Journal of Mathematical Physics, Analysis, Geometry
2014, vol. 10, No. 1, pp. 44–63
On Spectrum of Differential Operator with
Block-Triangular Matrix Coefficients
A.M. Kholkin
Pryazovskyi State Technical University
7 Universitets’ka Str., Mariupol 87500, Ukraine
E-mail: [email protected]
F.S. Rofe-Beketov
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail: [email protected]
Received November 5, 2012, revised July 15, 2013
For the Sturm–Louville equation with block-triangular matrix potential
that increases at infinity, both increasing and decreasing at infinity matrix
solutions are found. The structure of spectrum for the differential operator
with these coefficients is defined.
Key words: differential operator, spectrum, block-triangular matrix coefficients.
Mathematics Subject Classification 2010: 34K11, 47A10.
Dedicated to our Teacher, Academician Vladimir A. Marchenko on the occasion of his 90th birthday, with appreciation and admiration for his contribution
to spectral theory, differential equations and mathematical physics
The study of the relationship between spectral and oscillation properties
of non-selfadjoint differential operators with block-triangular matrix coefficients
that increase at infinity [1] includes the study of the structure of the spectra of
these operators. For the case of an operator with decreasing at infinity triangular
matrix potential and a bounded first moment, the structure of the spectrum in
the context of the inverse scattering problem was established in [2–4].
In [5, 6], V.A. Marchenko introduced a notion of the generalized spectral
function R for a Sturm–Liouville operator with arbitrary complex valued potential on the semiaxis. This result was generalized to the case of non-selfadjoint
systems [7, 8]. The distribution (the matrix one in the case of systems) acts on
c A.M. Kholkin and F.S. Rofe-Beketov, 2014
°
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
the topological space of test functions. It determines the formulas of expansion
in eigenfunctions and also allows one to solve the inverse problem of spectral
analysis for the non-selfadjoint case. In the case of selfadjoint problems, the distribution is generated by a non-negative measure (either a scalar or a matricial
one in the case of systems). We are interested in analyzing a specific form of a
spectrum and spectral matricial distribution R for some classes of non-selfadjoint
systems. In [3], while solving the inverse scattering problem on the semiaxis for
the case of triangular matrix potentials, the Parseval equality was established,
and thus a form of spectral matrix distribution of V.A. Marchenko type was
found (for the selfadjoint case, see [9]). We obtain conditions which guarantee
the discreteness of spectrum for a broad class of Sturm–Liouville operators on
the semiaxis with block-triangular matrix potentials, whose diagonal blocks are
Hermitian matrices. For these potentials, a form of V.A. Marchenko type spectral
matricial distribution is found. Note that this distribution, under the presence
of multiple poles for the resolvent, does not reduce to a matrix measure even in
the non-selfadjoint case. These results are applied to extending the Sturm type
oscillation theory from selfadjoint systems [10] to systems with triangular matrix
potentials [1].
Consider the equation with a block-triangular matrix potential
l [y] = −y 00 + V (x)y = λy,
0 ≤ x < ∞,
(1)
where

V (x) = v(x)Im + U (x),

U11 (x) U12 (x) . . . U1r (x)
 0
U22 (x) . . . U2r (x)
,
U (x) = 
 ...
...
...
... 
0
0
. . . Urr (x)
(2)
v(x) is a real scalar function such that 0 < v(x) → ∞ monotonically, as x → ∞,
with monotonic absolutely continuous derivative. Also, U (x) is a relatively small
perturbation, e.g., |U (x)|v −1 (x) → 0 as x → ∞ or |U |v −1 ∈ L∞ (R). The diagonal blocks Ukk (x), k = 1, r, are assumed to be mk × mk Hermitian matrices with
mk ≥ 1 (in particular, with mk = 1 these blocks are just real scalar functions).
r
P
Let
mk = m, and denote by Im the unit m × m matrix.
k=1
Denote by Hm the m-dimensional
Hilbert
space. A vector h ∈ Hm will be
¡
¢
written in the form h = col h1 , h2 , . . . , hr , where hk , k = 1, r, is a vector from
Hmk . Thus we have y = col (y 1 , y 2 , . . . , y r ), where y k ∈ Hmk .
I. Let us start with considering the case when
v(x) ≥ Cx2α ,
C > 0,
α > 1.
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
(3)
45
A.M. Kholkin and F.S. Rofe-Beketov
(The spectral properties of one-dimensional Schrödinger operator with polynomial potential is the subject of [11]).
Assume that the coefficients of (1) satisfy the conditions
Z∞
1
|U (t)|v − 2 (t)dt < ∞,
(4)
0
Z∞
− 25
Z∞
3
v 00 (t)v − 2 (t)dt < ∞.
(5)
−y 00 + (v(x) + q(x))y = ((λ + q(x)) Im − U (x)) y,
(6)
02
v (t)v
(t)dt < ∞,
0
0
Rewrite equation (1) in the form
where q(x) is given by (cf. [12, 13])
5
q(x) =
16
µ
v 0 (x)
v(x)
¶2
−
1 v 00 (x)
.
4 v(x)
(7)
Consider the functions

