On infinite-rank singular perturbations of the Schrödinger operator
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nasplib_isofts_kiev_ua-123456789-1644982025-02-09T17:42:10Z On infinite-rank singular perturbations of the Schrödinger operator Про сингулярні збурення оператора Шредінгера нескінченного рангу Kuzhel’, S. Vavrykovych, L. Статті Supported by DFFD of Ukraine (project 14.01/003). 2008 Article On infinite-rank singular perturbations of the Schrödinger operator / S. Kuzhel’, L. Vavrykovych // Український математичний журнал. — 2008. — Т. 60, № 4. — С. 487–496. — Бібліогр.: 17 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164498 519.21 en Український математичний журнал application/pdf Інститут математики НАН України |
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On infinite-rank singular perturbations of the Schrödinger operator |
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On infinite-rank singular perturbations of the Schrödinger operator |
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On infinite-rank singular perturbations of the Schrödinger operator |
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On infinite-rank singular perturbations of the Schrödinger operator |
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On infinite-rank singular perturbations of the Schrödinger operator |
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on infinite-rank singular perturbations of the schrödinger operator |
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On infinite-rank singular perturbations of the Schrödinger operator / S. Kuzhel’, L. Vavrykovych // Український математичний журнал. — 2008. — Т. 60, № 4. — С. 487–496. — Бібліогр.: 17 назв. — англ. |
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Український математичний журнал |
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AT kuzhels oninfiniteranksingularperturbationsoftheschrodingeroperator AT vavrykovychl oninfiniteranksingularperturbationsoftheschrodingeroperator AT kuzhels prosingulârnízburennâoperatorašredíngeraneskínčennogorangu AT vavrykovychl prosingulârnízburennâoperatorašredíngeraneskínčennogorangu |
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UDC 519.21
S. Kuzhel’* (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
L. Vavrykovych (Nizhin State Univ.)
ON INFINITE-RANK SINGULAR PERTURBATIONS
OF THE SCHRÖDINGER OPERATOR
ПРО СИНГУЛЯРНI ЗБУРЕННЯ ОПЕРАТОРА ШРЕДIНГЕРА
НЕСКIНЧЕННОГО РАНГУ
Schrödinger operators with infinite-rank singular potentials V =
∑∞
i,j=1
bij〈ψj , ·〉ψi are studied under
the condition that singular elements ψj are ξj(t)-invariant with respect to scaling transformations in R3.
Вивчається оператор Шредiнгера з сингулярними потенцiалами нескiнченного рангу V =
=
∑∞
i,j=1
bij〈ψj , ·〉ψi за умови, що сингулярнi елементи ψj є ξj(t)-iнварiантними вiдносно мас-
штабних перетворень в R3.
1. Introduction. Let −∆, D(∆) = W 2
2 (R3) be the Schrödinger operator in L2(R3)
and let U = {Ut}t∈(0,∞) be the collection of unitary operators Utf(x) = t3/2f(tx)) in
L2(R3) (so-called scaling transformations).
It is well known [1, 2] that −∆ is t−2-homogeneous with respect to U in the sense
that
Ut∆u = t−2∆Utu ∀t > 0, u ∈W 2
2 (R3). (1.1)
In other words, the set U determines the structure of a symmetry and the property
of −∆ to be t−2-homogeneous with respect to U means that −∆ possesses a symmetry
with respect to U.
Consider the heuristic expression
−∆ +
∞∑
i,j=1
bij〈ψj , ·〉ψi, ψj ∈W−2
2 (R3), bij = bji ∈ C. (1.2)
We will say that ψ ∈ W−2
2 (R3) is ξ(t)-invariant with respect to U if there exists a
real function ξ(t) such that
Utψ = ξ(t)ψ ∀t > 0, (1.3)
where Ut is the continuation of Ut onto W−2
2 (R3) (see Section 2 for details).
The aim of the paper is to study self-adjoint operator realizations of (1.2) assuming
that all ψj are ξj(t)-invariant with respect to the set of scaling transformations U.
It is well known, see e.g. [1 – 4] that the Schrödinger operators perturbed by potentials
homogeneous with respect to a certain set of unitary operators play an important role in
applications to quantum mechanics. To a certain extent this generates a steady interests
to the study of self-adjoint extensions with various properties of symmetry [5 – 11].
