Aϕ ψ 0 , ϕ, ψ ∈ D ( A ) .
U podal\ßomu nam znadobyt\sq takoΩ konstrukciq neskinçennoho lancgΩ-
ka hil\bertovyx prostoriv { }( )H Hk k kA≡ ∈R , qkyj nazyva[t\sq A-ßkalog
hil\bertovyx prostoriv.
Za oznaçennqm H Dk
kA: ( )/= 2 , k > 0, v pozytyvnij normi ⋅ k , qka vidpovi-
da[ skalqrnomu dobutku
( , )ϕ ψ k : = ( )/ /,A Ak k2 2
0ϕ ψ , ϕ, ψ ∈ D( )/A k 2 .
Prostir H−k z’qvlq[t\sq qk popovnennq H 0 vidnosno nehatyvno] normy
f k− : = A fk− /2
0
, f ∈ H 0 .
NevaΩko baçyty, wo koΩna trijka
H – k � H 0 � H k , k > 0, (2.3)
utvorg[ osnawenyj prostir, asocijovanyj z operatorom Ak /2
. Nexaj D k k− , :
H Hk k→ − poznaça[ operator kanoniçnoho unitarnoho izomorfizmu dlq trijky
(2.3). Oçevydno, wo D k k− , = ( ) ( )/ /A Ak k2 2cl ≡ D Dk k− , ,0 0 , de cl poznaça[ opera-
cig zamykannq. Zokrema, dlq k = 2 ma[mo D0 2, ≡ A : H H2 0→ ta D−2 0, ≡
≡ Acl :
H H0 2→ − .
3. Wil\nist\ vkladennq. Nexaj zadano osnawenyj prostir Hil\berta
H – � H 0 � H + . Prypustymo, wo pozytyvnyj prostir H + rozkladeno v or-
tohonal\nu sumu pidprostoriv: H + = M + � N + . Nastupna teorema da[ prostyj
kryterij wil\nosti vkladennq H 0 � M + .
Teorema32 [4]. Nexaj H + = M + � N + . Pidprostir M + [ wil\nym v H 0
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
SYNHULQRNI ZBURENNQ SAMOSPRQÛENYX OPERATORIV … 625
todi i til\ky todi, koly pidprostir N N− − + +=: ,D ma[ nul\ovyj pereriz z
H 0 :
H 0 � M + ⇔ N – ∩ H 0 = { 0 } . (3.1)
Dovedennq. Nexaj N – ∩ H 0 = { 0 } . Prypustymo, wo isnu[ vektor 0 ≠
≠ ψ ∈ H 0 takyj, wo ψ ⊥ M + . Oskil\ky M + [ pidprostorom v H + , to z ohlq-
du na te, wo ψ ∈ H – , ma[mo
0 = ( ),ψ M+ 0 =
〈 〉+ − +ψ, ,M =
( ), ,I+ − + +ψ M .
Tomu I+ −, ψ ∈ N + . Ce oznaça[, wo ψ ∈ N – , a ce supereçyt\ poçatkovomu pry-
puwenng. Navpaky, qkwo pidprostir M + [ wil\nym v H 0 , to prypuwennq pro
isnuvannq vektora 0 ≠ ω ∈ N – ∩ H 0 takoΩ pryvodyt\ do supereçnosti. Sprav-
di, oskil\ky N – =
D–,+ +N , ma[mo
〈 〉+ − +ω, ,M = ( ),ω M+ 0 =
( ), ,I+ − + +ω M = 0,
wo supereçyt\ spivvidnoßenng M + � H 0 , bo 0 ≠ ω ∈ H 0 .
Teoremu dovedeno.
Lehko zrozumity, wo spivvidnoßennq (3.1) moΩna zapysaty v ekvivalentnij
formi
H 0 � M + ⇔ N 0 ∩ H + = { 0 } , N 0 : =
D0,+ +N . (3.2)
Vvedemo do rozhlqdu rozßyrenyj osnawenyj prostir
H – – � H – � H 0 � H + � H + + ,
de H – – = H – 4 ( A ) , H + + = H 4 ( A ) = D ( A
2
) . Nexaj H + = N + � M + . Prypus-
tymo, wo H 0 � M + . Rozhlqnemo pidprostir M̃+ : = M + ∩ H + + . Vin [ zamkne-
nym v H + + . Spravdi, qkwo poslidovnist\ ϕn ∈ M̃+ [ zbiΩnog v H + + : ϕn →
→ ϕ ∈ H + + , to vona [ zbiΩnog i v H + zavdqky ⋅ + ≤ ⋅ ++ . OtΩe, ϕ ∈ M + ,
oskil\ky M + [ zamknenym pidprostorom v H + . Ce dovodyt\ zamknenist\ M̃+
v H + + .