1
γ0 (x, λ) = p
4
4v(x)
1
γ∞ (x, λ) = p
4
4v(x)

Zx p
exp −
v(u)du ,
(8)
0
 x

Z p
exp 
v(u)du .
(9)
0
It is easy to see that with x → ∞, one has
γ0 (x, λ) → 0,
γ∞ (x, λ) → ∞.
These solutions constitute a fundamental system of solutions of the scalar differential equation
−z 00 + (v(x) + q(x))z = 0
(10)
in such a way that for all x ∈ [0, ∞) one has
0
W (γ0 , γ∞ ) := γ0 (x, λ)γ∞
(x, λ) − γ00 (x, λ)γ∞ (x, λ) = 1.
46
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
Theorem 1. Under conditions (3), (4), (5), equation (1) has a unique decreasing at infinity m × m matrix solution Φ(x, λ), satisfying the conditions
Φ(x, λ)
= Im
γ0 (x, λ)
(11)
Φ0 (x, λ)
= Im .
x→∞ γ 0 (x, λ)
0
(12)
lim
x→∞
and
lim
Also, there exists a unique increasing at infinity m×m matrix solution Ψ(x, λ),
satisfying the conditions
Ψ(x, λ)
lim
= Im
(13)
x→∞ γ∞ (x, λ)
and
Ψ0 (x, λ)
= Im .
0 (x, λ)
x→∞ γ∞
(14)
lim
P r o o f. 1. Equation (6) allows one to derive the integral equation
Z∞
K(x, t, λ)Φ(t, λ)dt,
Φ(x, λ) = γ0 (x, λ)Im +
(15)
x
where
K(x, t, λ) = C(x, t)[(λ + q(t))Im − U (t)],
(16)
C(x, t) = γ∞ (x, λ)γ0 (t, λ) − γ∞ (t, λ)γ0 (x, λ),
(17)
with C(x, t) being the Cauchy function that in each variable satisfies equation
(10) and the initial conditions
C(x, t)|x=t = 0,
Cx0 (x, t)|x=t = 1,
Set
χ(x, λ) =
Ct0 (x, t)|x=t = −1.
Φ(x, λ)
γ0 (x, λ)
to rewrite equation (15) in the form
Z∞
χ(x, λ) = Im +
R(x, t, λ)χ(t, λ)dt,
(18)
x
where
R(x, t, λ) = K(x, t, λ)
γ0 (t, λ)
.
γ0 (x, λ)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
47
A.M. Kholkin and F.S. Rofe-Beketov
In view of
¯
¯
¯ ¯
¯
¯ ¯
¯
¯C(x, t) γ0 (t, λ) ¯ = ¯γ02 (t) γ∞ (x, λ) − γ0 (t, λ)γ∞ (t, λ)¯
¯
¯
¯
¯
γ0 (x, λ)
γ0 (x, λ)
¯
¯




¯
¯
Zt p
Zx p
¯ 1
¯
1
¯
= ¯¯ p
exp −2
v(u)du exp 2
v(u)du − p
2 v(t) ¯¯
¯ 2 v(t)
0
0
¯
¯


¯
¯
Zt p
¯
¯
1
¯
¯


p
=
exp
−2
v(u)du
−
1
¯
2 v(t) ¯¯
¯
x
µ
¶
Rt p
and since with x ≤ t one has exp −2
v(u)du ≤ 1, we deduce that
x
¯
¯
¯
¯
¯C(x, t) γ0 (t, λ) ¯ ≤ p 1 .
¯
γ0 (x, λ) ¯
v(t)
(19)
Hence
¯
¯
¯
¯
γ
(t,
λ)
0
|R(x, t, λ)| = ¯¯C(x, t)
[(λ + q(t))I − U (t)]¯¯
γ0 (x, λ)
1
(|λ| + |q(t)| + |U (t)|).
≤p
v(t)
By virtue of (3)–(5), (7),
1
p
(|λ| + |q(t)| + |U (t)|) ∈ L(0, ∞),
v(t)
(20)
which implies that the integral equation has a unique solution χ(x, λ), and
|χ(x, λ)| ≤ const. By (18), one has that lim χ(x, λ) = Im , which already imx→∞
plies (11).
Differentiate (15) to get
Φ0 (x, λ)
= Im +
γ00 (x, λ)
Z∞
S(x, t, λ)χ(t, λ)dt,
(21)
x
where
S(x, t, λ) = Kx0 (x, t, λ)
48
γ0 (t, λ)
γ0 (t, λ)
= Cx0 (x, t) 0
[(λ + q(t))Im − U (t)].
0
γ0 (x, λ)
γ0 (x, λ)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
Since
¯
¯ ¯
¯
0
¯
¯ 0
¯ ¯
¯Cx (x, t) γ0 (t, λ) ¯ = ¯γ02 (t)t γ∞ (x, λ) − γ∞ (t, λ)γ0 (t, λ)¯
0
0
¯
¯
¯
¯
γ0 (x, λ)
γ0 (x, λ)
¯
¯