In particular, an abstract framework to study finite rank singular perturbations with
symmetries for an arbitrary nonnegative operator was developed in [6].
*Supported by DFFD of Ukraine (project 14.01/003).
c© S. KUZHEL’, L. VAVRYKOVYCH, 2008
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4 487
488 S. KUZHEL’, L. VAVRYKOVYCH
In the present paper we generalize some results of [6] to the case of infinite rank
perturbations of the Schrödinger operator in L2(R3). In particular, the description of all
t−2-homogeneous extensions of the symmetric operator −∆sym is obtained. Another
interesting property studied here is the possibility to get the Friedrichs and the Krein –
von Neumann extension of −∆sym as solutions of a system of equations involving the
functions t−2 and ξ(t).
Throughout the paper D(A), R(A), and kerA denote the domain, the range, and the
null-space of a linear operator A, respectively, while A � D stands for the restriction of
A to the set D.
2. Auxiliary results. 2.1. Preliminaries. Since the Sobolev space W−2
2 (R3)
coincides with the completion of L2(R3) with respect to the norm
‖f‖W−2
2 (R3) =
∥∥(−∆ + I)−1f
∥∥ ∀f ∈ L2(R3), (2.1)
the resolvent operator (−∆+I)−1 can be continuously extended to an isometric mapping
(−∆ + I)−1 from W−2
2 (R3) onto L2(R3) (we preserve the same notation for the
extension). Hence, the relation
〈ψ, u〉 =
(
(−∆ + I)u, (−∆ + I)−1ψ
)
, u ∈W−2
2 (R3), (2.2)
enables one to identify the elements ψ ∈W−2
2 (R3) as linear functionals on W 2
2 (R3).
It follows from (1.1), (2.1) that the operators Ut ∈ U can be continuously extended
to bounded operators Ut in W−2
2 (R3) and for any ψ ∈W−2
2 (R3)
〈Utψ, u〉 = 〈ψ,U∗t u〉 = 〈ψ,U1/tu〉. (2.3)
Since the elements Ut of U have the additional multiplicative property Ut1Ut2 =
= Ut2Ut1 = Ut1t2 , relation (2.3) means that this relation holds for Ut also. But then,
equality (1.3) gives ξ(t1)ξ(t2) = ξ(t1t2) (ti > 0) that is possible only if ξ(t) = 0 or
ξ(t) = t−α (α ∈ R) [12] (Chap. IV). Hence, if an element ψ ∈ W−2
2 (R3) is ξ(t)-
invariant with respect to U, then ξ(t) = t−α (α ∈ R) (the case ξ(t) = 0 is impossible
because Ut has inverse).
2.2. Operator realizations of (2.1) in L2(R3). Let us consider (1.2) assuming that
all elements ψj are t−α-invariant with respect to U. This means that all elements of the
linear span X of {ψj}∞j=1 also satisfy (1.3) with ξ(t) = t−α. Obviously, the same is true
for the closure X of X in W−2
2 (R3). Hence, if ψ ∈ X , then Utψ = t−αψ. This implies
ψ ∈ W−2
2 (R3) \ L2(R3) (since the operator Ut = Ut � L2(R3) is unitary in L2(R3).
Thus X ∩ L2(R3) = {0}.
In that case, the perturbation V =
∑n
i,j=1
bij〈ψj , ·〉ψi turns out to be singular and
the formula
−∆sym = −∆ � D(−∆sym),
D(−∆sym) =
{
u ∈W−2
2 (R3) : 〈ψj , u〉 = 0, j ∈ N
} (2.4)
determines a closed densely defined symmetric operator in L2(R3).
Following [1] a self-adjoint operator realization −∆̃ of (1.2) in L2(R3) are defined
by
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
ON INFINITE-RANK SINGULAR PERTURBATIONS OF THE SCHRÖDINGER OPERATOR 489
−∆̃ = −∆R � D(−∆̃), D(−∆̃) =
{
f ∈ D(−∆∗
sym) : −∆Rf ∈ L2(R3)
}
, (2.5)
where
−∆R = −∆ +
∞∑
i,j=1
bij〈ψex
j , ·〉ψi (2.6)
is seen as a regularization of (1.2) defined onD(−∆∗
sym). Here 〈ψex
j , ·〉 denote extensions
of linear functionals 〈ψj , ·〉 onto D(−∆∗
sym).
