Singularly perturbed periodic and semiperiodic differential operators
Qualitative and spectral properties of the form sums $$S_{±}(V) := D^{2m}_{±} + V(x),\quad m ∈ N,$$ are studied in the Hilbert space $L_2(0, 1)$. Here, $(D_{+})$ is a periodic differential operator, $(D_{-})$ is a semiperiodic differential operator, $D_{±}: u ↦ −iu′$, and $V(x)$ is an arbitrary 1-p...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509419685543936 |
|---|---|
| author | Mikhailets, V. A. Molyboga, V. M. Михайлець, В. А. Молибога, В. М. |
| author_facet | Mikhailets, V. A. Molyboga, V. M. Михайлець, В. А. Молибога, В. М. |
| author_sort | Mikhailets, V. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:58Z |
| description | Qualitative and spectral properties of the form sums
$$S_{±}(V) := D^{2m}_{±} + V(x),\quad m ∈ N,$$
are studied in the Hilbert space $L_2(0, 1)$. Here, $(D_{+})$ is a periodic differential operator, $(D_{-})$ is a semiperiodic differential operator, $D_{±}: u ↦ −iu′$, and $V(x)$ is an arbitrary 1-periodic complex-valued distribution from the Sobolev spaces $H_{per}^{−mα},\; α ∈ [0, 1]$. |
| first_indexed | 2026-03-24T02:40:48Z |
| format | Article |
| fulltext |
UDC 517.984.5
V. A. Mikhailets, V. M. Molyboga (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
SINGULARLY PERTURBED PERIODIC
AND SEMIPERIODIC DIFFERENTIAL OPERATORS ∗
SYNHULQRNO ZBURENI PERIODYÇNI
TA NAPIVPERIODYÇNI DYFERENCIAL\NI OPERATORY
Qualitative and spectral properties of the form-sums
S±(V ) := D2m
± �V (x), m ∈ N,
in the Hilbert space L2(0, 1) are studied. Here, (D+) is the periodic differential operator, (D−) is the
semiperiodic differential operator, D± : u �→ −iu′, and V (x) is a 1-periodic complex-valued distribution
in the Sobolev spaces H−mα
per , α ∈ [0, 1].
DoslidΩeno qkisni ta spektral\ni vlastyvosti form-sum
S±(V ) := D2m
± �V (x), m ∈ N,
u hil\bertovomu prostori L2(0, 1). Tut (D+) ta (D−) — periodyçnyj ta napivperiodyçnyj dyfe-
rencial\ni operatory, D± : u �→ −iu′, a V (x) — dovil\na 1-periodyçna kompleksnoznaçna uzahal\-
nena funkciq z prostoriv Soboleva H−mα
per , α ∈ [0, 1].
1. Introduction and statement of results. In this paper, we study the operators S+(V )
and S−(V ) that are not selfadjoint in general and given on the Hilbert space L2(0, 1)
by two-terms differential expressions of an even order, with a 1-periodic complex-valued
potential V (x), which is a distribution in D′
1, and periodic and semiperiodic boundary
conditions,
S±u ≡ S±(V )u := D2m
± u + V (x)u,
D± := −i d
dx
, Dom(D±) = H1
±, D2m
± := |D±|2m,
Dom(D2m
± ) = H2m
± , m ∈ N,
V (x) =
∑
k∈Z
V̂ (2k)ei 2kπx ∈ D′
1,
u ∈ Dom(S±).
Here by the H1
± ≡ H1
±[0, 1] and H2m
± ≡ H2m
± [0, 1] we denote the Sobolev spaces of
functions that are 1-periodic and 1-semiperiodic on the interval [0, 1], and D′
1 denotes the
space of 1-periodic distributions [1, p. 115].
In this paper, we give sufficient conditions for the operators S±(V ) to exist as form-
sums, conduct a detailed study of their qualitative properties, prove theorems about their
approximation and spectrum decomposition. The approximation theorem gives another
definition of the operators S±(V ) as a limit, in the generalized convergence sense [2]
(Ch. IV, § 2.6), of a sequence of operators with smooth potentials.
Earlier in [3 – 5], the authors have carried out a detailed study of the differential op-
erators L±(V ) generated on the finite interval by the same differential expressions as the
operators S±(V ) but defined on the negative Sobolev spaces H−m
± . The case m = 1 for
operators L±(V ) was treated in [6, 7] (see also closely related papers [8 – 11]).