Prypustymo, wo vykonu[t\sq umova
( ) ,N H−
−−cl ∩ 0 = { 0 } , (3.3)
de N – : = D–,+ +N , a cl, – – poznaça[ zamykannq v H – – .
Teorema33. Qkwo pidprostir M + [ wil\nym v H 0 , H 0 � M + , i dodat-
kovo vykonu[t\sq umova (3.3), to pereriz M̃+ : = M + ∩ H + + takoΩ [ wil\-
nym v H 0 :
H 0 � M̃+ . (3.4)
Zokrema, pidprostir M̃+ [ wil\nym v H 0 , qkwo rozmirnist\ N + skinçenna:
dim N + < ∞ .
Dovedennq. Vykorystovugçy oznaçennq pidprostoru M̃+ u vyhlqdi
M̃+ =
{ }( , ) ,ϕ ϕ ψ ψ∈ = ∈++ + +H N0 ,
zhidno z vlastyvostqmy A-ßkaly ma[mo
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
626 R. V. BOÛOK, V. D. KOÍMANENKO
( , )ϕ ψ + = 〈 〉+ −ϕ ω, , = 〈 〉++ −−ϕ ω, , ,
de ω = D− +, ψ , ψ ∈ N + . Zvidsy vyplyva[
( ) ,N −
−−cl =
Ñ − : =
{ ˜ }, ,,ω ϕ ω ϕ∈ 〈 〉 = ∈−− ++ −− +H M0 . (3.5)
Dali, oskil\ky M + [ wil\nym v H 0 , to na pidstavi (3.1) i zavdqky umovi (3.3)
ma[mo Ñ H− ∩ 0 = { 0 } . Tomu H 0 � M̃+ zhidno z teoremogH2.
Zaverßugçy dovedennq, zauvaΩymo, wo iz spivvidnoßennq H 0 � M + umova
(3.3) vyplyva[ avtomatyçno, qkwo dim N 0 = dim N + < ∞ .
Zrozumilo, wo teoremaH3 zalyßyt\sq spravedlyvog, qkwo umovu (3.3) zapy-
saty u vyhlqdi ( ) ,N H−
−−
−
cl ∩ = N – .
4. Pro operator
�
A. Nexaj H – � H 0 � H + poznaça[ osnawenyj hil\-
bertiv prostir, qkyj [ asocijovanym z samosprqΩenym operatorom A ≥ 1 u tomu
sensi, wo H + = D ( A ) v normi ⋅ + = A⋅ 0 . Pry c\omu operator A2
zbiha[t\-
sq iz zvuΩennqm D–, :+ H + → H – na H + + ≡ H 4 ( A ) : A2 = D–,+ ++H . Prypu-
stymo, wo pozytyvnyj prostir H + rozkladeno v ortohonal\nu sumu H + =
= M + � N + u takyj sposib, wo pidprostir M + [ wil\nym v H 0 , H 0 � M + .
Rozhlqnemo novyj osnawenyj prostir
�
H − � H 0 �
�
H + , (4.1)
de
�
H+ ≡ M + . My xoçemo pobuduvaty samosprqΩenyj operator
�
A , qkyj aso-
cijovanyj z lancgΩkom (4.1) v takyj sposib, wo oblast\ vyznaçennq D( )
�
A zbi-
ha[t\sq z
�
H+ .
Nahada[mo, wo nehatyvnyj prostir
�
H− vyznaça[t\sq qk popovnennq H 0 v
novij normi:
�
f
−
: = sup ( , )
ϕ
ϕ
+ =1
0f , ϕ ∈ M + . (4.2)
Pry c\omu dlq dovil\noho fiksovanoho f ∈ H 0 vykonu[t\sq nerivnist\
�
f
−
≤ f − , (4.3)
de
f − : = sup ( , )
ϕ
ϕ
+ =1
0f , ϕ ∈ H + .
Zrozumilo, wo prostir H 0 wil\no i neperervno vklada[t\sq qk v H – , tak i
v
�
H− . Ale bulo b pomylkog dumaty, wo z (4.3) vyplyva[ vkladennq H – v
�
H−
qk vlasno] pidmnoΩyny.
TverdΩennq31. Zamykannq totoΩnoho vidobraΩennq
O : H – � f → f ∈
�
H − , f ∈ H 0 ,
[ neperervnym i ma[ netryvial\nyj nul\-pidprostir:
Ker Ocl = N – , N – =
I− + +, N ,
de cl poznaça[ zamykannq.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
SYNHULQRNI ZBURENNQ SAMOSPRQÛENYX OPERATORIV … 627
Dovedennq. Neperervnist\ vidobraΩennq O vyplyva[ bezposeredn\o z (4.3).