In what follows, the elements {ψj}∞j=1 in (1.2) are supposed to be a Riesz basis of
the subspace X ⊂ W−2
2 (R3). Then the vectors hj = (−∆ + I)−1ψj , j ∈ N, form a
Riesz basis of the defect subspace H = ker(−∆∗
sym + I) ⊂ L2(R3) of the symmetric
operator −∆sym (see (2.2) and (2.4)).
Let {ej}∞1 be the canonical basis of the Hilbert space l2 (i.e., ej = (. . . , 0, 1, 0, . . .),
where 1 occurs on the j th place only). Putting Ψej := ψj , j ∈ N, we define an injective
linear mapping Ψ: l2 →W−2
2 (R3) such that R(Ψ) = X .
Let Ψ∗ : W 2
2 (R3) → Cn be the adjoint operator of Ψ (i.e., 〈u,Ψd〉 = (Ψ∗u, d)l2
∀u ∈W 2
2 (R3) ∀d ∈ l2). It is easy to see that
Ψ∗u =
(
〈ψ1, u〉, . . . , 〈ψj , u〉, . . .
)
∀u ∈W 2
2 (R3). (2.7)
It follows from (2.7) that the extended functionals 〈ψex
j , ·〉 in (2.6) are completely
defined by an extension Ψ∗
R of Ψ∗ onto D(−∆∗
sym), i.e.,
Ψ∗
Rf =
(
〈ψex
1 , f〉, . . . , 〈ψex
j , f〉, . . .
)
∀f ∈ D(−∆∗
sym). (2.8)
Since D(−∆∗
sym) = W 2
2 (R3)+̇H, where H = ker(−∆∗
sym + I) the formula (2.8) can
be rewritten as
Ψ∗
Rf = Ψ∗
R
(
u+
∞∑
k=1
dkhk
)
= Ψ∗u+Rd ∀f ∈ D(−∆∗
sym), (2.9)
where u ∈ W 2
2 (R3), d = (d1, d2, . . .) ∈ l2, and R is an arbitrary bounded operator
acting in l2.
Using the definition of Ψ and Ψ∗
R, the regularization (2.6) takes the form
−∆R = −∆ + ΨBΨ∗
R, (2.10)
where the self-adjoint operator B is defined in l2 by the infinite-dimensional Hermitian
matrix B = ‖bij‖∞i,j=1.
2.3. Description in terms of boundary triplets. The formulas (2.5) and (2.10) do
not provide an explicit description of operator realizations −∆̃ of (1.2) through the
parameters bij of the singular perturbation V. To get the required description the method
of boundary triplets is now incorporated.
Definition 2.1 [13]. Let Asym be a closed densely defined symmetric operator in
a Hilbert space H. A triplet (N,Γ0,Γ1), where N is an auxiliary Hilbert space and
Γ0, Γ1 are linear mappings of D(A∗sym) into N, is called a boundary triplet of A∗sym if
(A∗symf, g)− (f,A∗symg) = (Γ1f,Γ0g)N − (Γ0f,Γ1g)N for all f, g ∈ D(A∗sym) and the
mapping (Γ0,Γ1) : D(A∗sym) → N ⊕N is surjective.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
490 S. KUZHEL’, L. VAVRYKOVYCH
The next two results (Lemma 2.1 and Theorem 2.3) are some ‘folk-lore’ of the
extension theory (see, e.g., [14 – 16]). Basically their proofs are the same as in [14],
where the case of finite defect numbers has been considered.
Lemma 2.1. Let R in (2.9) be a bounded self-adjoint operator in l2. Then the
triplet (l2,Γ0,Γ1), where the linear operators Γi : D(−∆∗
sym) → l2 are defined by the
formulas
Γ0f = Ψ∗
Rf, Γ1f = −Ψ−1(−∆ + I)h, (2.11)
(where f = u+ h, u ∈W 2
2 (R3), h ∈ H) is a boundary triplet of −∆∗
sym.
Theorem 2.1. The operator realization −∆̃ of (1.2) defined by (2.5) and (2.10)
is a self-adjoint extension of −∆sym which coincides with the operator
−∆B = −∆∗
sym � D(∆B), D(∆B) =
{
f ∈ D(∆∗
sym) : BΓ0f = Γ1f
}
, (2.12)
where Γi are defined by (2.11) and a self-adjoint operator B is defined in l2 by the
Hermitian matrix B = ‖bij‖∞i,j=1.