∗ The investigation of the first author has been supported in part by the DFFD under grant 14.1/003.
c© V. A. MIKHAILETS, V. M. MOLYBOGA, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6 785
786 V. A. MIKHAILETS, V. M. MOLYBOGA
So, for an arbitrary s ∈ R, the Sobolev spaces of 1-periodic and 1-semiperiodic func-
tions or distributions are defined in a natural fashion by means of their Fourier coefficients,
Hs
+ ≡ Hs
+[0, 1] :=
{
f =
∑
k∈Z
f̂(2k)ei2kπx
∣∣∣ ‖ f ‖Hs
+
< ∞
}
,
‖ f ‖Hs
+
:=
(∑
k∈Z
〈2k〉2s
∣∣f̂(2k)
∣∣2)1/2
, 〈k〉 := 1 + |k|,
f̂(2k) :=
〈
f, ei2kπx
〉
+
, k ∈ Z,
and
Hs
− ≡ Hs
−[0, 1] :=
{
f =
∑
k∈Z
f̂(2k + 1)ei(2k+1)πx
∣∣∣ ‖ f ‖Hs
−
< ∞
}
,
‖ f ‖Hs
−
:=
(∑
k∈Z
〈2k + 1〉2s
∣∣f̂(2k + 1)
∣∣2)1/2
, 〈k〉 = 1 + |k|,
f̂(2k + 1) :=
〈
f, ei(2k+1)πx
〉
−, k ∈ Z.
By 〈·, ·〉+ and 〈·, ·〉− we denote the sesquilinear forms that define the pairing between
the dual spaces Hs
± and H−s
± with respect to the zero space L2(0, 1); these pairings are
obtained by extending the inner product in L2(0, 1) by continuity [12, p. 47],
(f, g) =
1∫
0
f(x)g(x) dx, f, g ∈ L2(0, 1).
It will be useful to notice that the two-sided scales of Sobolev spaces {Hs
±}s∈R coin-
cide up to equivalent norms with scales generated by powers of the non-negative selfad-
joint operators |D±| [13] (Ch. II, § 2.1).
The Sobolev spaces
Hs
per ≡ Hs
per[−1, 1] :=
{
f =
∑
k∈Z
f̂(k)eikπx
∣∣∣ ‖ f ‖Hs
per
< ∞
}
, s ∈ R,
of 2-periodic elements (functions or distributions) are defined in a similar way.
Now, we are ready to formulate the main results obtained in the paper. But first recall
that an operator A on a Hilbert space is said to be m-sectorial if its numerical range Θ(A),
i.e., the set
Θ(A) := (Au, u), u ∈ Dom(A), ‖u‖ = 1,
is contained in a sector of the complex plane,
Θ(A) ⊆ Sect(γ, θ),
Sect(γ, θ) :=
{
λ ∈ C |
∣∣ arg(λ− γ)
∣∣ ≤ θ
}
, 0 ≤ θ <
π
2
,
and the exterior of the sector Sect(γ, θ) belongs to the resolvent set Resol(A) of the
operator A [2] (Ch. V, § 3.10).
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
SINGULARLY PERTURBED PERIODIC AND SEMIPERIODIC DIFFERENTIAL OPERATORS 787
Theorem 1. Let a 1-periodic complex-valued distribution V(x) be in the space
H−m
+ . Then the operators S±(V ) are well defined on the Hilbert space L2(0, 1) as m-
sectorial operators — form-sums,
S±(V ) = D2m
± � V (x),
associated with densely defined, closed, sectorial sesquilinear forms defined on L2(0, 1)
by
tS± [u, v] ≡ t±[u, v] :=
〈
D2m
± u, v
〉
± +
〈
V (x)u, v
〉
±, Dom(tS±) = Hm
± ,
and act on the dense domains
Dom(S±) =
{
u ∈ Hm
±
∣∣D2m
± u + V (x)u ∈ L2(0, 1)
}
as
S±(V )u = D2m
± u + V (x)u, u ∈ Dom(S±).
Let us remark that, in virtue of the convolution lemma (see Lemma 1 below), a 1-
periodic complex-valued distribution V (x) ∈ H−m
+ defines, on the Hilbert spaceL2(0, 1),
two sesquilinear forms,
t+V [u, v] :=
〈
V (x) · u, v
〉
+
, u, v ∈ Hm
+ ,
t−V [u, v] :=
〈
V (x) · u, v
〉
−, u, v ∈ Hm
− ,
where V (x) ·u denotes the formal product, which converges in the Sobolev spaces H−m
± ,
of the Fourier series of the distribution V (x) ∈ H−m
+ and the function u ∈ Hm
± .
If the distribution V (x) has additional smoothness in the scale {Hs
±}s∈R of the Hilbert
spaces, then functions in the domains of the operators S±(V ) have an additional re-
gularity.
Theorem 2. Let V (x) ∈ H−mα
+ , α ∈ [0, 1]. Then the inclusion
Dom(S±) ⊆ H
m(2−α)
±
holds.