√
¯
¯
Zt p
Zx p
1 − 54 0
¯
¯
v
v
−
v
1
4
¯exp −2
= p
v(u)du
exp 2
v(u)du − 1¯¯
√
5
¯
1 −4 0
2 v(t) ¯
v + v
¯
4v
0
0




Zt p
1
1
exp −2
v(u)du + 1 ≤ p
,
≤ p
2 v(t)
v(t)
x
one has
|S(x, t, λ)| ≤ p
1
v(t)
£
¤
|λ| + |q(t)| + |U (t)| ∈ L(0, ∞).
Therefore, (21) now implies (12).
2. Denote by

Ψ11 (x, λ) Ψ12 (x, λ)

0
Ψ22 (x, λ)
b
Ψ(x,
λ) = 
 ...
...
0
0

. . . Ψ1r (x, λ)
. . . Ψ2r (x, λ)

...
... 
. . . Ψrr (x, λ)
the matrix solution of (1) that increases at infinity. The diagonal blocks Ψkk (x)
are the mk × mk -matrices with mk ≥ 1, k = 1, r. Equation (6) is equivalent to
the integral equation
Zx
b
Ψ(x,
λ) = γ∞ (x, λ)Im −
b λ)dt,
K(x, t, λ)Ψ(t,
(22)
0
where, just as in (15), the kernel K(x, t, λ) is given by (16). Now set
χ(x, λ) =
b
Ψ(x,
λ)
γ∞ (x, λ)
to rewrite equation (22) in the form
Zx
χ(x, λ) = Im −
R(x, t, λ)χ(t, λ)dt,
(23)
0
where
R(x, t, λ) = C(x, t)
γ∞ (t, λ)
[(λ + q(t))Im − U (t)].
γ∞ (x, λ)
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
49
A.M. Kholkin and F.S. Rofe-Beketov
Since t ≤ x, one has
¯
¯ ¯
¯
¯
¯ ¯
γ0 (x, λ) ¯¯
2
¯C(x, t) γ∞ (t, λ) ¯ = ¯γ0 (t, λ)γ∞ (t, λ) − γ∞
(t, λ)
¯
γ∞ (x, λ) ¯ ¯
γ∞ (x, λ) ¯
¯



¯
¯
¯
Zt p
Zx p
¯
¯
1
¯




= p
1 − exp −2
v(u)du exp −2
v(u)du ¯¯
¯
2 v(t) ¯
¯
0
0




Zx p
1
1


= p
1 − exp −2
v(u)du ≤ p
,
2 v(t)
v(t)
t
and therefore
1
|R(x, t, λ)| ≤ p
[|λ| + |q(t)| + |U (t)|] ∈ L(0, ∞).
v(t)
It follows that integral equation (23) has a unique solution χ(x, λ), and |χ(x, λ)| ≤
const.
Pass in (23) to a limit as x → ∞ to get
e
lim χ(x, λ) = Im + C(λ),
x→∞
e
where C(λ)
is a triangular matrix, that is,
b
Ψ(x,
λ)
e
= Im + C(λ).
x→∞ γ∞ (x, λ)
lim
Now consider another matrix solution that increases

e 11 (x, λ) Ψ
e 12 (x, λ) . . .
Ψ

e 22 (x, λ) . . .
0
Ψ

e
Ψ(x,
λ) = 
...
...
 ...
0
0
...
(24)
at infinity

e 1r (x, λ)
Ψ
e 2r (x, λ)
Ψ

,
... 
e rr (x, λ)
Ψ
where
Zx
e kk (x, λ) = Φkk (x, λ)
Ψ
∗
−1
Φ−1
kk (t, λ)(Φkk (t, λ)) dt,
k = 1, r,
(a ≥ 0),
a
Φkk (x, λ) are the diagonal blocks of the matrix solution Φ(x, λ) as in Section 1.
50
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
In view of (11) and the definitions of γ0 (x, λ), γ∞ (x, λ), one has
Zx
e kk (x, λ) = γ0 (x, λ) (Im + ox (Im ))
Ψ
k
k
0
Zx
= γ0 (x, λ)
0
Hence
Imk + ot (Imk )
dt
γ02 (t, λ)
dt
(Imk + o (Imk )) = γ∞ (x, λ) (Imk + o (Imk )) . (25)
γ02 (t, λ)
e kk (x, λ)
Ψ
= Imk ,
x→∞ γ∞ (x, λ)
lim
k = 1, r.
(26)
b
e
Since Ψ(x,
λ) and Ψ(x,
λ) are the matrix solutions of (1) that increase at
infinity,
b
e
Ψ(x,
λ) = Ψ(x,
λ) + Φ(x, λ)C0 (λ),
(27)
where C0 (λ) is a block-triangular matrix. Additionally, one has
b
e
Ψ(x,
λ)
Ψ(x,
λ)
= lim
.
x→∞ γ∞ (x, λ)
γ∞ (x, λ)
lim
Hence, by virtue of (26),
lim
x→∞
Ψkk (x, λ)
= Imk ,
γ∞ (x, λ)
k = 1, m,
and in (24) one has