3. tα-Invariant singular perturbations of −∆. 3.1. Description of all tα-
invariant elements. An additional study of Ut allows one to restrict the variation of the
parameter α for t−α-invariant elements.
Theorem 3.1 [6]. t−α-Invariant elements ψ ∈ W−2
2 (R3) with respect to scaling
transformations exist if and only if 0 < α < 2.
Proof. For the convenience of the reader we briefly outline the principal stages of
the proof. Consider a family of self-adjoint operators on L2(R3)
Gt = (−t−2∆ + I)(−∆ + I)−1, t > 0. (3.1)
It follows from (1.1), (2.2), and (2.3) that for all u ∈W 2
2 (R3)
〈Utψ, u〉 =
(
(−∆ + I)U1/tu, h
)
=
(
U1/t(−t−2∆ + I)u, h
)
=
=
(
(−t−2∆ + I)u, Uth
)
=
(
Gt(−∆ + I)u, Uth
)
=
(
(−∆ + I)u,GtUth
)
, (3.2)
where h = (−∆ + I)−1ψ. On the other hand, if ψ is t−α-invariant, then
〈Utψ, u〉 = t−α〈ψ, u〉 =
(
(−∆ + I)u, t−αh
)
.
Combining the obtained relation with (2.3) one gets that an element ψ is t−α-invariant
with respect to scaling transformations if and only if
GtUth = t−αh, t > 0, h = (A0 + I)−1ψ. (3.3)
The formula for Gt in (3.1) with an evident reasoning leads to the estimates
α(t)‖h‖ = α(t)‖Uth‖ < ‖GtUth‖ < β(t)‖Uth‖ = β(t)‖h‖,
where α(t) = min{1, t−2} and β(t) = max{1, t−2}. Therefore α(t) < t−α < β(t) for
all t > 0. This estimation can be satisfied for 0 < α < 2 only.
To complete the proof it suffices to construct t−α-invariant elements ψ for 0 < α < 2.
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
ON INFINITE-RANK SINGULAR PERTURBATIONS OF THE SCHRÖDINGER OPERATOR 491
Fix m(w) ∈ L2(S2), where L2(S2) is the Hilbert space of square-integrable functi-
ons on the unit sphere S2 in R3, and determine the functional ψ(m,α) ∈W−2
2 (R3) by
the formula
〈ψ(m,α), u〉 =
∫
R3
m(w)
|y|3/2−α(|y|2 + 1)
(
|y|2 + 1
)
û(y)dy
(
y = |y|w ∈ R3
)
, (3.4)
where û(y) =
1
(2π)3/2
∫
R3
eix·yu(x)dx is the Fourier transformation of u(·) ∈W 2
2 (R3).
It is easy to verify that
̂(U1/tu)(y) =
1
(2πt)3/2
∫
R3
eiy·xu(x/t)dx = Utû(y) = t3/2û(ty). (3.5)
Using (3.4) and (3.5), one obtains 〈ψ(m,α), U1/tu〉 = t−α〈ψ(m,α), u〉 for all u ∈
∈ W 2
2 (R3). By (1.3) and (2.3) this means that ψ(m,α) is t−α-invariant with respect
to U.
Theorem 3.1 is proved.
The next statement describes all t−α-invariant elements for a fixed α ∈ (0, 2).
Proposition 3.1. An element ψ ∈ W−2
2 (R3) is t−α-invariant with respect to
scaling transformations if and only if ψ = ψ(m,α) where ψ(m,α) is defined by (3.4).
Proof. Let ψ ∈ W−2
2 (R3) be t−α-invariant with respect to U = {Ut}t∈(0,∞). This
means that (3.3) holds for h = (A0 + I)−1ψ. Using (3.5) one can rewrite (3.3) as
t−2|y|2 + 1
|y|2 + 1
t−3/2ĥ
(y
t
)
= t−αĥ(y), t > 0, (3.6)
where the equality is understood in the sense of L2(R3). Setting t = |y|, (w = y/|y|)
one derives that (3.6) holds if and only if
ĥ(y) =
m(w)
|y|3/2−α(|y|2 + 1)
, m(w) = 2ĥ(w), (3.7)
where m(w)∈L2(S2) (because ĥ(w) ∈ L2(R3)). Combining (3.7) with (2.2) and (3.4)
one concludes that ψ = ψ(m,α).
