On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations
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Інститут математики НАН України
2005
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nasplib_isofts_kiev_ua-123456789-1657402025-02-09T13:29:46Z On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations Застосування розсіяння Лакса - Філліпса в теорії сингулярних збурень Kuzhel', S.A. Matsyuk, L.V. Статті 2005 Article On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations / S.A. Kuzhel', L.V. Matsyuk // Український математичний журнал. — 2005. — Т. 57, № 5. — С. 679–688. — Бібліогр.: 15 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165740 517.9 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті Kuzhel', S.A. Matsyuk, L.V. On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations Український математичний журнал |
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Article |
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Kuzhel', S.A. Matsyuk, L.V. |
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Kuzhel', S.A. Matsyuk, L.V. |
| author_sort |
Kuzhel', S.A. |
| title |
On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations |
| title_short |
On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations |
| title_full |
On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations |
| title_fullStr |
On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations |
| title_full_unstemmed |
On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations |
| title_sort |
on an application of the lax-phillips scattering approach in the theory of singular perturbations |
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Інститут математики НАН України |
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2005 |
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Статті |
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https://nasplib.isofts.kiev.ua/handle/123456789/165740 |
| citation_txt |
On an Application of the Lax-Phillips Scattering Approach in the Theory of Singular Perturbations / S.A. Kuzhel', L.V. Matsyuk // Український математичний журнал. — 2005. — Т. 57, № 5. — С. 679–688. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
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AT kuzhelsa onanapplicationofthelaxphillipsscatteringapproachinthetheoryofsingularperturbations AT matsyuklv onanapplicationofthelaxphillipsscatteringapproachinthetheoryofsingularperturbations AT kuzhelsa zastosuvannârozsíânnâlaksafíllípsavteoríísingulârnihzburenʹ AT matsyuklv zastosuvannârozsíânnâlaksafíllípsavteoríísingulârnihzburenʹ |
| first_indexed |
2025-11-26T05:16:19Z |
| last_indexed |
2025-11-26T05:16:19Z |
| _version_ |
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| fulltext |
UDC 517.9
S. O. Kuzhel’* (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
L. V. Matsyuk (Univ. “KROK”, Kyiv)
ON AN APPLICATION OF THE LAX – PHILLIPS SCATTERING
APPROACH IN THEORY OF SINGULAR PERTURBATIONS
ZASTOSUVANNQ ROZSIQNNQ LAKSA – FILLIPSA
V TEORI} SYNHULQRNYX ZBUREN|
For a singular perturbation A = A0 +
i j
n
ij j it
,
,
=∑ ⋅
1
ψ ψ , n ≤ ∞, of a positive self-adjoint operator A0
with Lebesgue spectrum, the spectral analysis of the corresponding self-adjoint operator realizations AT
is carried out and the scattering matrix
� A AT , ( )
0( ) δ is calculated in terms of parameters tij under some
additional restrictions on singular elements ψ j that provides the possibility of application of the Lax –
Phillips approach in the scattering theory.
Dlq synhulqrnoho zburennq A = A0 +
i j
n
ij j it
,
,
=∑ ⋅
1
ψ ψ , n ≤ ∞, dodatnoho samosprqΩenoho
operatora A0 iz spektrom Lebeha provedeno spektral\nyj analiz vidpovidnyx samosprqΩenyx
realizacij AT . Krim toho, obçysleno matrycg rozsiqnnq � A AT , ( )
0( ) δ çerez parametry tij pry
deqkyx dodatkovyx obmeΩennqx na synhulqrni elementy ψ j . OderΩani rezul\taty dozvolqgt\
zastosovuvaty sxemu Laksa – Fillipsa v teori] rozsiqnnq.