PokaΩemo, wo koΩen
η− −∈N [ nul\-vektorom dlq Ocl. Nexaj poslidovnist\
fn ∈H0 zbiha[t\sq v H – do fiksovanoho η− −∈N . Todi zavdqky (4.3) cq
poslidovnist\ bude zbiΩnog v
�
H − takoΩ. Ale u prostori
�
H − cq poslidov-
nist\ zbiha[t\sq do nulq. Ce vyplyva[ z toho, wo
( , )fn ϕ 0 = 〈 〉− +fn, ,ϕ → 〈 〉− − +η ϕ, , = 0, ϕ ∈ M + ,
oskil\ky N – � M + i M + [ wil\nym v H 0 . OtΩe, η – ∈ Ker Ocl
.
ZauvaΩymo, wo Ωoden vektor 0 ≠ f ∈ H 0 ne naleΩyt\ do Ker Ocl
. Prostir
H 0 vklada[t\sq v
�
H − bez defektu.
TverdΩennq32. Dlq koΩnoho 0 ≠ f ∈ H 0
�
f
−
=
P fM− −
≠ 0, (4.4)
de PM−
poznaça[ ortohonal\nyj proektor na M – v H – .
Dovedennq. Spravedlyvist\ rivnosti v (4.4) vyplyva[ z oznaçennq normy u
prostori
�
H − (dyv. (4.2)) ta spivvidnoßennq
( , )f ϕ 0 = 〈 〉− +f , ,ϕ =
〈 〉
− − +P fM , ,ϕ , ϕ ∈ M + ,
v qkomu vykorystano ortohonal\nist\ pidprostoriv M – � N + u sensi dual\no-
ho skalqrnoho dobutku. Zaznaçymo, wo dlq vsix 0 ≠ f ∈ H 0
P fM−
≠ 0, (4.5)
oskil\ky z P fM−
= 0 vyplyva[, wo f ∈ N – , ale N – ∩ H 0 = { 0 } zavdqky
H 0 � M + (dyv. teoremuH1).
Z tverdΩennqH2 vyplyva[, wo zvuΩennq vidobraΩennq O cl
na pidprostir
M – : = D–,+ +M [ unitarnym operatorom. OtΩe, prostory
�
H − , M – unitarno
ekvivalentni.
OtΩe, nezvaΩagçy na te, wo normy u prostorax
�
H − i H – zadovol\nqgt\
nerivnist\ (4.3), a prostir
�
H + ≡ M + [ pravyl\nog çastynog prostoru H + ,
prostir H – ne mistyt\sq v
�
H − qk çastyna:
�
H − � H – .
Nexaj
� � �
D–, :+ + −→H H poznaça[ kanoniçnyj unitarnyj izomorfizm v osna-
wenomu prostori Hil\berta (4.1). Rozhlqnemo operator
L : =
�
D L− +, ( )D , D ( )L : = { },ϕ ϕ∈ ∈+ − +
� �
H HD 0 . (4.6)
NevaΩko perekonatysq (dyv. nyΩçe dovedennq teoremyH4), wo operator L [
symetryçnym i joho oblast\ znaçen\ zbiha[t\sq z usim prostorom H 0 . Tomu vin [
samosprqΩenym operatorom v H 0 z oblastg vyznaçennq D ( L ) ⊂ M + =
�
H + .
Osnovnym rezul\tatom ci[] statti [ nastupna teorema.
Teorema34. Nexaj oblast\ vyznaçennq samosprqΩenoho v H 0 operatora
A ≥ 1 rozkladeno v ortohonal\nu sumu D ( A ) = H + = M + � N + . Prypusty-
mo, wo pidprostir M + [ wil\nym v H 0 , a N – : = D− + +, N zadovol\nq[
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
628 R. V. BOÛOK, V. D. KOÍMANENKO
umovu (3.3). Todi operator L , vyznaçenyj u (4.6), dopuska[ nastupnyj qvnyj
opys u terminax A-ßkaly ta operatora A :
D ( L ) =
PM H
+ ++ ,
LPM +
ϕ = A2 ϕ, ϕ ∈ H ++ ≡ D ( )A2 , (4.7)
de
PM +
— ortohonal\nyj proektor na M + v H + . Bil\ß toho, operator L
[ rozßyrennqm za Fridrixsom symetryçnoho operatora
L̇ : = A
2 M̃ + , M̃ + = M + ∩ H ++ . (4.8)
Pry c\omu oblast\ vyznaçennq operatora
�
A : = L1 2/
v toçnosti zbiha[t\sq z pidprostorom M + :
D ( )
�
A = M + =
�
H + . (4.9)
Dovedennq. PokaΩemo, wo vidobraΩennq
L : PM +
ϕ → A2 ϕ, ϕ ∈ H ++ ,
[ symetryçnym operatorom v H 0 . Spravdi, dlq usix ϕ, ψ ∈ H ++ ma[mo
( ),LP PM M+ +
ϕ ψ 0 = ( ),A P2
0ϕ ψM +
= 〈 〉− + − ++
D P, ,,ϕ ψM =
=
〈 〉
− +− + − +P D PM M, ,,ϕ ψ =
〈 〉− + − ++ +
D P P, ,,M Mϕ ψ =
〈 〉
+ +− + + −P D PM Mϕ ψ, , , =
= 〈 〉
+ − − + + −P P DM Mϕ ψ, , , = 〈 〉
+ − + + −P DM ϕ ψ, , , = 〈 〉
+ + −P AM ϕ ψ, ,
2 =
=
( ),P LPM M+ +
ϕ ψ 0 .