In the case α �= 0, i.e., for
V (x) ∈ H−mα
+ , α ∈ (0, 1],
the question about locality of the operators S±(V ) is meaningful. Let us recall that an
operator A on a function space is called local if
supp(Au) ⊆ supp(u), u ∈ Dom(A).
For the Hilbert space L2(0, 1), this is equivalent to the following:
u
∣∣
(α,β)
= 0 ⇒ Au
∣∣
(α,β) = 0, u ∈ Dom(A), (α, β) ⊂ [0, 1].
Theorem 3. If V (x) ∈ H−m
+ , the operators S+(V ) and S−(V ) are local.
The following theorem describes qualitative properties of the operators S±(V ).
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
788 V. A. MIKHAILETS, V. M. MOLYBOGA
Theorem 4. Let a 1-periodic complex-valued distribution V (x) be in the space
H−m
+ .
a) The operators S±(V ) are m-sectorial with respect to an arbitrary angle contain-
ing the positive half-axis.
b) The operators S±(V ) are selfadjoint if and only if the distribution V (x) is real-
valued, i.e., if
V̂ (2k) = V̂ (−2k), k ∈ Z.
c) The operators S±(V ) have discrete spectra.
The following theorem allows to give another alternative definition of the operators
S±(V ) described in Theorem 1.
Theorem 5. Let Vn(x), n ∈ N, and V (x) be defined on the space H−m
+ , and
suppose that
Vn(x)
H−m
+−→ V (x), n → ∞.
Then the operators S(n)
± ≡ S±(Vn) converge to the operators S± ≡ S±(V ) in the uni-
form resolvent convergent sense,∥∥R(λ, S(n)
± ) −R(λ, S±)
∥∥ → 0, n → ∞.
So, by virtue of Theorem 5, the operators S±(V ) can be defined as a limit of a se-
quence of the operators S(n)
± with smooth potentials Vn(x) in the generalized convergence
sense [2] (Ch. IV, § 2.6).
As an example, consider
V (x) =
∑
k∈Z
V̂ (2k)ei 2kπx ∈ H−m
+ ,
the trigonometric polynomials
Vn(x) =
∑
|k|≤n
V̂ (2k)ei 2kπx ∈ H∞
+ ,
form the necessary sequence,
Vn(x)
H−m
+−→ V (x), n → ∞,
which yields the convergence∥∥R(λ, S(n)
± ) −R(λ, S±)
∥∥ → 0, n → ∞.
Due to Theorem 5 we also have that
σ(S(n)
± ) → σ(S±), n → ∞,
where the convergence of spectra is upper semicontinuous in general [2] (Ch. IV, § 3.1)
and, for real-valued potentials, it is continuous [14] (Theorems VIII.23 and VIII.24); by
σ(S(n)
± ) and σ(S±) we denote unordered spectra of the corresponding operators.
Now, let us consider, on the Hilbert space L2(−1, 1), the m-sectorial operators —
form-sums S(V ) with 1-periodic complex-valued potentials that are distributions V (x) ∈
∈ H−m
per , i.e., V̂ (2k + 1) = 0 ∀k ∈ Z,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
SINGULARLY PERTURBED PERIODIC AND SEMIPERIODIC DIFFERENTIAL OPERATORS 789
S ≡ S(V ) := D2m � V (x),
D := −i d
dx
, Dom(D) = H1
per, D2m := |D|2m, Dom(D2m) = H2m
per ,
V (x) =
∑
k∈Z
V̂ (2k)ei 2kπx ∈ H−m
per ,
Dom(S) =
{
u ∈ Hm
per
∣∣D2mu + V (x)u ∈ L2(−1, 1)
}
, m ∈ N.
Analogs of Theorems 2 – 4 and 5 hold for the operators S(V ). In particular, they have
discrete spectra.
Let us study the structure of spectra of the operators S(V ), S+(V ), and S−(V ) in
more details.
Denote by spec(A) the discrete spectrum of the operator A, taking into account the
algebraic multiplicity of the eigenvalues that ordered lexicographically. Namely, we will
say that an eigenvalue λk precedes an eigenvalue λk+1 for k ∈ Z+ if
Reλk < Reλk+1, or Reλk = Reλk+1 and Imλk ≤ Imλk+1, k ∈ Z+.
It is easy to see that
spec(D2m) ={
0; 1, 1; 22m, 22m; . . . ; (2k − 1)2m, (2k − 1)2m; (2k)2m, (2k)2m; . . .
}
· π2m,
spec(D2m
+ ) =
{
0; 22m, 22m; . . . ; (2k)2m, (2k)2m; . . .