1. Statement of the problem. Let A0 be a positive self-adjoint operator acting in a
Hilbert space � and let
� � � � �2 0 1 0 1 0 2 0( ) ( ) ( ) ( )A A A A⊂ ⊂ ⊂ ⊂− −
be the standard scale of Hilbert spaces associated with A0 [1]. Precisely, �2 0( )A =
= D ( )A0 , �1 0( )A = D A0
1 2/( ) with the norms u k = A I uk
0
2+( ) / , k = 1, 2, and
the conjugated spaces �−k A( )0 can be defined as the completions of � with respect
to the norms
u A I uk
k
−
−= +( )0
2/ ∀ ∈u � . (l)
By (1), the operator A I0
1+( )− can be continuously extended to an isometric
mapping A0
1+( )−I of �−2 0( )A onto �. Thus, for any ψ ∈ −� 2 0( )A , the element
A0
1+( )−I ψ belongs to � and the relation
ψ ψ, ,u A I u I= +( ) +( )( )−
0 0
1
A ∀ ∈u A�2 0( ) (2)
enables one to consider any element ψ ∈ −� 2 0( )A as a linear continuous functional
on �2 0( )A .
Let us fix an orthonormal system ψ j j
n{ } =1
n ∈ ∞{ }( )N, in �−2 0( )A and
consider the formal expression
A t
i j
n
ij j i0
1
+ ⋅
=
∑
,
,ψ ψ , t tij ji= , n ∈ ∞{ }N, . (3)
* Partially supported by CRDF (grant No. UM1-2567-OD-03) and DFFD of Ukraine (project
No. 01.07/027).
© S. O. KUZHEL’, L. V. MATSYUK, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5 679
680 S. O. KUZHEL’, L. V. MATSYUK
In what follows we suppose that the linear subspace X of �−2 0( )A generated by
the basis ψ j j
n{ } =1
satisfies the condition X ∩ � = {0} (i.e., elements ψj are � -
independent). In this case, the potential V =
i j
n
ij j it
,
,=∑ ⋅
1
ψ ψ is singular; the
symmetric operator
A A
Asym
sym
:= ( )0�D
, D A u D A u j njsym( ) = ∈ = ≤ ≤{ }( ) , ,0 0 1ψ (4)
is closed densely defined in � and the deficiency indices of A sym are equal to n.
Using approaches developed in the theory of singular perturbations [2], we can
associate with the formal expression (3) its self-adjoint operator realization AT acting
in � (see Theorem 1) and, as a result, to reduce the scattering problem for (3) to the
study of the scattering operator
S A A W A A W A AT T T, , ,*
0 0 0( ) = ( ) ( )+ − (5)
for perturbed AT and unperturbed A0 operator realizations of (3), where the wave
operators W A A±( )B, 0 are defined as follows:
W A A s e e
t
i A t i A t
±
→±∞
−( ) = −T
T, : lim0
0 .
In the case n < ∞, operators AT and A0 are different self-adjoint extensions of
the symmetric operator A sym with finite deficiency indices. On the basis of this fact,
the scattering matrix
� A AT , ( )
0( ) δ (in other words, the image of the scattering operator
S A AT, 0( ) in the spectral representation of A0 ) was expressed in terms of parameters
of the Krein’s resolvent formula with the use of the stationary approach in the
scattering theory (see [3])*.
In the present paper, we apply one of the well-developed nonstationary scattering
approaches (the Lax – Phillips approach) for the study of spectral and scattering
properties of operator realizations AT of the formal expression (3), where, in general,
the singular perturbation is not assumed to be of finite rank. In particular, for finite
rank singular perturbations, we obtain a representation of the scattering matrix
� A AT , ( )
0( ) δ directly in terms of parameters tij of the singular potential V.
Of course, in order to employ the Lax – Phillips approach we have to impose some
restrictions on the unperturbed operator A0 and singular elements ψj . An example
of such restrictions and the spectral analysis of the corresponding operator realizations
of (3) are contained in Section 3. In Section 2, we discuss the problem of realization of
the heuristic expression (3) as a self-adjoint operator in � and present a simple
description of such realizations in terms of parameters tij . The expression of the
scattering matrices � A AT , ( )
0( ) δ for nonnegative self-adjoint operator realization AT
of (3) is presented in Section 4. Section 5 contains an application of the obtained
results to the case of one-dimensional Schrödinger operator with symmetric zero-range
potentials.