Z c\oho vyplyva[, wo L [ samosprqΩenym operatorom, oskil\ky joho oblastg
znaçen\ [ uves\ hil\bertiv prostir: R ( L ) = R ( A
2
) = H 0 .
Dovedemo teper, wo operator L, vyznaçenyj v (4.7), zbiha[t\sq z operatorom
L u (4.6). Dlq c\oho spoçatku pokaΩemo, wo ci operatory zbihagt\sq na mno-
Ωyni M̃+ , a potim perekona[mos\, wo L [ rozßyrennqm za Fridrixsom symet-
ryçnoho operatora L̇ (dyv. (4.8)). Qk promiΩnyj rezul\tat dovedemo, wo vidob-
raΩennq
�
D− +, , D–,+ zbihagt\sq na pidprostori M̃+ = M + ∩ H + + i pry c\omu
]x znaçennq naleΩat\ H 0 :
�
D− +, ϕ = D− +, ϕ ∈ H 0 , ϕ ∈ M̃ + . (4.10)
Oçevydno takoΩ, wo
PM M
+ +
˜ = M̃ + . Z c\oho vyplyva[ vklgçennq M̃ + ⊂
⊂ D ( )L̇ ta rivnist\
˙ ˜LM+ = A2 M̃+ . Dlq dovedennq (4.10) nahada[mo, wo
H + + ≡ H4 ( )A = D( )A2 , a M̃+ = M + ∩ H + + . OtΩe, vektor f : = D–,+ϕ =
= A2ϕ ∈ H 0 dlq koΩnoho ϕ ∈ H + + . Dali, rozhlqnemo dlq fiksovanoho
ϕ ∈ M̃+ dva funkcionaly:
lϕ ψ( ) : = 〈 〉− + − +D , ,,ϕ ψ , ψ ∈ H + ,
ta
�
lϕ ψ( ) : = 〈 〉− + − +
� �
D , ,,ϕ ψ , ψ ∈ M + .
Funkcional lϕ ψ( ) [ neperervnym na H 0 ta lϕ ψ( ) = ( , )f ψ 0 = ( , )f ψ + dlq
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
SYNHULQRNI ZBURENNQ SAMOSPRQÛENYX OPERATORIV … 629
usix ψ ∈ M + . Funkcional
�
lϕ ψ( ) takoΩ [ neperervnym na H 0 , oskil\ky
M + =
�
H+ , ta
�
lϕ ψ( ) = ( , )ϕ ψ + =
〈 〉
+ +
ϕ ψ, ,H H0
,
�
lϕ ψ( ) ≤ c ψ 0 ,
de c = ϕ ++ . OtΩe,
�
lϕ ψ( ) = ( ),
�
f ψ 0 z deqkym
�
f ∈H0. My stverdΩu[mo, wo
f =
�
f . Spravdi, zhidno z pobudovog ( , )f ψ 0 = ( , )ϕ ψ + = ( ),
�
f ψ 0 dlq usix ψ ∈
∈ M + . Tomu vektory f ta
�
f zbihagt\sq, oskil\ky pidprostir M + [ wil\nym v
H 0 . OtΩe, (4.10) vstanovleno.
Dovedemo, wo operator L z (4.6) [ rozßyrennqm za Fridrixsom symetryçnoho
operatora L̇ . Nahada[mo, wo oblast\ vyznaçennq D( )L̇ = M̃+ [ wil\nog v
H 0 . Naspravdi z umovy (3.3) vyplyva[, wo pidprostir M̃+ [ wil\nym v M + .
Spravdi, qkwo φ ∈ M + ta φ � M̃+ , to D–,+φ � N – i D–,+φ �
Ñ − . OtΩe,
φ ≡ 0, oskil\ky
Ñ − = N – zavdqky (3.3). Dokladniße, nexaj M + =
=
˜ ˜M M+ +
⊥� ta φ ∈ +
⊥M̃ . Todi
ω φ: –,
˜= ∈+ −
⊥D M , de M̃−
⊥ = M M− −� ˜ . Tomu
ma[mo
〈 〉+ − +ω, ˜
,M = 0 =
〈 〉+ −− ++ω, ˜
,M ⇒ ω ∈
Ñ − = N – .
Ale ce moΩlyvo, lyße qkwo φ = 0, oskil\ky φ ∈ M + ta D–,+φ � N – . OtΩe,
M M+ +� ˜ .