}
· π2m,
spec(D2m
− ) =
{
1, 1; 32m, 32m; . . . ; (2k − 1)2m, (2k − 1)2m; . . .
}
· π2m.
And thus we get
spec(D2m) = spec(D2m
+ ) � spec(D2m
− ) (the disjoint sum).
The following theorem about spectra decomposition is a non-trivial generalization of the
last equality for the perturbed m-sectorial operators — form-sums S(V ), S+(V ), and
S−(V ).
Theorem 6. Let S(V ), S+(V ) and S−(V ) be the m-sectorial operators, where the
potential V (x) is a 1-periodic complex-valued distribution from the Sobolev spaces H−m
per
and H−m
+ for the first and the second two operators, respectively. Then
S(V ) = S+(V ) ⊕ S−(V ),
and we have the decomposition
spec(S) = spec(S+) ∪ spec(S−).
A part of results are announced in [15] and contained in [16].
2. The proofs. At first, we will recall some known facts and results that will be
necessary.
Consider the Hilbert spaces of two-sided weighted sequences,
hs ≡ hs(Z; C), s ∈ R,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
790 V. A. MIKHAILETS, V. M. MOLYBOGA
hs :=
{
a = (a(k))k∈Z
| ‖a‖hs < ∞
}
,
(a, b)hs :=
∑
k∈Z
〈k〉2sa(k) b(k), 〈k〉 = 1 + |k|,
‖a‖hs :=
(∑
k∈Z
〈k〉2s|a(k)|2
)1/2
.
The Fourier transform establishes an isometric isomorphisms between the Sobolev spaces
Hs
per, H
s
± and the Hilbert spaces hs of two-sided weighted sequences,
F : Hs
per � f �→ (f̂) =
(
f̂(k)
)
k∈Z
∈ hs,
F+ : Hs
+ � f �→ (f̂) =
(
f̂(2k)
)
k∈Z
∈ hs,
F− : Hs
− � f �→ (f̂) =
(
f̂(2k + 1)
)
k∈Z
∈ hs.
This, together with the convolution lemma (see bellow), allows to give sufficient condi-
tions of existence of the formal product
V (x) · u(x) =
∑
k∈Z
∑
j∈Z
V̂ (k − j)û(j) ei kπx.
To this end, introduce in the scale of the Hilbert spaces of two-sided weighted sequences
{hs}s∈R a commutative convolution operation. For arbitrary sequences
a = (a(k))k∈Z
and b = (b(k))k∈Z
,
it is defined in a natural fashion,
(a, b) �→ a ∗ b,
(a ∗ b)(k) :=
∑
j∈Z
a(k − j)b(j).
The following known lemma (see, for example, [7], Lemma 1.5.4) holds.
Lemma 1 (Convolution lemma). Let s, r ≥ 0, and t ∈ R with t ≤ min(s, r).
(I) If s + r − t > 1/2, then the convolution map
(a, b) �→ a ∗ b
is continuous when viewed as the maps
(a) hr × hs → ht,
(b) h−t × hs → h−r.
(II) If s + r − t < 1/2, then this statement fails to hold.
2.1. Proof of Theorem 1. Due to the convolution lemma for
V (x) ∈ H−m
+ and u(x) ∈ Hm
± ,
the products V (x) · u(x) are well defined in the Sobolev spaces H−m
± . Therefore, the
sesquilinear forms
t+V [u, v] = 〈V (x)u, v〉+, Dom(t+V ) = Hm
+ ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
SINGULARLY PERTURBED PERIODIC AND SEMIPERIODIC DIFFERENTIAL OPERATORS 791
t−V [u, v] = 〈V (x)u, v〉−, Dom(t−V ) = Hm
− ,
are well defined in the Hilbert space L2(0, 1).
Further, set
τ+[u, v] := 〈D2m
+ u, v〉+, Dom(τ+) = Hm
+ ,
τ−[u, v] := 〈D2m
− u, v〉−, Dom(τ−) = Hm
− .
The sesquilinear forms τ±[u, v] are well defined in the Hilbert space L2(0, 1), they are
densely defined, closed, and nonnegative.
The following assertion is true.
Proposition 1. The sesquilinear forms t±V [u, v] are τ±-bounded with τ±-boundary
that equals zero, i.e., we have V (x) ≺≺ D2m
± .
Proof. Represent the 1-periodic distribution
V (x) =
∑
k∈Z
V̂ (2k)ei 2kπx ∈ H−m
+
as the sum
V (x) = V0(x) + Vδ(x), (1)
where V0(x) is a smooth function and Vδ(x) is a distribution with an arbitrarily small
norm,
V0(x) ∈ Hm
+ ,
and
Vδ(x) ∈ H−m
+ with ‖Vδ‖H−m
+
≤ δ
Cm
.