Let us make a remark about notations. In what follows, any Hilbert space is
assumed to be separable. D ( )A and ker A denote the domain and the null-space of a
linear operator A, respectively. A�D means the restriction of A onto a set D.
2. Operator realizations of singular perturbations. To define a self-adjoint
operator realization of (3) in � with a given singular perturbation
* A survey of further development of the stationary scattering approach in the theory of singular
perturbations can be found in [2].
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
ON AN APPLICATION OF THE LAX – PHILLIPS SCATTERING APPROACH IN THEORY … 681
V =
i j
n
ij j it
,
,
=
∑ ⋅
1
ψ ψ ,
we use an approach suggested initially in [4] (see also [2]) for the case of finite rank
singular perturbations and its generalization to the infinite dimensional case [5]. The
main idea consists in the construction of some regularization
AT : ,
,
= + ⋅
=
∑A t
i j
n
ij j i0
1
ψ ψex , (6)
of (3) that is well defined as an operator from D Asym
*( ) to �−2 0( )A . In this case, the
corresponding operator realization AT of (3) is determined by the formula
AT = AT T
�D A( ) ,
D DA f A fT T( ) ∈ ( ) ∈{ }= sym
*
A � . (7)
Let us clarify the meaning of components A0 and ψ j
ex in (6). First of all we
observe that A0 is the continuation of A0 as a bounded linear operator acting from
� into �−2 0( )A and this continuation is determined by the formula
A A0 0
1 1
f I f f:= +( )[ ] −− −
∀ ∈f �. (8)
Next, the linear functionals ψ j
ex are extensions of the functionals ψj onto
D Asym
*( ). Using the well-known relation
D Asym
*( ) = D HA0( ) +̇ , where H = ker *A Isym +( ) , (9)
we arrive at the conclusion that ψj can be extended onto D Asym
*( ) if we know their
values on H.
It follows from (1), (2), and (4) that the vectors
h Ij j= +( )−A0
1ψ , j = 1, … , n, (10)
form an orthonormal basis of the Hilbert space H. Hence, ψ j
ex , 1 ≤ j ≤ n, are well-
defined by the formula
ψ ψ αj j
p
n
p jpf u rex, : ,= +
=
∑
1
(11)
for all elements f = u +
p
n
p ph=∑ 1
α , u A∈ ( )D 0 , α p ∈C , from D Asym
*( ) if we
determine the entries
r I hjp j p j p: , ,= +( ) =−ψ ψ ψA0
1
of a (in general, infinite-dimensional) matrix R = rjp j p
n( ) =, 1
.
If all ψ j A∈ −� 1 0( ), then rjp are well defined and R is defined uniquely
(see [2]). In other cases, the most appropriate choice of R has to be determined by
imposing additional requirements related to the nature of a perturbation (see, e.g., [2]).
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
682 S. O. KUZHEL’, L. V. MATSYUK
We recall (see, e.g., [6]) that a matrix X = xij i j
n( ) =, 1
is called the matrix
decomposition with respect to the basis hj j
n{ } =1
of a bounded operator X acting in
H if its entries xij are defined by the expansions
Xh x hj
i
n
ij i=
=
∑
1
, 1 ≤ j ≤ n.
In what follows we assume that R is already chosen as a matrix decomposition
(with respect to the basis hj
n{ }1
) of a bounded self-adjoint operator R acting in H.
Our aim now is to describe operator realizations AT of (3) in terms of parameters
tij of the singular perturbation V. To do this, the method of boundary triplets (see [7]
and references therein) can be used.
We recall that a triplet H , ,Γ Γ0 1( ) , where Γ0 , Γ1 are linear mappings of
D Asym
*( ) into H, is called a boundary triplet of Asym
* if
A f gsym
* ,( ) – f A g, *
sym( ) = Γ Γ1 0f g,( ) – Γ Γ0 1f g,( ) , f g A, *∈ ( )D sym , (12)
and for any F0 , F1 ∈H there exists an element f A∈ ( )D sym
* such that Γ0 f = F0 ,
Γ1 f = F1.
