Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 21, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
SELFADJOINT SINGULAR DIFFERENTIAL OPERATORS OF
FIRST ORDER AND THEIR SPECTRUM
ZAMEDDIN I. ISMAILOV, PEMBE IPEK
Dedicated to Prof. E. S. Panakhov on his 60-th birthday
Abstract. Based on Calkin-Gorbachuk method, we describe all selfadjoint
extensions of the minimal operator generated by linear multipoint singular
symmetric differential-operator, as a direct sum of weighted Hilbert space of
vector-functions. Another approach to the investigation of this problem has
been done by Everitt, Zettl and Markus. Also we study the structure of
spectrum of these extensions.
1. Introduction
The first works devoted to the general theory of selfadjoint extensions of symmetric operators having equal deficiency indexes in any Hilbert space belong to von
Neumann [12]. Generalization of this theory to normal extensions has been done
Coddington [3]. Applications of this theory to two-point differential operators in
Hilbert spaces can be found in works of many mathematicians; see for example
[4, 6, 7, 13].
It is known that for the existence of selfadjoint extension of any linear closed
densely defined symmetric T in a Hilbert space H the necessary and sufficient condition is a equality of deficiency indexes m(T ) = n(T ), where m(T ) = dim ker(T ∗ +
iE), n(T ) = dim ker(T ∗ − iE).
In the multipoint cases it may be faced with different views. Let T1 and T2 be
minimal operators generated by the linear singular differential expression
l(u) = itu0 (t)
in the weighted Hilbert L2α (−∞, a) and L2α (b, ∞), α(t) = t, a, b ∈ R of functions,
respectively. In this case it is clear that
(m(T1 ), n(T1 )) = (0, 1),
(m(T2 ), n(T2 )) = (1, 0)
Consequently, the operators T1 and T2 are maximal symmetric. Hence they are not
any selfadjoint extensions. However, direct sum T = T1 ⊕ T2 of operators T1 and T2
in the direct sum H = L2α (−∞, a) ⊕ L2α (b, ∞) have an equal defect numbers (1,1).
Then by the general theory it has at least one selfadjoint extension [12].
2010 Mathematics Subject Classification. 47A10, 47B38.
Key words and phrases. Singular Selfadjoint differential operator; spectrum.
c
2016
Texas State University.
Submitted August 25, 2015. Published January 12, 2016.
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Z. I. ISMAILOV, P. IPEK
EJDE-2016/21
Indeed it can be easily shown that the multipoint differential expression
l(u) = (itu01 (t), itu02 (t)) ,
u = (u1 , u2 ),
u1 ∈ D(T1∗ ),
u2 ∈ D(T2∗ )
with boundary condition u2 (b) = u1 (a) generates selfadjoint extension of T1 ⊕ T2 .
There are different methods for the description all selfadjoint extensions of minimal symmetric operators. For example, in mathematical literature the EwerittZettl-Markus and Calkin-Gorbachuk methods are very important of them.
The models of many physical and technical phenomena are expressed as differential operators. Therefore operator theory plays an exceptionally important role
in modern mathematics, especially in the modelling of processes of multi-particle
quantum mechanics, quantum field theory, the multipoint boundary value problems
for differential equations [1, 15].
Although the first studies of the theory multipoint differential operators were
performed at the beginning of twentieth century, most of them which are about the
investigation of the theory and application to spectral problems, have been found
since 1950 [5, 8, 9, 10, 15, 16].
In the article by Everitt and Zettl [5] in the scalar case, all selfadjoint extensions of the minimal operator generated by Lagrange-symmetric any order quasidifferential expression with equal deficiency indexes in terms of boundary conditions
are described by Glazman-Krein-Naimark method for regular and singular cases in
the direct sum of corresponding Hilbert spaces of functions.