Dali, oçevydno, wo operator L̇ z oblastg vyznaçennq D( )L̇ = M̃+ [ zamk-
nenym v H 0 , tomu wo pidprostir M̃+ [ zamknenym v H + + . My stverdΩu[mo,
wo joho oblast\ znaçen\ takoΩ [ wil\nog v
�
H− . Teper zauvaΩymo, wo na
pidstavi (4.10) oblast\ znaçen\ operatora L̇ zbiha[t\sq z pidprostorom M̃− =
= A2 M̃+ = A
2( )M H+ ++∩ = M – ∩ H 0 , qkyj [ wil\nym v
�
H− zavdqky tomu,
wo
� � �
D–, :+ + −→H H — unitarnyj operator.
Qkwo M̃+ � M + , to prostir
�
H+ [ popovnennqm M̃+ vidnosno skalqrno-
ho dobutku ( , ) : ˙ , ( , ) ( , )( )ϕ ψ ϕ ψ ϕ ψ ϕ ψ�
H+
= = = +L A A0 0 , ϕ ψ, ˜∈ +M . Tomu ope-
rator L [ samosprqΩenym rozßyrennqm symetryçnoho operatora L̇ . Za prove-
denog pobudovog ce [ rozßyrennqm za Fridrixsom operatora L̇ , oskil\ky my
vΩe vstanovyly vykonannq wil\noho i neperervnoho vkladennq: M̃+ � M + .
Nareßti, rivnist\ (4.9) [ pravyl\nog, oskil\ky popovnennq mnoΩyny
D(( ) )
�
A 2 = D ( L ) za normog ⋅ +
�
: = L1 2
0
/ ⋅ zbiha[t\sq z M + . Spravdi,
oskil\ky M̃+ [ wil\nym u M + , to dosyt\ lyße nahadaty, wo ( ),Lϕ ψ 0 =
= ( , )ϕ ψ + , ϕ ψ, ˜∈ +M . OtΩe, za oznaçennqm L ma[mo
( , )Lϕ ψ 0 = ( )/ /,L L1 2 1 2
0ϕ ψ = ( ),A2
0ϕ ψ = ( , )ϕ ψ + = (( ) ),
�
A2 2
0ϕ ψ
dlq usix ϕ, ψ ∈ M̃+ . Takym çynom, M + = H1( )L i, otΩe, M + = H2 ( )
�
A =
= D( )
�
A =
�
H+ . Ce zaverßu[ dovedennq teoremy.
5. Zahal\na konstrukciq. U c\omu punkti my pobudu[mo operator typu
�
A HH(qkyj budemo poznaçaty çerez
�
D ) u vypadku, koly wil\nyj v H 0 pidprostir
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
630 R. V. BOÛOK, V. D. KOÍMANENKO
M + [ netryvial\nog çastynog prostoru H k z A-ßkaly pry dovil\nomu
znaçenni k > 0.
OtΩe, nexaj H + = H k , k > 0, rozkladeno v ortohonal\nu sumu H + =
= M + � N + . Prypustymo, wo M + � H 0 . Porqd z
H – ≡ H – k � H 0 � H k ≡ H +
rozhlqda[mo osnawenyj prostir
�
H− ≡ ( )M+ − � H 0 � M + ≡
�
H+
i asocijovanyj z nym operator
� � ��
D D D: ( ) ( )H D H R+ = → =0 , qkyj [ samo-
sprqΩenym v H 0 . Vstanovymo zv’qzok miΩ
�
D ta operatorom Ak /2 , dlq qkoho
H k [ oblastg vyznaçennq: A
k : H k → H 0 .
Lema31. Dlq koΩnoho wil\noho v H 0 pidprostoru M + iz H k vidobra-
Ωennq
L : PM+
ϕ → A
k
ϕ , ϕ ∈ H 2 k
( PM+
poznaça[ ortohonal\nyj proektor v H + na M + ) [ samosprqΩenym ope-
ratorom v H 0 .
Dovedennq. VidobraΩennq L [ korektno oznaçenym operatorom. Spravdi,
qkwo
PM+
ϕ = 0, to ϕ ∈ N + = H + � M + . Ale N + ∩ H + = N – ∩ H 0 = { 0 } ,
oskil\ky M + � H 0 (dyv. teoremuH1). OtΩe, ϕ = 0.