The constant Cm is defined from the convolution lemma and is fixed. The decomposition
(1) is possible, since Hm
+ is densely embedded into the space H−m
+ .
So, for
u ∈ Dom(τ±) ⊂ Dom(t±V ),
we have ∣∣t±V [u]
∣∣ = |〈V (x)u, u〉±| ≤ |〈V0(x)u, u〉±| + |〈Vδ(x)u, u〉±| ≤
≤ ‖V0(x)u‖L2(0,1) ‖u‖L2(0,1) + ‖Vδ(x)u‖H−m
±
‖u‖Hm
±
≤
≤ Cm ‖V0(x)‖Hm
+
‖u‖2
L2(0,1) + δ ‖u‖2
Hm
±
.
Taking into account that
‖u‖2
Hm
±
≤ ‖u‖2
L2(0,1) + ‖u(m)‖2
L2(0,1) = ‖u‖2
L2(0,1) + 〈D2m
± u, u〉±
for an arbitrary δ > 0 we obtain the necessary estimate,∣∣t±V [u]
∣∣ ≤ δτ±[u] +
(
Cm ‖V0‖Hm
+ [0,1] + δ
)
‖u‖2
L2(0,1). (2)
The proof is complete.
Proposition 1, together with [2] (Theorem IV.1.33) yields the following corollary.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
792 V. A. MIKHAILETS, V. M. MOLYBOGA
Corollary. The sesquilinear forms
t±[u, v] := 〈D2m
± u, v〉± + 〈V (x)u, v〉±, Dom(t±) = Hm
± ,
are densely defined, closed, and sectorial in the Hilbert space L2(0, 1).
According to the first representation theorem [2] (Theorem VI.2.1), there exist m-
sectorial operators S±(V ) associated with the forms t±[u, v] such that
i) Dom(S±) ⊆ Dom(t±) and
t±[u, v] = (S±u, v)
for every u ∈ Dom(S±) and v ∈ Dom(t±);
ii) Dom(S±) are cores of t±[u, v];
iii) if u ∈ Dom(t±), w ∈ L2(0, 1), and
t±[u, v] = (w, v)
holds for every v belonging to the cores of t±[u, v], then u ∈ Dom(S±) and
S±(V )u = w.
The m-sectorial operators S±(V ) are uniquely defined by condition i).
Now, investigate the operators S±(V ) associated with the forms t±[u, v] in more de-
tails.
Let
u ∈ Dom(S±) and v ∈ Dom(t±).
Then we have
t±[u, v] = 〈D2m
± u, v〉± + 〈V (x)u, v〉± = 〈D2m
± u + V (x)u, v〉± =
= (S±u, v) = 〈S±u, v〉±.
This shows that we have the equality
〈D2m
± u + V (x)u, v〉± = 〈S±u, v〉±, v ∈ Hm
± ,
of linear forms. So, we can conclude that
S±(V )u = D2m
± u + V (x)u ∈ L2(0, 1), u ∈ Dom(S±),
and that the inclusions
Dom(S±) ⊆
{
u ∈ Hm
±
∣∣D2m
± u + V (x)u ∈ L2(0, 1)
}
hold. It remains to verify that the inverse inclusions hold.
Let
u ∈
{
u ∈ Hm
±
∣∣D2m
± u + V (x)u ∈ L2(0, 1)
}
and v ∈ Dom(t±).
Then
t±[u, v] = 〈D2m
± u, v〉± + 〈V (x)u, v〉± =
= 〈D2m
± u + V (x)u, v〉± =
(
D2m
± u + V (x)u, v
)
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
SINGULARLY PERTURBED PERIODIC AND SEMIPERIODIC DIFFERENTIAL OPERATORS 793
and using the first representation theorem iii) (see above) we get the necessary estimate,
u ∈ Dom(S±),
which implies that{
u ∈ Hm
±
∣∣D2m
± u + V (x)u ∈ L2(0, 1)
}
⊆ Dom(S±)
and
S±(V )u = D2m
± u + V (x)u ∈ L2(0, 1).
So,
Dom(S±) =
{
u ∈ Hm
±
∣∣D2m
± u + V (x)u ∈ L2(0, 1)
}
and
S±(V )u = D2m
± u + V (x)u ∈ L2(0, 1), u ∈ Dom(S±).
Theorem 1 is proved completely.
Remark 1. Throughout the rest of the paper we will often use the notations
tS± [u, v] ≡ t±[u, v]
to underline the dual relations between the sesquilinear forms t±[u, v] and the associated
with them operators S±(V ), see [2] ([Theorem VI.2.7).