0 C12 (λ)
0
0
e
C(λ)
=
. . .
...
0
0

. . . C1r (λ)
. . . C2r (λ)
.
...
... 
...
0
³
´−1
b
e
is subject to conThe solution Ψ(x, λ) given by Ψ(x, λ) = Ψ(x,
λ) I + C(λ)
dition (13).
e 0 (x, λ) as
Use (11), (12) to differentiate (27), then find the asymptotics of Ψ
x → ∞ similarly to (25) to obtain (14). Theorem 1 is proved.
II. Now consider the case when v(x) = x2α , 0 < α ≤ 1. Suppose that the
coefficients of equation (1) satisfy the condition
Z∞
|U (t)|t−α dt < ∞.
(28)
0
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
51
A.M. Kholkin and F.S. Rofe-Beketov
Rewrite (1) in the form
¡
¢
−y 00 + x2α − λ + q(x, λ) y = (q(x, λ)Im − U (x))y,
(29)
where q(x, λ) is defined similarly to q(x) in (7):
5
q(x, λ) =
16
i.e.,
5α2
q(x, λ) =
4
µ
µ
v 0 (x)
v(x) − λ
x2α−1
x2α − λ
¶2
−
¶2
−
1 v 00 (x)
,
4 v(x) − λ
α (2α − 1) x2α−2
.
2 (x2α − λ)
(30)
Consider the functions

1
γ0 (x, λ) = p
4
4 (x2α − λ)
1
γ∞ (x, λ) = p
4
4 (x2α − λ)

Zx p
exp −
u2α − λdu ,
a
 x

Z p
exp 
u2α − λdu .
a
These solutions form a fundamental system of solutions for the scalar differential
equation
¡
¢
−z 00 + x2α − λ + q(x, λ) z = 0,
(31)
such that for all x
0
W (γ0 , γ∞ ) := γ0 (x, λ)γ∞
(x, λ) − γ00 (x, λ)γ∞ (x, λ) = 1.
We are about to establish the asymptotics? of γ0 (x, λ) as x → ∞:
 x

µ
µ
¶− 1
¶1
Z
4
2
1
λ
λ
exp − uα 1 − 2α
du .
γ0 (x, λ) = (2xα )− 2 1 − 2α
x
u
a
After expanding here the integrand, we obtain the exponential as follows:
 x