2. Statement of the problem
Let H be a separable Hilbert space and let a, b ∈ R. In the Hilbert space
L21/α (H, (−∞, a))⊕L21/β (H, (b, +∞)) of a vector-functions we consider the following
linear multipoint differential-operator expression for first order in a form
l(w) = (k(u), m(v)),
(2.1)
where w = (u, v), k(u) = iα(t)u0 (t) + Au(t), m(v) = iβ(t)v 0 (t) + Bv(t), α ∈
C(−∞, a), β ∈ C(b, +∞) and there exist positive numbers α1 , α2 , β1 , β2 such that
α1 ≤ α(t) ≤ α2 for any t < a, β1 ≤ β(t) ≤ β2 for any t > b and for simplicity
assumed that A and B are linear bounded selfadjoint operators in H.
In similar way in [7] the minimal K0 (M0 ) and maximal K(M ) operators associated with differential expression k(m) in L21/α (H, (−∞, a))(L21/β (H, (b, +∞))) can
be constructed.
The operators L0 = K0 ⊕ M0 and L = K ⊕ M in the Hilbert space H =
L21/α (H, (−∞, a)) ⊕ L21/β (H, (b, +∞)) are called minimal and maximal operators
associated with differential expression (2.1), respectively. It is clear that operator
L0 is a symmetric and L∗0 = L in H. The minimal operator L0 is not maximal.
Indeed, differential expression (2.1) with boundary condition u(a) = v(b) generates
a selfadjoint extension of L0 .
The main goal in this article is to describe all selfadjoint extensions of minimal
operator L0 in H in terms of boundary values (Sec. 3). In section 4 the structure
of spectrum of these extensions will be investigated.
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3
3. Description of Selfadjoint Extensions
In this section using by Calkin-Gorbachuk method will be investigated the general representation of selfadjoint extensions of minimal operator L0 . Firstly, prove
the following proposition.
Lemma 3.1. The deficiency indexes of the operators K0 and M0 are of the form
(m(K0 ), n(K0 )) = (dim H, 0),
(m(M0 ), n(M0 )) = (0, dim H).
Proof. From the conditions under the weight functions α(·) and β(·) imply that
1
1
1
≤
≤
, t < a and
α2
α(t)
α1
From these relations we have
Z a
−∞
dt
=
α(t)
Z
b
∞
1
1
1
≤
≤
, t > b.
β2
β(t)
β1
dt
= +∞
β(t)
On the other hand, it is clear that the general solutions of differential equations
0
iα(t)u0± ± iu± = 0, t < a and iβ(t)v±
± iv± = 0, t > b
in the L21/α (H, (−∞, a)) and L21/β (H, (b, +∞)) are
Z t ds u± (t) = exp ∓
f, f ∈ H, t < a
−∞ α(s)
and
∞
Z
ds g, g ∈ H, t > b
β(s)
t
respectively From these representations we have
Z a
1
2
ku+ (t)k2H dt
ku+ kL2 (H,(−∞,a)) =
1/α
α(t)
−∞
Z a
Z t
1
ds =
exp − 2
dtkf k2H
−∞ α(t)
−∞ α(s)
Z
Z a
Z t
ds t ds exp − 2
=
d
kf k2H
α(s)
α(s)
−∞
−∞
−∞
1
2
= kf kH < ∞
2
Consequently,
dim ker(K + iE) = dim H
v± (t) = exp ±
On the other hand it is clear that for any f ∈ H the solutions
Z t
ds u− (t) = exp
f∈
/ L21/α (H, (−∞, a)).
−∞ α(s)
Hence dim ker(K − iE) = 0.
In a similar way we can show that
m(M0 ) = dim ker(M + iE) = 0
This completes the proof of theorem.
and n(M0 ) = dim ker(M − iE) = dim H
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Z. I. ISMAILOV, P. IPEK
EJDE-2016/21
From last Lemma 3.1 we have
m(L0 ) = m(K0 ) + m(M0 ) = dim H
and n(L0 ) = n(K0 ) + n(M0 ) = dim H
Consequently, (m(L0 ), n(L0 )) = (dim H, dim H). Consequently, the minimal operator L0 has a selfadjoint extensions; see [12].