2.2. Proof of Theorem 2. Let the 1-periodic distribution V (x) belong to the space
H−mα
+ , α ∈ [0, 1]. Then for any u ∈ Dom(S±), due to the convolution lemma, we have
V (x)u ∈ H−mα
+ ,
and therefore
D2m
± u ∈ H−mα
± .
From this we conclude that
u ∈ H
m(2−α)
± .
2.3. Proof of Theorem 3. Let
u ∈ Dom(S±)
and
u
∣∣
(α,β) = 0 with (α, β) ⊂ [0, 1],
and let
ϕ(x) ∈ C∞
0 [0, 1] with supp(ϕ) � (α, β).
Then we have
(S±u, ϕ) = 〈S±u, ϕ〉± = 〈D2m
± u + V (x)u, ϕ〉± = 〈D2m
± u, ϕ〉± + 〈V (x)u, ϕ〉± =
= 〈u,D2m
± ϕ〉± + 〈V (x), uϕ〉± = 〈V (x), 0〉± = 0,
which yields the necessary statement,
(S±u)
∣∣
(α,β) = 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
794 V. A. MIKHAILETS, V. M. MOLYBOGA
2.4. Proof of Theorem 4. (a) The m-sectoriality of the operators S±(V ) have been
proved in Theorem 1. Let us prove the second part of the assertion, i.e., we need to show
that for any ε > 0 and some constant cε ≥ 0 the following estimates hold:
|arg ((S± + cεId)u, u)| ≤ ε, u ∈ Dom(S±).
For this we have to make sure that
|Im(S±u, u)| ≤ εRe(S±u, u) + cε‖u‖2
L2(0,1), u ∈ Dom(S±),
for any ε > 0 and some constant cε ≥ 0.
So, take 0 < ε < 1/2. From Proposition 1 (see (2)) we get∣∣t±V [u]
∣∣ ≤ ε
2
τ±[u] +
(
Cm ‖V0(x)‖Hm
+
+
ε
2
)
‖u‖2
L2(0,1), u ∈ Dom(τ±),
and, hence,
−εRe t±V [u] ≤ ε
2
τ±[u] +
(
Cm ‖V0(x)‖Hm
+
+
ε
2
)
‖u‖2
L2(0,1), u ∈ Dom(τ±).
Further, taking into account that
Re(S±u, u) = 〈D2m
± u, u〉± + Re〈V (x)u, u〉±,
Im(S±u, u) = Im〈V (x)u, u〉±, u ∈ Dom(S±),
we obtain the necessary estimates,
|Im(S±u, u)| ≤ |〈V (x)u, u〉±| ≤
ε
2
τ±[u] +
(
Cm ‖V0(x)‖Hm
+
+
ε
2
)
‖u‖2
L2(0,1) ≤
≤ ε
(
τ±[u] + Re t±V [u]
)
+
(
2Cm ‖V0(x)‖Hm
+
+ ε
)
‖u‖2
L2(0,1) =
= εRe(S±u, u) + cε‖u‖2
L2(0,1), u ∈ Dom(S±).
(b) Let the 1-periodic distribution V (x) be real-valued. Then the sesquilinear forms
tS± [u, v] are symmetric and, consequently, in virtue of [2] (Theorem VI.2.7) (also see the
KLMN theorem [17] (Theorem X.17)), the operators are self-adjoint.
Conversely, let the operators S±(V ) be selfadjoint. In the case of a non-real-valued
distribution V (x), the operators S±(V ) are not symmetric either. This contradiction al-
lows to make conclusion that the distribution V (x) is real-valued.
(c) From [2] (Theorem VI.3.4) and Proposition 1 we immediately obtain the follow-
ing proposition.
Proposition 2. The resolvent sets of the operators S±(V ) are non-empty. Moreover,
the resolvents R(λ, S±(V )) of the operators S±(V ) are compact.
Proposition 2 implies that the operators S±(V ) have discrete spectra.
2.5. Proof of Theorem 5. The proof is based on the following proposition.
Proposition 3. Let the 1-periodic distributions Vn(x), n ∈ N, and V (x) be in the
Sobolev space H−m
+ . For
Vn(x)
H−m
+−→ V (x), n → ∞,
the operators
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
SINGULARLY PERTURBED PERIODIC AND SEMIPERIODIC DIFFERENTIAL OPERATORS 795
S
(n)
± ≡ S±(Vn) := D2m
± � Vn(x),
Dom(S(n)
± ) =
{
u ∈ Hm
±
∣∣D2m
± u + Vn(x)u ∈ L2(0, 1)
}
,
converge to the operators
S± ≡ S±(V ) = D2m
± � V (x),
Dom(S±) =
{
u ∈ Hm
±
∣∣D2m
± u + V (x)u ∈ L2(0, 1)
}
,
in the generalized convergence sense [2] (Ch. IV, § 2.6).