Denote
Γ̂0 0
f P fA= ,
ˆ *Γ1 f P A I f= +( )H sym , f A∈ ( )D sym
* , (13)
where PH is the orthogonal projector onto H in � and PA0
is the projector onto
H with respect to the decomposition (9). The triplet H , ˆ , ˆΓ Γ0 1( ) is an example of the
well-known boundary triplet that is used widely in the Krein – Birman – Vishik
extension theory.
Lemma 1. The triplet H , ,Γ Γ0 1( ) , where
Γ Γ Γ0 1 0f f R f= +ˆ ˆ , Γ Γ1 0f f= − ˆ , f A∈ ( )D sym
* , (14)
forms a boundary triplet of Asym
* .
Proof. Since H , ˆ , ˆΓ Γ0 1( ) is a boundary triplet, we get
A f g f A g f g f gsym sym
* *, , ˆ , ˆ ˆ , ˆ( ) ( ) ( ) ( )− = −Γ Γ Γ Γ1 0 0 1 . (15)
Expressing Γ̂i in terms of Γi with the use of (14), substituting the obtained
expressions into (15), and taking into account that R is a self-adjoint operator in H,
we establish (12) for Γi .
Let F0 , F1 be arbitrary elements of H. Since, H , ˆ , ˆΓ Γ0 1( ) is a boundary triplet,
there exists f A∈ ( )D sym
* such that Γ̂0 f = −F1 and Γ̂1 f = F0 + RF1. Comparing
these relations with (14), we get Γ0 f = F0 and Γ1 f = F1. Thus, H , ,Γ Γ0 1( ) is a
boundary triplet.
Lemma 1 is proved.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
ON AN APPLICATION OF THE LAX – PHILLIPS SCATTERING APPROACH IN THEORY … 683
Theorem 1. Let the coefficient matrix T = tij i j
n( ) =, 1
of the singular perturbation
V =
i j
n
ij j it
,
,=∑ ⋅
1
ψ ψ in (3) be a matrix decomposition (with respect to the basis
hj
n{ }1
) of a bounded self-adjoint operator T acting in H , then the corresponding
self-adjoint operator realization AT of (3) is defined as follows:
A AT T
�D ( ) , D DA f A T f fT( ) = ∈ ( ) ={ }sym
* Γ Γ0 1 . (16)
Proof. Representing f A∈ ( )D sym
* in the form f = u +
i i ih=∑ 1
α , where
u A∈D( )0 , hi ∈H , αi ∈C and employing (2), (10), (11), (13), (14), we get
T fΓ0 = T R fˆ ˆΓ Γ1 0+( ) =
i j p
n
ij j p jp it u r h
, ,
,
=
∑ +( )
1
ψ α =
i j
n
ij j it f h
,
,
=
∑
1
ψex
and Γ1 f = − =∑i
n
i ih
1
α . Using these relations and taking (6), (8), and (10) into
account, we obtain
AT f = A u0 –
i
n
i ih
=
∑
1
α +
i j
n
ij j it f
,
,
=
∑
1
ψ ψex +
i
n
i i
=
∑
1
α ψ =
= A fsym
* + A0
1 1
0 1+( )[ ] −( )− −
I T fΓ Γ . (17)
Since A0
1 1
+( )[ ]− −
I maps H onto the subspace X of �−2 0( )A generated by
the basis ψ j j
n{ } =1
and such that X ∩ H = {0}, equalities (7) and (18) imply that
f A∈D ( ) if and only if T fΓ0 – Γ1 f = 0. Therefore, the operator realization AT of
(3) is determined by (16). The self-adjointness of AT follows from the fact that T is
self-adjoint and the general properties of boundary triplets [7].
Theorem 1 is proved.
Remark 1. For the case of finite rank singular perturbations, Lemma 1 and
Theorem 1 were proved in [8].