Proposition 3.1 is proved.
Remark 3.1. Proposition 3.1 generalizes Proposition 3.1 in [9] where the case
α = 3/2 was considered.
3.2. t−2-Homogeneous extensions of −∆sym transversal to −∆. Denote
−∆R = −∆∗
sym � ker Γ0, where Γ0 is defined by (2.11). Since (l2,Γ0,Γ1) is a
boundary triplet of −∆∗
sym and the initial operator −∆ coincides with −∆∗
sym � ker Γ1,
one concludes that −∆R and −∆ are transversal self-adjoint extensions of −∆sym, i.e.,
D(−∆R) ∩ D(−∆) = D(−∆sym) and D(−∆R) +D(−∆) = D(−∆∗
sym) [13].
In view of (1.3) and (2.3) the t−αj -invariance of an element ψj in (1.2) is equivalent
to the relation
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
492 S. KUZHEL’, L. VAVRYKOVYCH
t−αj 〈ψj , u〉 = 〈ψj , U1/tu〉 ∀u ∈W 2
2 (R3), t > 0. (3.8)
It turns out that the preservation of (3.8) for the extended functionals 〈ψex
j , ·〉 is equivalent
to the t−2-homogeneity of −∆R.
Proposition 3.2. Let ψex
j be defined by (2.8). Then the relations
t−αj 〈ψex
j , f〉 = 〈ψex
j , U1/tf〉 ∀j ∈ N ∀t > 0 (3.9)
hold for all f ∈ D(−∆∗
sym) if and only if the operator −∆R is t−2-homogeneous with
respect to U = {Ut}t∈(0,∞).
Proof. It follows from (2.2) and (2.3) that
〈ψj , Utu〉 = 〈U1/tψj , u〉 = tαj 〈ψj , u〉 = 0
for every u ∈ D(−∆sym). Thus Ut : D(−∆sym) → D(−∆sym) and, by (1.1) and (2.4),
the symmetric operator −∆sym is t−2-homogeneous: Ut∆sym = t−2∆symUt. But then
the adjoint −∆∗
sym of −∆sym is also t−2-homogeneous. This means that a self-adjoint
extension −∆̃ of −∆sym is t−2-homogeneous with respect to U = {Ut}t∈(0,∞) if and
only if UtD(−∆̃) = D(−∆̃) for all t > 0. Since UtU1/t = I the last equality is
equivalent to the inclusion
UtD(−∆̃) ⊂ D(−∆̃) ∀t > 0. (3.10)
Using (2.8) one can rewrite relations (3.9) as follows:
Ξ(t)Ψ∗
Rf = Ψ∗
RU1/tf ∀f ∈ D(−∆∗
sym) ∀t > 0, (3.11)
where a bounded invertible operator Ξ(t) in l2 is defined by the formulas
Ξ(t)ej = t−αjej , j ∈ N. (3.12)
Since D(−∆0) = ker Γ0 = kerΨ∗
R, (3.11) implies that D(−∆R) satisfies (3.10).
Thus −∆R is t−2-homogeneous with respect to U.
Conversely, assume that −∆R is t−2-homogeneous. According to (2.9) and (3.10)
this is equivalent to the relation
Ψ∗
RU1/tf = 0 ∀f = u+
∞∑
j=1
djhj ∈ D(−∆R) ∀t > 0. (3.13)
Let us study (3.13) more detail. Using (3.1) and (3.3) it is seen that
U1/thj = t−2G1/tU1/thj + (I − t−2G1/t)U1/thj =
=
t−2
t−αj
hj + (1− t−2)(−∆ + I)−1U1/thj ,
where hj = (−∆ + I)−1ψj . Therefore,
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 4
ON INFINITE-RANK SINGULAR PERTURBATIONS OF THE SCHRÖDINGER OPERATOR 493
U1/tf = v +
∞∑
j=1
tαj−2djhj , (3.14)
where the element v = U1/tu + (1 − t−2)(−∆ + I)−1U1/t
∑∞
i=1
djhj belongs to
D(−∆). Substituting the obtained expression for U1/tf into (3.13) and using (2.9) one
gets
Ψ∗U1/tu+ (1− t−2)Ψ∗(−∆ + I)−1U1/t
∞∑
j=1
djhj + t−2RΞ−1(t)d = 0. (3.15)
