Integral Representations for Spectral Functions of Some Nonself-Adjoint Jacobi Matrices
We study a Jacobi matrix J with complex numbers an, n belongs Z+, in the main diagonal such that r0 ≤ Im an ≤ r1, r0, r1 belongs Z+. We obtain an integral representation for the (generalized) spectral function of the matrix J. Themethod of our study is similar to Marchenko’s method for nonself-adjoi...
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| Zitieren: | Integral representations for spectral functions of some nonself-adjoint Jacobi matrices / S.M. Zagorodnyuk // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 91-100. — Бібліогр.: 7 назв. — англ. |
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| citation_txt | Integral representations for spectral functions of some nonself-adjoint Jacobi matrices / S.M. Zagorodnyuk // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 91-100. — Бібліогр.: 7 назв. — англ. |
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| description | We study a Jacobi matrix J with complex numbers an, n belongs Z+, in the main diagonal such that r0 ≤ Im an ≤ r1, r0, r1 belongs Z+. We obtain an integral representation for the (generalized) spectral function of the matrix J. Themethod of our study is similar to Marchenko’s method for nonself-adjoint differential operators.
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Methods of Functional Analysis and Topology
Vol. 15 (2009), no. 1, pp. 91–100
INTEGRAL REPRESENTATIONS FOR SPECTRAL FUNCTIONS OF
SOME NONSELF-ADJOINT JACOBI MATRICES
S. M. ZAGORODNYUK
Abstract. We study a Jacobi matrix J with complex numbers an, n ∈ �+, in
the main diagonal such that r0 ≤ Im an ≤ r1, r0, r1 ∈ �. We obtain an integral
representation for the (generalized) spectral function of the matrix J . The method of
our study is similar to Marchenko’s method for nonself-adjoint differential operators.
1. Introduction
The main object of our present investigation will be a three-diagonal semi-infinite
complex number matrix of the following form:
(1) J =
a0 b0 0 0 . . .
b0 a1 b1 0 . . .
0 b1 a2 b2 . . .
...
...
...
...
. . .
,
where bn > 0, and
(2) an ∈ C : r0 ≤ Im an ≤ r1,
for some r0, r1 ∈ R, n ∈ Z+.
Thus, in the case r0 = r1 = 0 we obtain the classical Jacobi matrix. The spectral
theory of Jacobi matrices is classic, see [1], [2], [3]. For the Jacobi matrix J , there is a
corresponding non-decreasing function σ(x), x ∈ R, which is called a spectral function.
The procedure of a construction of σ(x) provides a solution of the direct spectral problem
for J . The inverse spectral problem is to reconstruct J from σ. The corresponding
procedure is well-known and simple.
Recently, we have introduced a notion of a spectral function for some nonself-adjoint
semi-infinite banded matrices, see [4], [5]. The spectral function is a bilinear (that means
linear with respect to the both arguments) functional σ(u, v), u, v ∈ P, defined on a set of
complex polynomials P. We will use methods which were applied by Marchenko to some
nonself-adjoint Sturm-Liouville operators (see [6]) and obtain an integral representation
for the spectral function σ(u, v) of the matrix J from (1).
Notations. As usual, we denote by R, C, N, Z, Z+ the sets of real, complex, positive
integer, integer, non-negative integer numbers, respectively. By P we denote the set
of all polynomials with complex coefficients. By l2 we denote a space of vectors x =
(x0, x1, x2, . . .), xn ∈ C, n ∈ Z+, such that ‖x‖ := (
∑
∞
n=0 |xn|
2)
1
2 < ∞. By l2fin we
denote a subset of l2 which consists of finite vectors, i.e., vectors x = (x0, x1, x2, . . .),
xn ∈ C, n ∈ Z+, with only a finite number of nonzero elements xn.
2000 Mathematics Subject Classification. Primary 47B36; Secondary 47B39, 39A70.
Key words and phrases. Jacobi matrix, semi-infinite matrix, spectral function, difference equation.
91
92 S. M. ZAGORODNYUK
2. Polynomials of the first and of the second kinds
Let J be the semi-infinite matrix from (1), (2). Consider the following difference
equations:
(3) a0y0 + b0y1 = λy0,
(4) bn−1yn−1 + anyn + bnyn+1 = λyn, n ∈ N,
where yn are unknowns and λ is a complex parameter.
By Pn(λ), n ∈ Z+, we denote a solution of (3), (4) with the initial condition P0 = 1.
Polynomials Pn(λ) we will call polynomials of the first kind. Denote by Qn(λ), n ∈ Z+,
a solution of (4) with the initial conditions Q0 = 0, Q1 = 1
b0
. Polynomials Qn(λ) we will
call polynomials of the second kind.
If we write relation (4) for Pn and then multiply it by Qn, we will get
(5) bn−1Pn−1Qn + anPnQn + bnPn+1Qn = λPnQn, n ∈ N.
In a similar manner we will get
(6) bn−1Qn−1Pn + anQnPn + bnQn+1Pn = λQnPn, n ∈ N.
Subtract (6) from (5) to get
bn−1(Pn−1Qn − PnQn−1) = bn(PnQn+1 − Pn+1Qn), n ∈ N.
Using the initial conditions we obtain
(7) Pn−1(λ)Qn(λ) − Pn(λ)Qn−1(λ) =
1
bn−1
, n ∈ N.
We will use relation (7) in the sequel.
Proposition 1. Let yn = yn(λ), n ∈ Z+, be an arbitrary solution of difference equa-
tion (4). The following relation holds true:
(8)
n−1∑
j=1
(Im aj − Im λ)|yj(λ)|2 = bn−1 Im(yn−1(λ)yn(λ)) − b0 Im(y0(λ)y1(λ)),
n = 2, 3, . . .