Dali, perekona[mos\, wo vidobraΩennq L z oblastg vyznaçennq D ( L ) =
=
P kM H
+ 2 [ symetryçnym operatorom v H 0 . Spravdi,
( ),LP PM M+ +
ϕ ψ 0 =
( ),A Pk ϕ ψM+ 0 =
〈 〉
+ −Ak
k kPϕ ψ, ,M 2 2 =
=
〈 〉
− + −P Pk
k kM MA ϕ ψ, ,2 2 =
〈 〉
+ + −Ak
k kP PM Mϕ ψ, ,2 2 =
〈 〉
+ + + −P Pk
M Mϕ ψ, ,A =
=
〈 〉
+ − −P P k
k kM Mϕ ψ, ,A 2 2 =
〈 〉
+ −P k
k kM ϕ ψ, ,A 2 2 =
〈 〉
+
P k
M ϕ ψ, A 0 =
=
( ),P LPM M+ +
ϕ ψ 0 ,
de
PM−
— ortoproektor v
H−2k na pidprostir M− : =
D− + +, M , a Ak
— za-
mykannq operatora A
k : H 0 → H – 2k ; tut bulo vykorystano spivvidnoßennq
Ak PM +
=
P k
M −
A . Teper samosprqΩenist\ L vyplyva[ z toho, wo joho oblast\
znaçen\ R ( L ) = R ( A
k
) = H 0 .
Lemu dovedeno.
Rozhlqnemo v H 0 porqd z L we j operator
�
L , porodΩenyj kanoniçnym
unitarnym izomorfizmom
�
D− +, , qkyj vidobraΩa[
�
H + v
�
H − :
�
L : =
� �
D L− +, ( )D , D( )
�
L : = { },
� � ��
ϕ ϕ∈ ∈+ − +H HD 0 .
Lema32. Za umovy M + � H 0 operatory L ta
�
L zbihagt\sq.
Dovedennq. Nexaj
�
ϕ ∈ D( )
�
L ⊂ M + ≡
�
H+ . Todi H 0 �
�
f =
� �
D− +, ϕ i
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
SYNHULQRNI ZBURENNQ SAMOSPRQÛENYX OPERATORIV … 631
〈 〉− + − +
� � �
D , ,,ϕ ψ = ( ),
� �
ϕ ψ + = ( ),
�
f ψ 0, ψ ∈ M + .
Oskil\ky M + =
�
H+ [ pidprostorom v H + , to ( ),
� �
ϕ ψ + = ( ),
�
ϕ ψ + =
= ( )/ /,A Ak k2 2
0
�
ϕ ψ . Vzahali, vektor
�
ϕ ne naleΩyt\ do oblasti vyznaçennq ope-
ratora Ak, ale zavdqky rivnosti ( ),
�
ϕ ψ + = ( ),
�
f ψ 0 i vnaslidok wil\nosti pid-
prostoru M + v H 0 isnu[ vektor ϕ ∈ H + takyj, wo
�
f = A k ϕ. OtΩe, ma[mo
( ),
�
ϕ ψ + = ( ),
�
f ψ 0 = ( ),Ak ϕ ψ 0 = 〈 〉− + − +D , ,,ϕ ψ =
= ( , ) ,ϕ ψ − + = ( ), ,PM+ − +ϕ ψ , ψ ∈ M + .
Ce oznaça[, znovu vnaslidok wil\nosti M + v H 0 , wo
�
ϕ =
PM+
ϕ i
� �
L ϕ =
=
� �
D− +, ϕ =
�
f = Ak ϕ =
LPM+
ϕ .
Lemu dovedeno.
Vvedemo pidprostir M̃+ : = M H+ ∩ 2k .
TverdΩennq33. Na pidprostori M̃+ operatory L ta Ak
digt\ odnako-
vo:
L M̃+ = Ak M̃+ .
Dovedennq. Cej fakt vyplyva[ z lemyH2, oskil\ky
PM M
+ +
˜ = M̃+ .
OtΩe,
L M̃+ =
�
L M̃+ = A
k M̃+ ,
i, bil\ß toho, za umovy M̃+ � H 0 operator L [ samosprqΩenym rozßyrennqm
wil\no vyznaçenoho symetryçnoho operatora
L̇ : = A
k M̃+ .
ZauvaΩymo, wo wil\nist\ M̃+ v H 0 harantu[ umova typu (3.3).
TverdΩennq34. Qkwo M̃+ � M + , to operator L [ rozßyrennqm za
Fridrixsom symetryçnoho operatora L̇ .
Dovedennq. Na pidstavi poperedn\oho tverdΩennq kvadratyçna forma
γ ϕ ψ( , ) : = ( )˙ ,Lϕ ψ 0 zbiha[t\sq z formog ( ),Ak ϕ ψ 0 =
( , )ϕ ψ M+
na vektorax ϕ,
ψ ∈ M̃+ . Zavdqky wil\nosti pidprostoru M̃+ v M + zamykannq formy γ
zbiha[t\sq iz skalqrnym dobutkom M + . Tomu
�
L [ rozßyrennqm za Fridrixsom
operatora L̇ . Ale my vΩe vstanovyly, wo L =
�
L .
Vvedemo operator
�
D : =
�
L1 2/ = L1 2/ .