Here Ψ∗U1/tu = Ξ(t)Ψ∗u by (2.3) and (2.7). Moreover Ψ∗u = −Rd since the vector
f = u+
∑∞
j=1
djhj belongs to D(−∆R) = kerΨ∗
R. Thus Ψ∗U1/tu = −Ξ(t)Rd.
On the other hand, employing (2.2) and (2.7), one gets
Ψ∗(−∆ + I)−1U1/t
∞∑
j=1
djhj = Ktd,
where Kt is a bounded operator in l2 that is defined by the infinite-dimensional matrix
K = ‖kij‖∞i,j=1, kij = (hj , Uthi) with respect to the canonical basis {ej}∞1 (see
Subsection 2.2). The obtained relations allow one to rewrite (3.15) as follows:[
− Ξ(t)R+ t−2RΞ−1(t) + (1− t−2)Kt
]
d = 0 ∀t > 0,
where d is an arbitrary element from l2 (it follows from the presentation f ∈ D(−∆R)
in (3.13) and the transversality −∆ and −∆R with respect to −∆sym). Therefore, the
t−2-homogeneity of −∆R is equivalent to the operator equality in l2:
Ξ(t)R− t−2RΞ−1(t) = (1− t−2)Kt ∀t > 0. (3.16)
Finally, employing (2.9) and (3.15) it is easy to see that equality (3.16) is equivalent
to (3.11). Therefore, the extended functionals 〈ψex
j , ·〉 satisfy (3.9).
Proposition 3.2 is proved.
Remark 3.2. The result similar to Proposition 3.2 was proved in [6] for the case of
finite rank perturbations of a self-adjoint operator acting in an abstract Hilbert space H.
Theorem 3.2. Let αj ∈ (1, 2) for any t−αj -invariant element ψj in the definiti-
on (2.4) of −∆sym. Then there exists a unique t−2-homogeneous self-adjoint extension
of −∆sym transversal to −∆.
Proof. It follows from the general theory of boundary triplets [13, 17] that an
arbitrary self-adjoint extension −∆̃ of −∆sym transversal to −∆ coincides with −∆R
for a certain choice of a bounded self-adjoint operator R in l2. As was shown in the proof
of Proposition 3.2, −∆R is t−2-homogeneous with respect to scaling transformations if
and only if the operator R is a solution of (3.16) that does not depend on t > 0. Using
(3.12) and the definition of Kt one can rewrite (3.16) componentwise as follows:
(t−αi − tαj−2)rij = (1− t−2)(hj , Uthi), R = ‖rij‖∞i,j=1 (3.17)
where the infinite-dimensional matrix R is the matrix presentation of R with respect to
the canonical basis {ej}∞1 .
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494 S. KUZHEL’, L. VAVRYKOVYCH
Let us calculate (hj , Uthi) in (3.17). According to Proposition 3.1, t−αj -invariant
elements ψj in (1.2) have the form ψj = ψ(mj , αj), where mj(·) ∈ L2(S2) and
elements hj = (−∆ + I)−1ψ(mj , αj) are defined by (3.7).
It follows from (3.5) that
Ûthi(y) = t−3/2ĥ
(y
t
)
= t2−αi
mi(w)
|y|3/2−αi(|y|2 + t2)
.
Hence,
(hj , Uthi) = t2−αi
∫
R3
mj(w)mi(w)
|y|3−(αj+αi)(|y|2 + t2)(|y|2 + 1)
dy =
= (mj ,mi)L2
∞∫
0
t2−αi
|y|1−(αi+αj)(|y|2 + t2)(|y|2 + 1)
d|y| =
= cij
tαj − t2−αi
t2 − 1
(mj ,mi)L2 ,
where cij =
∫ ∞
0
|y|3−(αi+αj)
|y|2 + 1
d|y| and (mi,mj)L2 =
∫
S2
mi(w)mj(w)dw is the scalar
product in L2(S2). Substituting the obtained expression for (hj , Uthi) into (3.17) one
finds rij = −cij(mj ,mi)L2 . The matrix R = ‖rij‖∞i,j=1 determined in such a way is
the matrix representation of a unique solution R of (3.16) that does not depend on t > 0.
Theorem 3.2 is proved.