3. A sufficient condition of the applicability of the Lax – Phillips approach and
spectral analysis of AT. In what follows we suppose that the unperturbed operator
A0 in (3) has absolutely continuous spectrum on R+ = 0, ∞[ ) with the same
multiplicity m ≤ ∞ at each point of R+ ( i.e., the spectrum σ( )A0 is Lebesgue and
σ( )A0 = R+ ). This condition is equivalent (see [9]) to the existence of a simple
maximal symmetric operator B in � such that A0 is a self-adjoint extension of
the symmetric operator B2 and
A u u B u0
2
, *( ) = ∀ ∈u AD ( )0 . (18)
(Note that the non-zero deficiency index of B coincides with m and B2 is a densely
defined symmetric operator with deficiency indices m.)
We also suppose that the symmetric operator Asym defined by (4) coincides with
B2 (if n = m ) or Asym is a symmetric extension of B2 (if n < m ).
It should be noted that such a situation is typical for Schrödinger operators with
point interactions and for cases where singular elements ψ j in (3) possess the
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
684 S. O. KUZHEL’, L. V. MATSYUK
homogeneity property with respect to the scaling transformations in L p
2 R( ) (see,
e.g., [10, 11]).
It has been established in [10, 12] that the restrictions imposed above on A0 and
Asym are sufficient for the applicability of the well-developed methods of the Lax –
Phillips scattering approach to the spectral analysis of AT .
Note also that the case n < m can be reduced to the case n = m by the supplement
of elements ψ j of the basis ψ j j n
m{ } = +1
of ker *B I2 +( ) � H with zero entries bij
in (3). Thus, in what follows, without loss of generality, we suppose that n = m and,
hence, Asym = B2 and H = ker *B I2 +( ).
Assume that − ∈ ( )1 ρ AT and denote
C A I B B I0 0
1 1
: *= +( ) − +( )− −
, C A I A IT T:= +( ) − +( )− −1
0
1. (19)
Since A0 , B B* , and AT are self-adjoint extensions of B2 , the operators C0
and CT are self-adjoint in the Hilbert space H. Moreover, taking (18) into account
and using Lemma 3.5 in [12] (Chapter 4), we get that the spectrum σ( )C0 of C0 is a
pure point (i.e., σ( )C0 = σ p C( )0 ) and it may consists of only points 0 and 1 / 2.
Theorem 2. For any self-adjoint operator realization AT of (3) defined by (7)
the following statements are true:
1. The point spectrum σρ( )AT has empty intersection with R+ .
2. If AT is nonnegative, then the wave operators W A A±( )T, 0 exist and are
unitary operators in �.
3. For the case of finite rank singular perturbations (n < ∞), AT is non-
negative if and only if ker I RT+( ) = {0} and
0 ≤ C0 – T I RT+( )−1 ≤ 1
2
I .
Proof. Statement 1 follows from Corollary 3.3 in [12] (Chapter 4) and statement 2
is a particular case of Proposition 2 in [10].
Let us prove statement 3. Recalling the well-known result [13] on extremal
properties of the Friedrichs B B* and Krein – von Neumann BB* extensions of B2 ,
we arrive at the conclusion that AT is nonnegative if and only if − ∈ ( )1 ρ AT and
B B I* +( )−1
≤ A IT +( )−1 ≤ BB I* +( )−1
.
Using (19), we rewrite this relation as follows:
0 ≤ CT + C0 ≤ CN = BB I* +( )−1
– B B I* +( )−1
.
It follows from Lemma 3.5 in [12] (Chapter 4) that CN = 1
2
I . Hence,
AT ≥ 0 ⇔ − ∈ ( )1 ρ AT and 0 ≤ CT + C0 ≤ 1
2
I . (20)
Let us show that conditions − ∈ ( )1 ρ AT and ker I RT+( ) = {0} are equivalent.