Definition 3.2 ([7]). Let H be any Hilbert space and S : D(S) ⊂ H → H be
a closed densely defined symmetric operator in the Hilbert space H having equal
finite or infinite deficiency indexes. A triplet (B, γ1 , γ2 ), where B is a Hilbert space,
γ1 and γ2 are linear mappings from D(S ∗ ) into B, is called a space of boundary
values for the operator S if for any f, g ∈ D(S ∗ )
(S ∗ f, g)H − (f, S ∗ g)H = (γ1 (f ), γ2 (g))B − (γ2 (f ), γ1 (g))B
while for any F1 , F2 ∈ B, there exists an element f ∈ D(S ∗ ) such that γ1 (f ) = F1
and γ2 (f ) = F2 .
It is known that for any symmetric operator with equal deficiency indexes have
at least one space of boundary values [7].
Lemma 3.3. If u ∈ D(K) and v ∈ D(M ), then limt→−∞ u(t) = 0, limt→+∞ v(t) =
0 and u(a), v(b) ∈ H.
Proof. Under the assumptions for α(·) and β(·) we have
1
1
1
≤
≤
, t < a,
α2
α(t)
α1
1
1
1
≤
≤
, t > b.
β2
β(t)
β1
Then for any pair of functions x(·) ∈ L2 (H, (−∞, a)), y(·) ∈ L2 (H, (b, +∞)), from
the above relations,
Z a
Z a
1
1
2
0≤
kx(t)kH dt ≤
kx(t)k2H dt,
α1 −∞
−∞ α(t)
Z ∞
Z ∞
1
1
2
ky(t)kH dt ≤
ky(t)k2H dt
0≤
β(t)
β1 b
b
Consequently,
L2 (H, (−∞, a)) ⊂ L21/α (H, (−∞, a))
and L2 (H, (b, +∞)) ⊂ L21/β (H, (b, +∞)) .
Similarly, if x(·) ∈ L21/α (H, (−∞, a)) and y(·) ∈ L21/β (H, (b, +∞)), then from the
conditions on the weight functions we have
Z a
Z a
1
1
kx(t)k2H dt ≤
kx(t)k2H dt,
α2 −∞
−∞ α(t)
Z ∞
Z ∞
1
1
ky(t)k2H dt ≤
ky(t)k2H dt
β2 b
β(t)
b
Hence,
L21/α (H, (−∞, a)) ⊂ L2 (H, (−∞, a))
and L21/β (H, (b, +∞)) ⊂ L2 (H, (b, +∞)).
Therefore it is obtained that
L21/α (H, (−∞, a)) = L2 (H, (−∞, a))
and L21/β (H, (b, +∞)) = L2 (H, (b, +∞)).
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SELFADJOINT SINGULAR DIFFERENTIAL OPERATORS
5
On the other hand, from the relations
D(K) ⊂ L2 (H, (−∞, a))
and D(M ) ⊂ L2 (H, (b, +∞)),
we have limt→−∞ u(t) = 0 and limt→+∞ v(t) = 0 for any u ∈ D(K) and v ∈ D(M ).
Now we assume that u ∈ D(K). Then
u ∈ L21/α (H, (−∞, a))
and αu0 ∈ L21/α (H, (−∞, a))
This implies
1
√ u ∈ L2 (H, (−∞, a))
α
and
√
αu0 ∈ L2 (H, (−∞, a))
Then for any t < a it holds
(u, u)0H (t) = (u0 , u)H (t) + (u, u0 )H (t) =
√
1
√ 1 αu0 , √ u (t) + √ u, αu0 (t)
α H
α
H
and from this,
ku(a)k2H − ku(−∞)k2H Z a Z a √ √ 0 1
1
≤
αu , √ u (t)dt +
√ u, αu0 (t)dt
α
α
H
H
−∞
−∞
1/2 Z a
1/2
Z a
√ 0 2
1
k √ uk2H (t)dt
≤2
k αu kH (t)dt
α
−∞
−∞
√ 0
u
= 2k √ kL2 (H,(−∞,a)) k αu kL2 (H,(−∞,a)) < ∞
α
So u(a) ∈ H. In a similar way we can show that v(b) ∈ H.
The following theorem defines the space of boundary condition.
Theorem 3.4. The triplet (H, γ1 , γ2 ), with
1
γ1 : D(L) → H, γ1 (w) = √ (u(a) + v(b)),
i 2
1
γ2 : D(L) → H, γ2 (w) = √ (u(a) − v(b)), w = (u, v) ∈ D(L),
2
is a space of boundary values of the minimal operator L0 in H.