3.3. The Friedrichs and Krein – von Neumann extensions. As was shown in the
proof of Proposition 3.2, the symmetric operator −∆sym is t−2-homogeneous with
respect to scaling transformations. According to general results obtained in [6, 10],
the Friedrichs −∆F and the Krein – von Neumann −∆N extensions of −∆sym are also
t−2-homogeneous.
Theorem 3.3. Let αj ∈ (1, 2) for any t−αj -invariant element ψj in the definiti-
on (2.4) of −∆sym and let the spectrum of −∆R, where R is a unique solution of (3.16)
does not cover real line R. Then the Krein – von Neumann extension −∆N coincides
with −∆R and the Friedrichs extension −∆F coincides with the initial operator −∆.
Proof. A simple analysis of (3.7) shows that hj ∈ L2(R3)\W 1
2 (R3) for 1 ≤ α < 2,
i.e., singular elements ψj in (2.4) form a W−1
2 (R3)-independent system. This means
that the initial operator −∆ coincides with the Friedrichs extension −∆F .
Since −∆R is t−2-homogeneous and σ(−∆R) 6= R, the equality
Ut(−∆R − λI) = t−2(−∆R − t2λI)Ut, t > 0,
means that the spectrum of −∆R is nonnegative. Therefore, −∆R is a nonnegative
extension of−∆sym transversal to the Friedrichs extension−∆. But then the Krein – von
Neumann extension −∆N is also transversal to −∆. Since −∆N is t−2-homogeneous,
Theorem 3.2 gives −∆N = −∆R that completes the proof.
3.4. t−2-Homogeneous extensions of −∆sym. Let us consider the heuristic ex-
pression (1.2), where all elements ψj are assumed to be t−α-invariant with respect to
scaling transformations, i.e., ψj = ψ(mj , α), where α ∈ (1, 2) is fixed.
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ON INFINITE-RANK SINGULAR PERTURBATIONS OF THE SCHRÖDINGER OPERATOR 495
It follows from (1.3) and (2.3) that the singular potential V =
∑∞
i,j=1
bij〈ψj , ·〉ψi
in (1.2) is t−2α-homogeneous in the sense that
UtV u = t−2αV Utu ∀u ∈W 2
2 (R3).
Hence, the initial operator−∆ and its singular perturbation V possess the homogenei-
ty property with different index of homogeneity: t−2 and t−2α, respectively. In view
of this, it is natural to expect that any self-adjoint extension −∆̃ of −∆sym having the
t−2-homogeneity property (as well as−∆ and−∆R) is closely related to−∆ and−∆R.
Let (l2,Γ0,Γ1) be a boundary triplet of −∆∗
sym defined by (2.11), where R is a
unique solution of (3.16).
Theorem 3.4. Let all elements ψj be t−α-invariant with respect to scaling trans-
formations, where α ∈ (1, 2) is fixed. Then an arbitrary t−2-homogeneous self-adjoint
extension −∆̃ of −∆sym coincides with the restriction of −∆∗
sym onto the domain
D(−∆̃) = {f ∈ D(−∆∗
sym) : (I − V )Γ0f = i(I + V )Γ1f}, (3.18)
where V is taken from the set of unitary and self-adjoint operators in l2.
Proof. If Γ0 is a boundary operator defined by (2.11), where R is a unique solution
of (3.16), then formulas (3.11) and (3.12) give
Γ0U1/tf = t−αΓ0f ∀f ∈ D(−∆∗
sym) ∀t > 0. (3.19)
On the other hand, using (3.14), one derives
Γ1U1/tf = tα−2Γ1f ∀f ∈ D(−∆∗
sym) ∀t > 0. (3.20)
It is known [13] that an arbitrary self-adjoint extension −∆̃ of −∆sym is the restri-
ction of −∆∗
sym onto the domain (3.18) where V is a unitary operator in l2. By (3.19),
(3.20),
U1/tD(−∆̃) = {f ∈ D(−∆∗
sym) : tα(I − V )Γ0f = it2−α(I + V )Γ1f}. (3.21)
The operator −∆̃ is t−2-homogeneous if and only if its domain D(−∆̃) satisfies
(3.10). Comparing (3.18) and (3.21) and taking into account that α > 1, one concludes
that (3.10) holds if and only if Γ0D(−∆̃) = ker(I − V ) and Γ1D(−∆̃) = ker(I + V ).