Since AT is a finite rank self-adjoint extension of Asym, condition − ∈ ( )1 σ AT is
equivalent to the existence of an element f A∈D H( )T ∩ . By virtue of (13), Γ̂0 f ≠
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
ON AN APPLICATION OF THE LAX – PHILLIPS SCATTERING APPROACH IN THEORY … 685
≠ 0 and Γ̂1 f = 0. Using (14) and (16), it is easy to establish that the existence of such
f means that Γ̂0 f ∈ ker I T R+( ) and, hence, ker I RT+( ) also is a nontrivial
subspace of �. Thus, − ∈ ( )1 ρ AT ⇔ ker I RT+( ) = {0}.
To calculate CT in (20), we observe that condition − ∈ ( )1 ρ AT is equivalent (see,
e.g., [7]) to the presentation of D ( )AT in the form
D ( )AT = f A C f f∈ ( ) ={ }D sym
* ˆ ˆ
TΓ Γ1 0 .
Comparing this relation with (16) and taking Lemma 1 into account, we get CT =
= − +( )−T I RT 1. Substituting the obtained expression into (20), we establish
statement 3.
Theorem 2 is proved.
Remark 2. Another description of nonnegative self-adjoint extensions of a
nonnegative symmetric operator has been obtained recently in [14].
4. Scattering matrices. Since A0 has a Lebesgue spectrum on R+ , there exists
an isometric mapping � : � →onto
L N2 R+( ), (N is an auxiliary Hilbert space and its
dimension is equal to the multiplicity m of σ( )A0 ) such that
�A u0( )( )δ = δ δ2 �u( )( ), δ > 0, ∀ ∈u AD ( )0 .
The mapping � determines a modified spectral representation of the unperturbed
operator A0 in which the action of A0 corresponds to the multiplication by δ2 in
L N2 R+( ), . This representation is determined uniquely up to isometrics of N. Since
the dimensionalities of N and H coincide and are equal to m, without loss of
generality, we can choose N = H.
Let us consider a nonnegative operator realization AT of (3) defined by (16). By
Theorem 2, the wave operators W A A±( )T, 0 are complete and the image
� � �A A A AS
T T, ,:
0 0
1
( ) ( )
−=
of the scattering operator S A AT , 0( ) is a unitary operator in the modified spectral
representation L2 R+( ), H .
Denote JA0
= P Cker 0
– P C Iker /0 2−( ) , where PM is the orthogonal projector onto
M in H and C0 is defined by (19). It has been proved [10] that the operator
� A AT , 0( ) coincides with an operator of multiplication by the boundary value*
� A AT , ( )
0( ) δ of the contraction operator-valued function
� A AT , ( )
0( ) λ = J I i C I i CA0
2 1 2 1 1− −[ ] − +[ ]−( ) ( )λ λ , λ ∈ +C (21)
(here C = A IT +( )−1 – B B I* +( )−1
) analytic in the upper half-plane.
In the case of finite rank singular perturbations (n < ∞), Theorem 2 yields that C =
= C0 – T I RT+( )−1. Thus, formula (21) provides a representation of the analytic
continuation of the scattering matrix
� A AT , ( )
0( ) δ in terms of coefficients tij of the
singular perturbation. This formula becomes especially simple if A0 coincides with
the Friedrichs B B* or with the Krein – von Neumann BB* extensions of Asym. In
* In the sense of strong convergence.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
686 S. O. KUZHEL’, L. V. MATSYUK
particular, if A0 = B B* , then JA0
= I, C0 = 0 and, after simple transformations, we
get
�
A B BT , * ( )( ) λ = I R i I T I R i I T+ + −( )[ ] + + +( )[ ]−2 1 2 1 1( ) ( )λ λ , λ ∈ +C .
Similarly, if A0 = BB*, then JA0
= – I, C0 = I / 2 and
�
A BBT , * ( )( ) λ = i I i R i I T i I i R i I Tλ λ λ λ λ λ+ + −( )[ ] + − +( )[ ]−2 1 2 1 1( ) ( ) .