Proof. In this case for any w1 = (u1 , v1 ) and w2 = (u2 , v2 ) from D(L) can be easy
verified that
(Lw1 , w2 )H − (w1 , Lw2 )H = (Ku1 , u2 )L21/α (H,(−∞,a)) − (u1 , Ku2 )L21/α (H,(−∞,a))
+ (M v1 , v2 )L21/β (H,(b,∞)) − (v1 , M v2 )L21/β (H,(b,∞))
= (γ1 (w1 ), γ2 (w2 ))H − (γ2 (w1 ), γ2 (w2 ))H
Now let f and g be any elements from H. Find the function w = (u, v) ∈ D(L)
such that
1
1
γ1 (w) = √ (u(a) + v(b)) and γ2 (w) = √ (u(a) − v(b)).
i 2
2
From this it is obtained that
√
√
u(a) = (if + g)/ 2 and v(b) = (if − g)/ 2
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Z. I. ISMAILOV, P. IPEK
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If we choose the functions u(·) and v(·) in the form
p
Z
√
α(t) t s−a
u(t) = p
e ds(if + g)/ 2, t < a,
α(a) −∞
p
Z
√
β(t) ∞ b−t
v(t) = p
e ds(if − g)/ 2, t > b,
β(b) t
then it is clear that (u, v) ∈ D(L) and γ1 (w) = f , γ2 (w) = g.
Lastly, using the method in [7] it can be established the following results.
e is a selfadjoint extension of the minimal operator L0 in H, then
Theorem 3.5. If L
it generates by the differential-operator expression (2.1) and boundary condition
v(b) = W u(a),
where W : H → H is a unitary operator. Moreover, the unitary operator W in H
e i.e. L
e = LW and vice versa.
is determined uniquely by the extension L,
4. Spectrum of the selfadjoint extensions
In this section the structure of the spectrum of the selfadjoint extension LW in
H will be investigated. Firstly, prove the following results.
Theorem 4.1. The point spectrum of selfadjoint extension LW is empty, i.e.
σp (LW ) = ∅.
Proof. Consider the eigenvalue problem
l(w) = λw,
w = (u, v) ∈ H, λ ∈ R
with boundary condition v(b) = W u(a). From this it is obtained that
iα(t)u0 (t) + Au(t) = λu(t), t < a,
iβ(t)v 0 (t) + Bv(t) = λv(t), t > b,
v(b) = W u(a)
The general solutions of the above equations are
Z a
ds u(λ; t) = exp − i(A − λ)
fλ , fλ ∈ H, t < a,
t α(s)
Z
t
ds v(λ; t) = exp i(B − λ)
gλ , gλ ∈ H, t > b,
b β(s)
v(λ; b) = W u(λ; a)
It is clear that for fλ 6= 0 and gλ 6= 0 the solutions u(λ; ·) ∈
/ L21/α (H, (−∞, a))
and v(λ; ·) ∈
/ L21/β (H, (b, ∞)). Therefore, for every unitary operator W we have
σp (LW ) = ∅. Since the residual spectrum for any selfadjoint operator in a Hilbert
space is empty, then we study the continuous spectrum of selfadjoint extensions LW
of the minimal operator L0 . On the other hand from the general theory of linear
selfadjoint operators in Hilbert spaces for the resolvent set ρ(LW ) of any selfadjoint
extension LW is true
ρ(LW ) ⊃ {λ ∈ C : Im λ 6= 0}.
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SELFADJOINT SINGULAR DIFFERENTIAL OPERATORS
7
For the continuous spectrum of selfadjoint extensions the following proposition
is true.
Theorem 4.2. The continuous spectrum of the selfadjoint extension LW is of the
form
σc (LW ) = R.