Ã
µ
¶k !
Z
∞
X
1
λ
1
·
3
·
.
.
.
·
(2k
−
3)
λ
exp − uα 1 −
−
du .
2 u2α
u2α
k!2k
a
k=2
?
With α = 1 and α = 12 , that is, with v(x) = x2 and v(x) = x, the asymptotics of γ0 (x, λ)
and γ∞ (x, λ) are already known. See the monograph [12], where the Langer method is applied
to obtain the asymptotics in a different form by using the Hankel function.
52
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
In the case
α+1
1
= n ∈ N, i.e., α =
, after integration this expression has
2α
2n − 1
the form
µ
x1+α
λ x1−α
c exp −
+
1+α 21−α
!
n−1
X 1 · 3 · . . . · (2k − 3) µ λ ¶k x1−(2k−1)α
+
k!
2
1 − (2k − 1)α
k=2
µ
µ ¶n
¶
1 · 3 · . . . · (2n − 3) λ
× exp
ln x + o(1)
n!
2
µ
x1+α
λ x1−α
= c exp −
+
1+α 21−α
!
µ
¶
n−1
X 1 · 3 · . . . · (2k − 3) λ k x1−(2k−1)α
1·3·...·(2n−3) λ n
( 2 ) (1 + o(1)).
n!
+
x
k!
2
1 − (2k − 1)α
k=2
The asymptotics of γ0 (x, λ) as x → ∞ is as follows:
µ
λ x1−α
x1+α
+
γ0 (x, λ) = c exp −
1+α 21−α
!
n−1
X 1 · 3 · . . . · (2k − 3) µ λ ¶k x1−(2k−1)α
1·3·...·(2n−3) λ n α
(2) −2
n!
x
+
k!
2
1 − (2k − 1)α
k=2
× (1 + o(1)). (32)
With α = 1 (n = 1), γ0 (x, λ) has the following asymptotics at infinity:
µ 2¶
λ−1
x
(1 + o(1)).
(33)
γ0 (x, λ) = c x 2 exp −
2
·
¸
α+1
α+1
In the case
∈
/ N, set n =
+ 1, with [β] being the integral part
2α
2α
of β, to obtain the following asymptotics for γ0 (x, λ) at infinity:
µ
α
x1+α
λ x1−α
+
γ0 (x, λ) = c x− 2 exp −
1+α 21−α
!
n−1
X 1 · 3 · . . . · (2k − 3) µ λ ¶k x1−(2k−1)α
+
k!
2
1 − (2k − 1)α
k=2
µ ¶n −α ¶
µ
¡
¡
¢¢
x
1 · 3 · . . . · (2n − 3) λ
1 + o x−α . (34)
× exp −
n!
2
α
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
53
A.M. Kholkin and F.S. Rofe-Beketov
In particular, with α =
Ã
γ0 (x, λ) = cx
− 14
1
2
(n = 2) one has
1
2 3
exp − x 2 + λx 2 −
3
!
µ ¶2
³
³ 1 ´´
1
λ
x− 2
1 + o x− 2
.
2
(35)
A similar procedure allows one to establish the asymptotics of γ∞ (x, λ) as
α+1
1
x → ∞. If
= n ∈ N, i.e., α =
, then
2α
2n − 1
µ
x1+α
λ x1−α
−
1+α 21−α
! ³
´
n−1
1·3·...·(2n−3) λ n α
X 1 · 3 · . . . · (2k − 3) µ λ ¶k x1−(2k−1)α
−
(2) +2
n!
−
x
k!
2
1 − (2k − 1)α
γ∞ (x, λ) = c exp
k=2
× (1 + o(1)). (36)
With α = 1 (n = 1), this becomes
¶
x2
(1 + o(1)).
2
·
¸
α+1
α+1
∈
/ N, we set n =
+ 1 to get the asymptotics
In the case
2α
2α
µ
γ∞ (x, λ) = c x−
µ
α
γ∞ (x, λ) = c x− 2 exp
λ+1
2
exp
(37)
x1+α
λ x1−α
−
1+α 21−α
!
µ ¶
1 · 3 · . . . · (2k − 3) λ k x1−(2k−1)α
−
k!
2
1 − (2k − 1)α
k=2
µ ¶n −α ¶
µ
¡
¡
¢¢
x
1 · 3 · . . . · (2n − 3) λ
1 + o x−α . (38)
× exp
n!
2
α
n−1
X
1
2
(n = 2), one has
Ã
!
µ ¶2
³
³ 1 ´´
1
1
2 3
λ
−
− 14
x 2 − λx 2 +
x 2
γ∞ (x, λ) = cx exp
.
1 + o x− 2
3
2
In the case α =
(39)
Theorem 2. With 0 < α ≤ 1 and under condition (28), the claim of Theorem
1 is valid for equation (1).
54
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
The proof goes similarly to that of Theorem 1. It should be observed in this
context that
¯
¯
¯ ¯
¯
¯
¯ ¯
¯C(x, t, λ) γ0 (t, λ) ¯ = ¯γ02 (t, λ) γ∞ (x, λ) − γ0 (t, λ)γ∞ (t, λ)¯
¯
¯
¯
¯
γ0 (x, λ)
γ0 (x, λ)
¯