5. One-dimensional Schrödinger operator with symmetric zero-range
potentials. A one-dimensional Schrödinger operator corresponding to a general zero-
range symmetric potential at the point x = 0 can be given by the expression
A0 + a δ δ, ⋅ + b ′ ⋅δ δ, + c δ δ, ⋅ ′ + d ′ ⋅ ′δ δ, , (22)
where A0 = − d
d x
2
2 D( )A W0 2
2= ( )( )R acts in � = L2 R( ) , δ ′ is the derivative of the
Dirac δ-function (with support at 0), the parameters a, d are real, and b = −c .
In this case, the singular elements ψ1 = 2δ and ψ2 = 2δ ′ form an orthonormal
system in �−2 0( )A = W2
2− ( )R and the functions
A0
1
1+( )−I ψ = h x1( ) =
e x
e x
x
x
− >
<
, ,
, ,
0
0
A0
1
2+( )−I ψ = h x2( ) =
− >
<
−e x
e x
x
x
, ,
, ,
0
0
form an orthonormal basis of H = ker *A Isym +( ) , where Asym
* = − d
d x
2
2 , D Asym
*( ) =
= W2
2 0− ( )R \{ } and Asym = − d
d x
2
2 , D Asym( ) = {u x( ) ∈ W2
2
R( ) | u( )0 = ′u ( )0 = 0}.
Representing (22) in the form (3), we get
− d
d x
2
2 +
i j
n
ij j it
,
,
=
∑ ⋅
1
ψ ψ , (23)
where coefficients t a
11 4
= , t b
12 4
= , t c
21 4
= , t d
22 4
= form the Hermitian matrix T =
= tij i j{ } =, 1
2
.
To obtain the regularization AT of (23) it suffices to extend the distributions δ
and δ ′ onto W2
2 0R \{ }( ) . The most reasonable way (based on preserving of initial
homogeneity of δ and δ ′ with respect to scaling transformations, see, for details, [2])
leads to the following definition:
δex,
( ) ( )
f
f f= + + −0 0
2
, ′ = − ′ + + ′ −δex,
( ) ( )
f
f f0 0
2
(24)
for all f x( ) ∈ W2
2 0R \{ }( ) . In this case, applying Theorem 1, we immediately obtain
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ON AN APPLICATION OF THE LAX – PHILLIPS SCATTERING APPROACH IN THEORY … 687
the following description of self-adjoint operator realizations AT of (23) that has been
obtained for the first time in [15]: AT = − ( )
d
d x A
2
2 �D T
,
D AT( ) = f x W f f( ) \{ }∈ ( ) ={ }2
2
0 10R TΓ Γ , T = 1
4
a b
b d
, (25)
where
Γ0 f =
f f
f f
( ) ( )
( ) ( )
+ + −
− ′ + − ′ −
0 0
0 0
, Γ1 f = 1
2
0 0
0 0
′ + − ′ −
+ − −
f f
f f
( ) ( )
( ) ( )
.
Let us consider the following simple maximal symmetric operator B in the space
L2 R( ):
B i x d
d x
= ( )sign , D B( ) = u x W u u( ) \{ } ( ) ( )∈ ( ) − = + ={ }2
1 0 0 0 0R .
It is easy to verify that A0 satisfies (18) and Asym = B2 for such a choice of B.
Thus, we can apply the Lax – Phillips scattering approach to the investigation of
operator realizations of (22). In our case, R =
2 0
0 2−
and the matrix
decomposition C0 (with respect to the basis h xi i( ){ } =1
2 ) of the operator C0 defined
by (19) has the form C0 =
1
2
0
0 0
. Using Theorem 2, it is easy to prove the
following statement.
Proposition 1. A self-adjoint operator realizations AT of (23) is nonnegative if
and only if p : = b 2 – ad + 2( )a d− + 4 ≠ 0 and
0 ≤ 1 4 2 2
2 4 2p
d b
b p a
− −
− − −
≤
1 0
0 1
.
In this case the spectrum σ AT( ) is Lebesgue and it coincides with 0, ∞[ ). In the
opposite case, AT possesses at least one negative eigenvalue.