These relations give
ker(I − V )⊕ ker(I + V ) = l2 (3.22)
since −∆̃ is a self-adjoint operator and (l2,Γ0,Γ1) is a boundary triplet of −∆∗
sym. The
obtained identity implies that the unitary operator V also is self-adjoint.
Conversely, if V is unitary and self-adjoint, then (3.22) is satisfied. Hence, (3.10)
holds and −∆̃ is t−2-homogeneous.
Theorem 3.4 is proved.
Corollary 3.1. There are no t−2-homogeneous operators among nontrivial (6= −∆)
self-adjoint operator realizations of (1.2).
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496 S. KUZHEL’, L. VAVRYKOVYCH
Proof. According to Theorem 2.1 an operator realization −∆B of (1.2) is defined by
(2.12). It follows from (2.12) and (3.18) that B = −i(I − V )(I + V )−1. If the operator
V has the additional property (3.22) (the condition of t−2-homogeneity of −∆B), then
B = 0. Hence −∆B is t−2-homogeneous if and only if −∆B = −∆.
1. Albeverio S., Kurasov P. Singular perturbations of differential operators // Solvable Schrödinger
Type Operators (London Math. Soc. Lect. Note Ser. 271). – Cambridge: Cambridge Univ. Press,
2000.
2. Cycon H. L., Froese R. G., Kirsch W., Simon B. Schrödinger operators with applications to quantum
mechanics and global geometry. – Berlin: Springer, 1987.
3. Albeverio S., Dabrowski L., Kurasov P. Symmetries of Schrödinger operators with point interactions
// Lett. Math. Phys. – 1998. – 45. – P. 33 – 47.
4. Kiselev A. A., Pavlov B. S., Penkina N. N., Suturin M. G. Interaction symmetry in the theory of
extensions technique // Teor. i Mat. Phys. – 1992. – 91. – P. 179 – 191.
5. Bekker M. On non-densely defined invariant Hermitian contractions // Meth. Funct. Anal. and Top.
– 2007. – 13. – P. 223 – 235.
6. Hassi S., Kuzhel S. On symmetries in the theory of singular perturbations // Working Papers Univ.
Vaasa, 2006. – 29 p.; http://lipas.uwasa.fi/julkaisu/sis.html.
7. Jørgensen P. Commutators of Hamiltonian operators and nonabelian algebras // J. Math. Anal. and
Appl. – 1980. – 73. – P. 115 – 133.
8. Kochubei A. N. About symmetric operators commuting with a family of unitary operators // Funk.
Anal. i Pril. – 1979. – 13. – P. 77 – 78.
9. Kuzhel S., Moskalyova Ul. The Lax – Phillips scattering approach and singular perturbations of
Schrödinger operator homogeneous with respect to scaling transformations // J. Math. Kyoto Univ.
– 2005. – 45. – P. 265 – 286.
10. Makarov K. A., Tsekanovskii E. On µ-scale invariant operators // Meth. Funct. Anal. and Top. –
2007. – 13. – P. 181 – 186.
11. Phillips R. S. The extension of dual subspaces invariant under an algebra // Proc. Int. Symp. Linear
Spaces (Jerusalem, 1960). – Jerusalem Acad. Press, 1961. – P. 366 – 398.
12. Hille E., Phillips R. S. Functional analysis and semi-groups. – Providence: Amer. Math. Soc., 1957.
13. Gorbachuk M. L., Gorbachuk V. I. Boundary-value problems for operator-differential equations. –
Dordrecht: Kluwer, 1991.
14. Albeverio S., Kuzhel S., Nizhnik L. Singularly perturbed self-adjoint operators in scales of Hilbert
spaces // Ukr. Math. J. – 2007. – 59, № 6. – P. 723 – 744.
15. Arlinskii Yu. M., Tsekanovskii E. R. Some remarks of singular perturbations of self-adjoint operators
// Meth. Funct. Anal. and Top. – 2003. – 9. – P. 287 – 308.
16. Derkach V., Hassi S., de Snoo H. Singular perturbations of self-adjoint operators // Math. Phys.,
Anal., Geom. – 2003. – 6. – P. 349 – 384.
17. Derkach V. A., Malamud M. M. Generalized resolvents and the boundary value problems for
Hermitian operators with gaps // J. Funct. Anal. – 1991. – 95. – P. 1 – 95.
Received 26.12.07
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