Bezposeredn\o z lemH1, 2 ta tverdΩen\H3, 4 vyplyva[ spravedlyvist\ nastupno]
teoremy.
Teorema35. Za umovy M̃+ � M + samosprqΩenyj v H 0 operator
�
D
ma[ svo[g oblastg vyznaçennq pidprostir M + i zbiha[t\sq z kvadratnym ko-
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
632 R. V. BOÛOK, V. D. KOÍMANENKO
renem vid rozßyrennq za Fridrixsom wil\no vyznaçenoho symetryçnoho operato-
ra L̇ : = A
k M̃+ .
Dovedennq. Lyße zaznaçymo, wo rivnist\ D ( )
�
D = M + [ naslidkom wil\-
nosti M̃+ v M + .
Nasamkinec\ zauvaΩymo, wo znaçennq kvadratyçnyx form
�
γ ϕ ψ( , ) : = ( ),
�
Lϕ ψ 0 = ( ),
� �
D Dϕ ψ 0 = ( , )ϕ ψ �
+ ,
γ ϕ ψ( , ) : = ( ),Ak ϕ ψ 0 = ( )/ /,A Ak k2 2
0ϕ ψ = ( , )ϕ ψ +
odnakovi na M̃+ . Ale ci formy magt\ rizni zamkneni rozßyrennq u prostori
H 0 , z qkymy asocijovani rizni samosprqΩeni operatory, i tomu
�
D ≠ Ak /2 . Pry
c\omu zrozumilo, wo ostanni operatory ne moΩut\ buty rivnymy na bud\-qkij
wil\nij v H 0 mnoΩyni, nezvaΩagçy na te, wo
�
L ta A
k
zbihagt\sq na mnoΩy-
ni, qka [ wil\nog v H 0 .
1. Berezanskyj G. M. RazloΩenye po sobstvenn¥m funkcyqm samosoprqΩenn¥x operatorov.
– Kyev: Nauk. dumka, 1965. – 798 s.
2. Berezanskyj G. M. SamosoprqΩenn¥e operator¥ v prostranstvax funkcyj beskoneçnoho
çysla peremenn¥x. – Kyev: Nauk. dumka, 1978. – 360 s.
3. Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H. Solvable models in quantum mechanics. –
Berlin etc.: Springer, 1988. – 568 p.
4. Albeverio S., Karwowski W., Koshmanenko V. Square power of singularly perturbed operators //
Math. Nachr. – 1995. – 173. – P. 5 – 24.
5. Albeverio S., Karwowski W., Koshmanenko V. On negative eigenvalues of generalized Laplace
operator // Repts Math. Phys. – 2000. – 45, # 2. – P. 307 – 325.
6. Albeverio S., Koshmanenko V. Singular rank one perturbations of self-adjoint operators and Krein
theory of self-adjoint extensions // Potent. Anal. – 1999. – 11. – P. 279 – 287.
7. Albeverio S., Kurasov P. Singular perturbations of differential operators and solvable Schrödinger
type operators. – Cambridge: Univ. Press, 2000. – 265 p.
8. Albeverio S., Kurasov P. Rank one perturbations, approximations and self-adjoint extensions // J.
Funct. Anal. – 1997. – 148. – P. 152 – 169.
9. Kato T. Teoryq vozmuwenyj lynejn¥x operatorov. – M.: Myr, 1972. – 740 s.
10. Karwowski W., Koshmanenko V., Ôta S. Schrödinger operator perturbed by operators related to
null-sets // Positivity. – 1998. – 77, # 2. – P. 18 – 34.
11. Koshmanenko V. D. Towards the rank-one singular perturbations of self-adjoint operators // Ukr.
Math. J. – 1991. – 43, # 11. – P. 1559 – 1566.
12. Koßmanenko V. D. Synhulqrn¥e bylynejn¥e form¥ v teoryy vozmuwenyj samosoprqΩen-
n¥x operatorov. – Kyev: Nauk. dumka, 1993. – 176 s.
13. Koshmanenko V. Singular quadratic forms in perturbation theory. – Dordrecht etc.: Kluwer Acad.
Publ., 1999. – 308 p.
14. Gesztesy F., Simon B. Rank-one perturbations at infinite coupling // J. Funct. Anal. – 1995. – 128.
– P. 245 – 252.
15. Krejn M. H. Teoryq samosoprqΩenn¥x rasßyrenyj poluohranyçenn¥x πrmytov¥x
operatorov y ee pryloΩenyq. I // Mat. sb. – 1947. – 20(62), #H3. – S.H431 – 495.
16. Karwowski W., Koshmanenko V. Generalized Laplace operator in L n
2( )R // Stochast. Process.,
Phys. and Geom.: New Interplays. II. Can. Math. Soc. (Conf. Proc.). – 2000. – 29. – P. 385 – 393.