¯
¯
Zt p
Zx p
¯
¯
1
1
2α
2α
¯
¯




=¯ √
exp −2
u − λdu exp 2
u − λdu − √
2α
2α
2 t − λ¯
¯2 t − λ
a
a
¯

¯
¯
¯
Zt p
¯
¯
1
2α
¯


exp −2
= √
u − λdu − 1 ¯¯ .
¯
2α
2 t −λ¯
¯
x
µ
¶
Rt √
2α
As x ≤ t, one has exp −2
u − λdu ≤ 1, hence
x
¯
¯
¯
¯
¯C(x, t, λ) γ0 (t, λ) ¯ ≤ √ 1
,
¯
γ0 (x, λ) ¯
t2α − λ
and thus
(40)
¯
¯
¯
¯
γ0 (t, λ)
¯
[q(t, λ)Im − U (t)]¯¯
|R(x, t, λ)| = ¯C(x, t, λ)
γ0 (x, λ)
1
(|q(t, λ)| + |U (t)|).
≤√
2α
t −λ
It follows from (28) and (30) that
1
√
(|q(t, λ)| + |U (t)|) ∈ L(a, ∞).
t2α − λ
(41)
Hence integral equation (18) has a unique solution χ(x, λ) and |χ(x, λ)| ≤ const.
By (18), we have that lim χ(x, λ) = I, which implies (11).
x→∞
The rest of the claims of Theorem 1 can be obtained in a similar way.
Theorem 1, together with asymptotic formulas (33) and (37), allows one to
deduce
Corollary 1. With α = 1, i.e., v(x) = x2 , under condition (28), equation (1)
has a unique m × m matrix solution Φ(x, λ), which decreases at infinity, satisfies
the conditions
Φ(x, λ)
lim
= Im ,
(42)
x→∞ γ0 (x, λ)
and
Φ0 (x, λ)
= Im ,
x→∞ γ 0 (x, λ)
0
lim
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
55
A.M. Kholkin and F.S. Rofe-Beketov
³ 2´
λ−1
where γ0 (x, λ) = x 2 exp − x2 . Also, the equation has a unique m × m matrix
solution Ψ(x, λ), which increases at infinity, satisfies the conditions
lim
x→∞
and
Ψ(x, λ)
= Im
γ∞ (x, λ)
Ψ0 (x, λ)
= Im ,
0 (x, λ)
x→∞ γ∞
³ 2´
exp x2 .
lim
where γ∞ (x, λ) = x−
λ+1
2
Theorem 2, together with asymptotic formulas (35) and (39), implies
Corollary 2. If α = 12 , i.e., v(x) = x, under conditions (28), equation (1)
has a unique m × m matrix solution Φ(x, λ), which decreases at infinity, satisfies
the conditions
Φ(x, λ)
lim
= Im ,
x→∞ γ0 (x, λ)
and
Φ0 (x, λ)
= Im ,
x→∞ γ 0 (x, λ)
0
³
´
1
3
1
where γ0 (x, λ) = x− 4 exp − 23 x 2 + λx 2 . Also, the equation has an m × m
matrix solution Ψ(x, λ) which increases at infinity, satisfies the conditions
lim
Ψ(x, λ)
= Im
x→∞ γ∞ (x, λ)
lim
and
Ψ0 (x, λ)
= Im ,
0 (x, λ)
x→∞ γ∞
³ 3
´
1
exp 23 x 2 − λx 2 .
lim
1
where γ∞ (x, λ) = x− 4
R e m a r k 1. In the monograph [14], it was shown that the scalar equation
−φ00 + x2 φ = λφ,
(43)
with λ = 2n + 1, has a solution
µ 2¶
x
φn (x) = Hn (x) exp −
,
2
56
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
where Hn (x) is the Chebishev–Hermit polynomial. Note that this polynomial
has the following asymptotics as x → ∞: Hn (x) = (2x)n (1 + o(1)), and thus the
asymptotics of the solution φn (x) of (43) as x → ∞ is
µ 2¶
x
n
· (1 + o(1)).
(44)
φn (x) = (2x) exp −
2
In the case when in (2) one has U (x) = 0, v(x) = x2 , matrix equation (1)
splits into m scalar equations of the form (43). The matrix solution Φ(x, λ) in
this case appears to be diagonal. Denote by φ(x, λ) the diagonal elements of the
matrix Φ(x, λ). Then by virtue of (42), the solution φ(x, λ) has the following
asymptotics at infinity:
µ 2¶
λ−1
x
φ(x, λ) = (x) 2 exp −
(1 + o(1)).
(45)
2
In particular, with λ = 2n + 1 this allows one to derive a solution which is a
scalar multiple of φn (x).
III. Suppose at x = 0, we are given the boundary condition
By 0 (0) − Cy(0) = 0,
(46)
where B and C are the commuting block-triangular matrices of the same structure
as the coefficients of the differential equation