Proof. Set
t
S
(n)
±
[u, v] ≡ t
(n)
± [u, v] := (S(n)
± u, v), u ∈ Dom(S(n)
± ), v ∈ Dom(t(n)
± ) = Hm
± ,
and recall that
tS± [u, v] ≡ t±[u, v] = (S±u, v), u ∈ Dom(S±), v ∈ Dom(t±) = Hm
± .
Then, for every u ∈ Dom(t±) = Dom(t(n)
± ) = Hm
± ,∣∣∣t(n)
± [u] − t±[u]
∣∣∣ =
∣∣〈(Vn(x) − V (x))u, u〉±
∣∣ ≤ ‖(Vn(x) − V (x))u‖H−m
±
‖u‖Hm
±
≤
≤ Cm‖Vn(x) − V (x)‖H−m
+
(
‖u‖2
Hm
±
+ τ±[u]
)
,
where the constant Cm is defined due to the convolution lemma, and τ±[u, v], as above,
τ±[u, v] = 〈D2m
± u, v〉±, Dom(τ±) = Hm
± ,
are sesquilinear, densely defined, closed and nonnegative forms. Since the forms
t±V [u, v] = 〈V (x)u, v〉±, Dom(t±V ) = Hm
± ,
are τ±-bonded with zero τ±-boundary for an arbitrary 0 < ε ≤ 1/2, the following esti-
mates hold:
2
∣∣Re t±V [u]
∣∣ ≤ τ±[u] + 2
(
Cm ‖V0(x)‖Hm
+
+ ε
)
‖u‖2
L2(0,1),
and thus
2Re t±V [u] + τ±[u] + 2
(
Cm ‖V0(x)‖Hm
+
+ ε
)
‖u‖2
L2(0,1) ≥ 0.
Taking to account that
Re t±[u] = τ±[u] + Re t±V [u]
we get the needed estimates,∣∣∣t(n)
± [u] − t±[u]
∣∣∣ ≤ Cm‖Vn(x) − V (x)‖H−m
+
(
‖u‖2
Hm
±
+ τ±[u]
)
≤
≤ Cm‖Vn(x) − V (x)‖H−m
+
×
×
(
2Re t±V [u] + 2τ±[u] + 2
(
Cm ‖V0(x)‖Hm
+
+ ε + 1/2
)
‖u‖2
L2(0,1)
)
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
796 V. A. MIKHAILETS, V. M. MOLYBOGA
= an‖u‖2
L2(0,1) + bnRe t±[u],
where
an = 2
(
Cm ‖V0(x)‖Hm
+
+ 1
)
‖Vn(x) − V (x)‖H−m
+
≥ 0
and
bn = 2Cm‖Vn(x) − V (x)‖H−m
+
≥ 0
tend to zero as n → ∞.
To complete the proof it suffices to apply [2] (Theorem VI.3.6).
Proposition 3 and Theorem IV.2.25 [2] together with Proposition 2, give Theorem 5.
2.6. Proof of Theorem 6. Let the operators — form-sums S(V ), S+(V ), and S−(V )
be given with V (x) a 1-periodic complex-valued distribution from the Sobolev spaces
H−m
per and H−m
± , correspondingly.
For an arbitrary s ∈ R let us consider the Sobolev spaces
Hs
per =
{
f =
∑
k∈Z
f̂(k)eikπx
∣∣∣ ‖ f ‖Hs
per
< ∞
}
,
Hs
+ =
{
f =
∑
k∈Z
f̂(2k)ei2kπx
∣∣∣ ‖ f ‖Hs
+
< ∞
}
,
Hs
− =
{
f =
∑
k∈Z
f̂(2k + 1)ei(2k+1)πx
∣∣∣ ‖ f ‖Hs
−
< ∞
}
.
It should be remarked that
H0
per ≡ L2(−1, 1) and H0
+ ≡ H0
− ≡ L2(0, 1).
Set
Hs
per,+ :=
{
f ∈ Hs
per
∣∣∣ f̂(2k + 1) = 0 ∀k ∈ Z
}
,
Hs
per,− :=
{
f ∈ Hs
per
∣∣∣ f̂(2k) = 0 ∀k ∈ Z
}
,
and thus
Hs
per = Hs
per,+ ⊕Hs
per,−, s ∈ R.
Let
I± : Hs
± � f(x) �→ f(x) ∈ Hs
per,±, s ∈ R,
be extension operators that extend the elements f(x) ∈ Hs
± defined on the interval [0, 1]
to the elements f(x) ∈ Hs
per,± defined on the interval [−1, 1]. The operators I± establish
isometric isomorphisms between the spaces Hs
± and Hs
per,± for s ∈ R.