By virtue of the definition of C0 , the matrix decomposition JA0
of JA0
has the
form JA0
=
−
1 0
0 1
. Taking this fact into account and going over to the matrix
decomposition in (21), we get that the matrix decomposition of the analytic
continuation
� A AT , ( )
0( ) λ of the scattering matrix into the upper half-plane has the
form
S
TA A, ( )
0( ) λ =
=
− + −( ) −( )
− −( ) − −( )
− +( ) − +( )
− +( ) − +( )
−p i i
i p i
p i i
i p i
1 1
1 1
1 1
1 1
11 12
21 22
11 12
21 22
1λ α λ α
λ α λ α
λ α λ α
λ α λ α
,
where α11 = 4 – 2d, α12 = – 2b, α21 = −2b , α22 = p – 4 – 2a.
1. Berezansky Yu. M. Expansion in eigenfunctions of self-adjoint operators. – Kiev: Naukova Dumka,
1965 (in Russian). Engl. transl.: Providence, R. I.: AMS, 1968.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 5
688 S. O. KUZHEL’, L. V. MATSYUK
2. Albeverio S., Kurasov P. Singular perturbations of differential operators // Solvable Schrödinger
type operators: Lect. Notes Ser. – Cambridge: Cambridge Univ. Press, 2000. – 271.
3. Adamyan V., Pavlov B. Zero-radius potentials and M. G. Krein’s formula for generalized
resolvents // Zap. Nauchn. Sem. LOMI. – 1986. – 149. – P. 7 – 23.
4. Albeverio S., Kurasov P. Finite rank perturbations and distribution theory // Proc. Amer. Math.
Soc. – 1999. – 127, # 4. – P. 1151 – 1161.
5. Arlinskii Yu. M., Tsekanovskii E. R. Some remarks on singular perturbations of self-adjoint
operators // Meth. Funct. Anal. and Top. – 2003. – 9. – P. 287 – 308.
6. Berezansky Yu. M., Us G. F., Sheftel Z. G. Functional analysis. – Kiev: Vyshcha Shkola, 1990
(in Russian). Engl. transl.: Basel: Birkhäuser, 1996.
7. Gorbachuk M. L., Gorbachuk V. I. Boundary-value problems for operator-differential equations. –
Kiev: Naukova Dumka, 1984 (in Russian). Engl. transl.: Dordrecht: Kluwer, 1991.
8. Albeverio S., Kuzhel S. η-Hermitian operators and previously unnoticed symmetries in the theory
of singular perturbations. – Bonn, 2004. – 26 p. – (Preprint / Univ. Bonn).
9. Kuzhel S. On the determination of free evolution in the Lax – Phillips scattering scheme for
second-order operator-differential equations // Math. Notes. – 2000. – 68. – P. 724 – 729.
10. Kuzhel S. On elements of scattering theory for abstract Schrödinger equation Lax – Phillips
approach // Meth. Funct. Anal. and Top. – 2001. – 7, # 2. – P. 13 – 22.
11. Kuzhel S., Moskalyova Yu. The Lax – Phillips scattering approach and singular perturbations of
Schrödinger operator homogeneous with respect to scaling transformations (submitted to J. Math.
Kyoto Univ.).
12. Kuzhel A., Kuzhel S. Regular extensions of Hermitian operators. – Dordrecht: VSP, 1998.
13. Krein M. G. Theory of self-adjoint extensions of semibounded Hermitian operators and its
applications. I // Math. Trans. – 1947. – 20. – P. 431 – 495.
14. Arlinskii Yu. M., Tsekanovskii E. R. On the theory of nonnegative self-adjoint extensions of a
nonnegative symmetric operator // Repts Nat. Acad. Sci. Ukraine. – 2002. – # 11. – P. 30 – 37.
15. Albeverio S., Nizhnik L. A Schrödinger operator with a δ′-interaction on a Cantor set and Krein –
Feller operator. – Bonn, 2003. – 16 p. – (Preprint / Univ. Bonn, # 99).
Received 07.02.2005
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