B11 B12 . . . B1r
C11 C12 . . . C1r
 0 B22 . . . B2r 
 0 C22 . . . C2r 


B=
C=
(47)
. . . . . . . . . . . .  ,
... ... ... ...,
0
0 . . . Brr
0
0 . . . Crr
Bkk , Ckk , k = 1, r, are mk × mk Hermitian matrices and mk ≥ 1,
r
P
k=1
mk = m,
r
¡
¢
¡ 2
¢
Q
2
which satisfy the conditions det B 2 + C 2 =
det Bkk
+ Ckk
6= 0. It follows
k=1
from BC = CB that Bkk Ckk = Ckk Bkk , k = 1, r.
Lemma 1. The boundary condition (46) can be rewritten in the equivalent
form:
cos Ay 0 (0) − sin Ay(0) = 0,
(48)
where A is a block-triangular matrix of the same structure as the matrices B, C
in (47).
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
57
A.M. Kholkin and F.S. Rofe-Beketov
In fact, since B and C commute, each of the matrices commutes with the
1
matrix (B 2 + C 2 ) as well as with the matrix (B 2 + C 2 )− 2 . It follows that there
exists a block-triangular matrix A such that
1
1
sin A = (B 2 + C 2 )− 2 C.
cos A = (B 2 + C 2 )− 2 B,
1
After multiplying on the left by (B 2 + C 2 )− 2 , the boundary condition (46) has
the form of (48).
Together with problem (1) with boundary condition (48), consider the split
system
lk [y k ] = −y 00k + (v(x)Imk + Ukk (x))y k = λy k ,
k = 1, r,
with the boundary conditions
cos Akk · y 0k (0) − sin Akk · y k (0) = 0,
k = 1, r,
(49)
where Akk are the diagonal elements of the matrix A, Akk , k = 1, r are mk × mk
r
P
Hermitian matrices with mk ≥ 1,
mk = m.
k=1
Denote by L0 the minimal differential operator generated by the differential
expression l [y] and boundary condition (48). Let also Lk , k = 1, r, stand for the
minimal symmetric operators in L2 (Hmk , (0, ∞)) generated by the differential
expressions lk [y k ] and boundary conditions (49). By the assumptions on the
coefficients, along with the relatively small amount of the perturbations Ukk (x)
(|U (x)| · v −1 (x) → 0 as x → ∞ or |U | · v −1 ∈ L∞ (R)), we are in the case of
a limit point at infinity for each symmetric operator Lk , k = 1, r. Hence their
e k are given by the closures of Lk . The operators L
e k are
selfadjoint extensions? L
semibounded, and their spectra are discrete.
Denote by L the extension of the operator L0 given by the requirement that
the functions from the domain of L are in L2 (Hm , (0, ∞)).
Theorem 3. Consider equation (1). If either conditions (3), (4), (5) with
α > 1 or condition (28) with 0 < α ≤ 1 are satisfied, then the discrete spectrum
of L is real and coincides with the union of spectra of the selfadjoint operators
e k , k = 1, r, i.e.,
L
r
³ ´
[
ek .
σ L
(50)
σd (L) =
k=1
?
The selfadjointness property for the general (4-term) Sturm–Liouville differential equations
with matrix coefficients was studied in [24].
58
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
P r o o f. Recall that the matrix solutions Φ(x, λ), Ψ(x, λ), as in Theorems
1, 2, form a fundamental system of solutions of (1). It follows that every vector
solution y(x, λ) of (1) is representable in the form
y(x, λ) = Φ(x, λ)h + Ψ(x, λ)g,
with h and g being constant vectors. A solution y(x, λ) ∈ L2 (Hm , (0, ∞)) if and
only if g = 0, that is,
y(x, λ) = Φ(x, λ)h.
The eigenvalues of L coincide with zeros of the determinant ∆(λ) := det Ω(0, λ),
where
Ω(0, λ) = cos A Φ0 (0, λ) − sin A Φ(0, λ).
(51)
Since the matrices A, Φ(0, λ), Φ0 (0, λ) are block-triangular, one has ∆(λ) =
r
Q
k=1
∆k (λ), where ∆k (λ) := det Ωk (0, λ),
Ωk (0, λ) = cos Akk Φ0kk (0, λ) − sin Akk Φkk (0, λ).
On the other hand, zeros of ∆k (λ), k = 1, r, (cf. (49)) are the eigenvalues
e k and hence are real. It follows that the discrete
of the selfadjoint operator L
ek ,
spectrum of L is real and coincides with the union of spectra of the operators L
k = 1, r. The Theorem is proved.
R e m a r k 2. In the case α = 1 under the absence of condition (28), the
claim of Theorem 3 can fail to be true.
E x a m p l e 1. Consider the equation
µ 2
¶
x q(x)
00
l [y] = −y +
y = λy,
0 π 2 x2
0 ≤ x < ∞,
µ ¶
y1
,
y=
y2
(52)
with the boundary condition
y(0) = 0.
(53)
e 1 and L
e 2 generated by problem (52), (53) are
The eigenvalues of the operators L
e 2 , and y2 (x, λ0 ) be the corresponding
not the same. Let λ0 be an eigenvalue of L
eigenfunction. λ0 is an eigenvalue of L in the case when the solution y1 (x, λ0 ) of
the equation −y100 + x2 y1 + q(x)y2 = λ0 y1 , which satisfies the initial conditions
y1 (0, λ) = y10 (0, λ) = 0, is in L2 (0; ∞). This solution is given by
Zx
y1 (x, λ0 ) =
q(t)C(x, t, λ0 )y2 (t, λ0 )dt,
0
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
59
A.M. Kholkin and F.S. Rofe-Beketov
where C(x, t, λ0 ) is the Cauchy function of equation (43). With the coefficient
µ
q(x) = y2 (x)ex and µ > 2, one can prove (to be done in a subsequent work) that
R∞
the integral y12 (x, λ0 )dx diverges and hence λ0 ∈
/ σ(L).
0
It is to be shown in the separate paper
³ that
´ S the
³ (set
´ of) poles of the Green
³ ´
e1
e 2 . If some λ0 ∈ σ L
e1 ,
function G(x, t, λ) of L coincide with σ L
σ L
³ ´
e 2 is an eigenvalue of L if and only
then λ0 ∈ σ(L). On the other hand, λ0 ∈ σ L
³ ´
e 2 and y1 (x, λ0 ) ∈
/ L2 (0, ∞) as in Example 1,
if y1 (x, λ0 ) ∈ L2 (0, ∞). If λ0 ∈ σ L
then λ0 is a pole of the Green function G(x, t, λ) but not an eigenvalue. These
λ0 are called the spectral singularities of the operator L. The special features of
these points were initially discovered by M.A. Naimark in [15]. The term ‘spectral
singularity’ was coined later on by J. Schwartz [16] (see also the monograph by
M.A. Naimark [17], Supplement I by V.E. Ljance [18]).
IV. Suppose that in (1) one has v(x) ≡ 0, inf Ukk (x) → +∞ as x → ∞ for
all k, and Ujk (x) with j 6= k are compactly supported matrix valued functions
whose carrier is [0, b].
Denote by Φk (x, λ), Θk (x, λ) the matrix solutions of the equation
lk [y k ] = −y 00k + Ukk (x)y k = λy k ,
k = 1, r,
(54)
which satisfy the initial conditions
Φk (0, λ) = 0,
Φ0k (0, λ) = Imk ,
Θk (0, λ) = −Imk ,
Θ0k (0, λ) = 0.
For all nonreal λ there exists a matrix Weyl function? Mk (λ) such that for
any hk ∈ Hmk the solution
(Θk (x, λ) + Φk (x, λ)Mk (λ))hk
of equation (54) is in L2 (Hmk , (0, ∞)). The function Mk (λ) is meromorphic,
with its poles being on the real axis. These poles are just the points of spece k generated in L2 (Hm , (0, ∞)) by the differential expression lk [y k ]
trum of L
k
and the boundary condition y k (0, λ) = 0. The matrix valued meromorphic function Mk (λ) admits a representation in the form
Mk (λ) = Mk (λ)(Nk (λ))−1 ,
?
The theory of matrix Weyl functions is a subject for a large number of papers. An extended
bibliography on this topic can be found in the monograph [10]. Here we restrict ourselves to
referring to [19]–[23].
60
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients
where Mk (λ), Nk (λ) are entire matrix valued functions of λ. In fact, all the
(k)
matrix elements mij (λ) of Mk (λ) can be represented as ratios of entire functions
with no common zeros
(k)
µij (λ)
(k)
mij (λ) = (k)
.
νij (λ)
³
´mk
(k)
(k)
We choose Nk (λ) = nij (λ)
to be a diagonal matrix with njj (λ) =
m
Qk
i=1
i,j=1
(k)
νij (λ).
In this case, the matrix Mk (λ) is also an entire function of λ.
Then the function
Yk (x, λ) = Θk (x, λ)Nk (λ) + Φk (x, λ)Mk (λ)
is a matrix solution of equation (54). For any vector hk ∈ Hmk , the vector
function y k (x, λ) = Yk (x, λ)hk ∈ L2 (Hmk , (0, ∞)) and the matrix valued function
Yk (x, λ) are entire analytic in λ on the complex plane at every x ∈ [0, ∞).
The matrix