17. Koshmanenko V. D. Singular perturbations defined by forms // Lect. Notes Phys. Appl. Self-
adjoint Extens. in Quant. Phys. / Eds P. Exner, P. Seba.
�
– 1987. – 324. – P. 55 – 66.
18. Koshmanenko V. Singular operator as a parameter of self-adjoint extensions // Operator Theory.
Adv. and Appl. (Proc. Krein Conf. (Odessa, 1997)). – 2000. – 118. – P. 205 – 223.
19. Koshmanenko V. D. Regular approximations of singular perturbations of H−2 -class // Ukr. Math.
J. – 2000. – 52, # 5. – P. 626 – 637.
20. Posilicano A. A Krein-like formula for singular perturbations of self-adjoint operators and appli-
cations // J. Funct. Anal. – 2001. – 183. – P. 109 – 147.
OderΩano 17.01.2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
|
| id | umjimathkievua-article-3628 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:46:03Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f0/c9c076e9be5bc8ad9fd6b7f8bf9972f0.pdf |
| spelling | umjimathkievua-article-36282020-03-18T20:00:32Z Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces Сингулярні збурення самоспряжених операторів, асоційовані з оснащеними гільбертовими просторами Bozhok, R. V. Koshmanenko, V. D. Божок, Р. В. Кошманенко, В. Д. Let A be an unbounded self-adjoint operator in a Hilbert separable space \(H_0\) with rigging \(H_ - \sqsupset H_0 \sqsupset H_ +\) such that \(D(A) = H_ +\) in the graph norm (here, \(D(A)\) is the domain of definition of A). Assume that \(H_ +\) is decomposed into the orthogonal sum \(H_ + = M \oplus N_ +\) so that the subspace \(M_ +\) is dense in \(H_0\). We construct and study a singularly perturbed operator A associated with a new rigging \(H_ - \sqsupset H_0 \sqsupset \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ +\), where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ + = M_ + = D(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} )\), and establish the relationship between the operators A and A. Нехай $A$ є необмеженим самоспряженим оператором в сепарабельному гільбертовому просторі $\mathcal{H}_0$, який оснащено $\mathcal{H}_{-} \sqsupset \mathcal{H}_0 \sqsupset \mathcal{H}_+$ таким чином, що область визначення $D(A) = \mathcal{H}_+$ в нормі графіка. Припустимо, що $\mathcal{H}_+$ розкладено в ортогональну суму $\mathcal{H}_{+} = \mathcal{M}_+ \oplus \mathcal{N}_+$ так, що підпростір $\mathcal{M}_+$ є щільним в $\mathcal{H}_0$. У роботі будується і вивчається сингулярно збурений оператор A , асоційований з новим оснащенням $\breve{\mathcal{H}}_{-} \sqsupset \mathcal{H}_0 \sqsupset \breve{\mathcal{H}}_+$, де $\breve{\mathcal{H}}_{+} = \mathcal{M}_+ = \mathcal{D}(\breve{A})$. Встановлено зв'язок між операторами $A$ та $\breve{A}$. Institute of Mathematics, NAS of Ukraine 2005-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3628 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 5 (2005); 622–632 Український математичний журнал; Том 57 № 5 (2005); 622–632 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3628/3986 https://umj.imath.kiev.ua/index.php/umj/article/view/3628/3987 Copyright (c) 2005 Bozhok R. V.; Koshmanenko V. D. |
| spellingShingle | Bozhok, R. V. Koshmanenko, V. D. Божок, Р. В. Кошманенко, В. Д. Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces |
| title | Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces |
| title_alt | Сингулярні збурення самоспряжених операторів, асоційовані з оснащеними гільбертовими просторами |
| title_full | Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces |
| title_fullStr | Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces |
| title_full_unstemmed | Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces |
| title_short | Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces |
| title_sort | singular perturbations of self-adjoint operators associated with rigged hilbert spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3628 |
| work_keys_str_mv | AT bozhokrv singularperturbationsofselfadjointoperatorsassociatedwithriggedhilbertspaces AT koshmanenkovd singularperturbationsofselfadjointoperatorsassociatedwithriggedhilbertspaces AT božokrv singularperturbationsofselfadjointoperatorsassociatedwithriggedhilbertspaces AT košmanenkovd singularperturbationsofselfadjointoperatorsassociatedwithriggedhilbertspaces AT bozhokrv singulârnízburennâsamosprâženihoperatorívasocíjovanízosnaŝenimigílʹbertovimiprostorami AT koshmanenkovd singulârnízburennâsamosprâženihoperatorívasocíjovanízosnaŝenimigílʹbertovimiprostorami AT božokrv singulârnízburennâsamosprâženihoperatorívasocíjovanízosnaŝenimigílʹbertovimiprostorami AT košmanenkovd singulârnízburennâsamosprâženihoperatorívasocíjovanízosnaŝenimigílʹbertovimiprostorami |