Proof. For λ ∈ C with λi = Im λ > 0 and f = (fa , fb ) ∈ H the norm of function
Rλ (LW )f (t) in H satisfies
Z a
ds kRλ (LW ))f (t)k2H = k exp i(λ − A)
fλ
t α(s)
Z t
Z a
dτ fa (s)
exp i(A − λ)
+i
dsk2L2 (H,(−∞,a))
1/α
s α(τ ) α(s)
t
Z ∞
Z t
dτ fb (s)
exp i(B − λ)
+ ki
dsk2L2 (H,(b,∞))
1/β
t
s β(τ ) β(s)
Z ∞
Z t
dτ fb (s)
dsk2L2 (H,(b,∞))
≥ ki
exp i(B − λ)
1/β
t
s β(τ ) β(s)
The vector functions f ∗ (λ; t) in the form
Z
f ∗ (λ; t) = 0, exp − i(λ − B)
t
ds f ,
b β(s)
with λ ∈ C, λi = Im λ > 0, f ∈ H belong to H. Indeed,
Z ∞
Z t
1
ds 2
kf ∗ (λ; t)k2H =
k exp − i(λ − B)
f kH dt
β(t)
b
b β(s)
Z ∞
Z t
1
ds =
exp − 2λi
dtkf k2H
β(t)
β(s)
b
b
1
2
=
kf kH < ∞
2λi
For the such functions f ∗ (λ; ·) we have
kRλ (LW )f ∗ (λ; ·)k2H
Z t
Z s
Z ∞
dτ
dτ 1
exp i(B − λ)
− i(λ − B)
≥ ki
f dsk2L2 (H,(b,∞))
1/β
β(s)
β(τ
)
β(τ
)
s
b
t
Z t
Z t
dτ
dτ
= k exp − iλ
+ iB
β(τ
)
β(τ
)
b
Z ∞
Z s b
1
dτ
×
exp − 2λi
f dsk2L2 (H,(b,∞))
1/β
β(s)
β(τ
)
t
b
Z s
Z t dτ Z ∞ 1
dτ 2
= k exp λi
exp − 2λi
dskL2 (H,(b,∞)) kf k2H
1/β
β(τ
)
β(s)
β(τ
)
b
t
b
Z t
dτ
1
exp − λi
k2 2
kf k2H
=k
2λi
β(τ
) L1/β (H,(b,∞))
b
Z t
Z ∞
1
1
dτ = 2
exp − 2λi
dtkf k2H
4λi b β(t)
b β(τ )
1
= 3 kf k2H .
8λi
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Z. I. ISMAILOV, P. IPEK
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From this,
kf k2
1
kRλ (LW )f ∗ (λ; ·)kH ≥ √ H
kf ∗ (λ; t)kH ,
√ =
2λ
2 2λi λi
i
i.e., for λi = Im λ > 0 and f 6= 0 is valid
1
kRλ (LW )f ∗ (λ; ·)kH
≥
∗
kf (λ; t)kH
2λi
On the other hand it is clear that
kRλ (LW )k ≥
kRλ (LW )f ∗ (λ; ·)kH
,
kf ∗ (λ; t)kH
f 6= 0
Consequently,
kRλ (LW )k ≥
1
2λi
for λ ∈ C, λi = Im λ > 0.
Using the above theorem, the spectrum of the singular differential operator generated by differential expression
t2α + 1
t2β + 1
l((u, v)) = i 2α u0 (t, x) + xu(t, x), i 2β v 0 (t, x) + xv(t, x) , α, β > 0
t
t
with boundary condition
u(1, x) = eiϕ u(−1, x), ϕ ∈ [0, 2π)
in the direct sum L2(t2α +1)/t2α ((−∞, −1) × R) ⊕ L2(t2β +1)/t2β ((1, ∞) × R) is purely
continuous and coincides with R.
Note that another approach for the singular differential operators for nth order
in the scalar case has been given in [14]. In special case of functions α(·) and β(·)
the analogous results have been obtained in [2].
Acknowledgements. The authors would like to thank to Prof. A. Ashyralyev
(Fatih University, Istanbul) for his interesting and encouragement discussions.
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Zameddin I. Ismailov
Karadeniz Technical University, Department of Mathematics, 61080, Trabzon, Turkey
E-mail address: zameddin.ismailov@gmail.com
Pembe Ipek
Karadeniz Technical University, Department of Mathematics, 61080, Trabzon, Turkey
E-mail address: ipekpembe@gmail.com