Y1 (x, λ)
0
...
0
 0
Y2 (x, λ) . . .
0 

Y (x, λ) = 
 ...
...
...
... 
0
0
. . . Yr (x, λ)
is a solution of equation (1) for all x ≥ b; given any vector h ∈ Hm , one has
y(x, λ) = Y (x, λ)h ∈ L2 (Hm , (0, ∞)). Denote by Φ(x, λ) the matrix solution of
equation (1) which satisfies the following initial condition at x = b:
Φ(b, λ) = Y (b, λ),
Φ0 (b, λ) = Y 0 (b, λ).
Φ(x, λ) is a solution of equation (1) for all x ∈ [0, ∞). It has a block-triangular
form


Φ11 (x, λ) Φ12 (x, λ) . . . Φ1r (x, λ)

0
Φ21 (x, λ) . . . Φ2r (x, λ)
 ≡ Y (x, λ) with x ≥ b.
Φ(x, λ) = 
 ...
...
...
... 
0
0
. . . Φrr (x, λ)
Hence, for any vector h one has y(x, λ) = Φ(x, λ)h ∈ L2 (Hm , (0, ∞)). This
solution Φ(x, λ) is an entire analytic function of λ.
The eigenvalues of L coincide with zeros of the determinant ∆(λ) := det Ω(0, λ),
with the matrix Ω(0, λ) being given by (51).
Now we are in a position to reproduce the final part of the proof of Theorem
3 in order to deduce
Journal of Mathematical Physics, Analysis, Geometry, 2014, vol. 10, No. 1
61
A.M. Kholkin and F.S. Rofe-Beketov
Theorem 4. Suppose in (1) we have v(x) ≡ 0, inf Ukk (x) → +∞ as x → ∞
for all k, and Ujk (x) (j 6= k) are compactly supported functions whose carrier is
[0, b]. Then the spectrum of L is discrete, real, and coincides with the union of
e k , k = 1, r.
spectra of the operators L
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