Further, let us consider the operators S(V ). Since the potentials V (x) are 1-periodic
distributions from the space H−m
per , i.e., V (x) ∈ H−m
per,+, the operators S(V ) are reduced
by the space H−m
per,+ [18] (Ch. IV, § 40). So, we have
S(V ) = Sper,+(V ) ⊕ Sper,−(V ), (3)
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
SINGULARLY PERTURBED PERIODIC AND SEMIPERIODIC DIFFERENTIAL OPERATORS 797
where the operators Sper,±(V ) are defined on the Hilbert spaces H0
per,±. Taking into
account that
Hs
+
I+� Hs
per,+ and Hs
−
I−� Hs
per,−
for an arbitrary s ∈ R we conclude that the operators Sper,±(V ) and S±(V ) are unitary
equivalent,
S+(V )
I+� Sper,+(V ) and S−(V )
I−� Sper,−(V ).
From the latter relations and decomposition (3), we obtain the need statement,
S(V ) = S+(V ) ⊕ S−(V ),
which implies
spec(S) = spec(S+) ∪ spec(S−).
The proof of Theorem 6 is completed.
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Moscow: Mir, 1972. – 740 p.).
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with V-distribution // Meth. Funct. Anal. and Top. – 2003. – 9, # 2. – P. 163 – 178.
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differential operators // Ibid. – # 4. – P. 30 – 57.
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Int. Math. Res. Not. – 2003. – 37. – P. 2019 – 2031.
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(in Russian) // Uspekhi Mat. Nauk. – 2006. – 61, # 4. – P. 77 – 182.
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Soc. Transl., 1968. – 17. – 809 p. (Russian edition: Kiev: Naukova Dumka, 1965. – 800 p.).
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drecht etc.: Kluwer, 1991. – 347 p. (Russian edition: Kiev: Naukova Dumka, 1984. – 283 p.).
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Received 30.03.2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 6
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| id | umjimathkievua-article-3345 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:40:48Z |
| publishDate | 2007 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2a/d0a5b97efcd24755acdcb3a249379d2a.pdf |
| spelling | umjimathkievua-article-33452020-03-18T19:51:58Z Singularly perturbed periodic and semiperiodic differential operators Cингулярно збурені періодичні ta напівперіодичні диференціальні оператори Mikhailets, V. A. Molyboga, V. M. Михайлець, В. А. Молибога, В. М. Qualitative and spectral properties of the form sums $$S_{±}(V) := D^{2m}_{±} + V(x),\quad m ∈ N,$$ are studied in the Hilbert space $L_2(0, 1)$. Here, $(D_{+})$ is a periodic differential operator, $(D_{-})$ is a semiperiodic differential operator, $D_{±}: u ↦ −iu′$, and $V(x)$ is an arbitrary 1-periodic complex-valued distribution from the Sobolev spaces $H_{per}^{−mα},\; α ∈ [0, 1]$. Досліджено якісні та спектральні властивості форм-сум $$S_{±}(V) := D^{2m}_{±} + V(x),\quad m ∈ N,$$ у гільбертовому просторі $L_2(0, 1)$. Тут $(D_{+})$ та $(D_{-})$ — періодичний та напівперіодичний диференціальні оператори,$D_{±}: u ↦ −iu′$, а $V(x)$ — довільна 1-періодична комплекснозначна узагальнена функція з просторів Соболева $H_{per}^{−mα},\; α ∈ [0, 1]$. Institute of Mathematics, NAS of Ukraine 2007-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3345 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 6 (2007); 785–797 Український математичний журнал; Том 59 № 6 (2007); 785–797 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3345/3434 https://umj.imath.kiev.ua/index.php/umj/article/view/3345/3435 Copyright (c) 2007 Mikhailets V. A.; Molyboga V. M. |
| spellingShingle | Mikhailets, V. A. Molyboga, V. M. Михайлець, В. А. Молибога, В. М. Singularly perturbed periodic and semiperiodic differential operators |
| title | Singularly perturbed periodic and semiperiodic differential operators |
| title_alt | Cингулярно збурені періодичні ta напівперіодичні диференціальні оператори |
| title_full | Singularly perturbed periodic and semiperiodic differential operators |
| title_fullStr | Singularly perturbed periodic and semiperiodic differential operators |
| title_full_unstemmed | Singularly perturbed periodic and semiperiodic differential operators |
| title_short | Singularly perturbed periodic and semiperiodic differential operators |
| title_sort | singularly perturbed periodic and semiperiodic differential operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3345 |
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