Proof. Set ân = ân(λ) = an − λ, n ∈ N, and rewrite relation (4) in the following form:
(9) bn−1yn−1 + ânyn + bnyn+1 = 0, n ∈ N.
Apply the complex conjugation to the both sides of (9) to get
(10) bn−1yn−1 + ânyn + bnyn+1 = 0, n ∈ N.
Multiply relation (9) by yn, relation (10) by yn, and then subtract to obtain
(11) bn−1(yn−1yn − yn−1yn) + (ân − ân)ynyn + bn(yn+1yn − yn+1yn) = 0, n ∈ N.
Set
An = An(λ) = bn(ynyn+1 − ynyn+1) = bn2i Im(ynyn+1), n ∈ Z+.
Then we can write
(12) (ân − ân)ynyn = An − An−1, n ∈ N.
Summing up we obtain
(13)
n−1∑
j=1
(âj − âj)yjyj = An−1−A0 = 2ibn−1 Im(yn−1yn)−2ib0 Im(y0y1), n = 2, 3, . . .
Therefore relation (8) is true. �
INTEGRAL REPRESENTATIONS FOR SPECTRAL FUNCTIONS . . . 93
Corollary 1. Let Pn(λ) and Qn(λ), n ∈ Z+, be polynomials of the first and of the
second kinds for difference equations (3), (4), respectively. Polynomials Pn(λ) satisfy the
following relation:
(14)
n−1∑
j=0
(Im aj − Im λ)|Pj(λ)|2 = bn−1 Im(Pn−1(λ)Pn(λ)), n ∈ N.
Choose an arbitrary w ∈ C and consider the polynomials
(15) Ψn(λ, w) = wPn(λ) + Qn(λ), n ∈ Z+.
The polynomials Ψn(λ, w) satisfy the following relation:
(16)
n−1∑
j=0
(Im aj − Im λ)|Ψj(λ, w)|2 = bn−1 Im(Ψn−1(λ, w)Ψn(λ, w)) − Im w, n ∈ N.
Proof. To obtain relations (14), (16) for n ≥ 2, it is sufficient to write relation (8) for
the polynomials Pn(λ) and Ψn(λ, w), respectively, and to use the initial conditions. For
the case n = 1 relations (14), (16) can be verified using the initial conditions. �
Set
(17) Π = Π(r0, r1) = {λ ∈ C : r0 ≤ Im λ ≤ r1}.
Corollary 2. Let Pn(λ), n ∈ Z+, be polynomials of the first kind for difference equa-
tions (3), (4). The roots of polynomials Pn(λ) lie in the strip Π(r0, r1).
Proof. For an arbitrary root λ0 ∈ C of Pn−1(λ), n = 2, 3, . . . , by (14) we obtain
(18)
n−1∑
j=0
(Im aj − Im λ0)|Pj(λ0)|
2 = 0.
Suppose that Imλ0 > r1. By (2) we obtain
Im aj − Im λ0 < 0, j ∈ Z+.
Then (18) leads to a contradiction since P0 = 1. If we suppose that Im λ0 < r0, we will
get
Im aj − Im λ0 > 0, j ∈ Z+.
That contradicts relation (18) as well. �
3. Weyl’s discs
Like in the classical case (see [1]), an important role in our further considerations will
play the following function:
(19) wn(λ, τ) = −
Qn(λ) − τQn−1(λ)
Pn(λ) − τPn−1(λ)
,
where λ, τ ∈ C, n ∈ N (Pn, Qn are polynomials of the first and of the second kinds for
difference equations (3), (4)). We set
(20) Π+ = Π+(r1) = {λ ∈ C : Im λ > r1}, Π− = Π−(r0) = {λ ∈ C : Im λ < r0},
(21) Π0 = Π0(r0, r1) = Π+(r1) ∪ Π−(r0).
1) Choose an arbitrary λ ∈ Π+(r1) and n ∈ N. By virtue of Corollary 2, relations (14)
and (2) we get
bn−1 Im(Pn−1(λ)Pn(λ)) = bn−1|Pn−1(λ)|2 Im
((
Pn(λ)
Pn−1(λ)
))
< 0.
94 S. M. ZAGORODNYUK
Thus, we have
(22) Im
(
Pn(λ)
Pn−1(λ)
)
> 0.
So, a pole of the map wn(λ, τ) (for the fixed λ ∈ Π+, n ∈ N) lies in the upper half-plane
C′
+ = {λ ∈ C : Im λ > 0}. In particular, this means that the real line R is mapped on a
circle Cn(λ) in the w-plane (the complex plane of the variable w). The lower half-plane
C− = {τ ∈ C : Im τ ≤ 0} is mapped on a disc Dn(λ). The inverse map for wn(λ, τ) has
the following form:
(23) τn(λ, w) =
wPn(λ) + Qn(λ)
wPn−1(λ) + Qn−1(λ)
=
Ψn(λ, w)
Ψn−1(λ, w)
.
For an arbitrary w ∈ C : Ψn−1(λ, w) �= 0 (this means that w �= −Qn−1(λ)
Pn−1(λ) ) by virtue of
relation (16) we can write
(24)
n−1∑
j=0
(Im aj − Im λ)|Ψj(λ, w)|2 = −bn−1|Ψn−1(λ)|2 Im(
Ψn(λ, w)
Ψn−1(λ)
) − Im w.
In our case we have Im aj − Im λ < 0, j ∈ Z+. Therefore
(25)
n−1∑
j=0
| Im aj − Im λ||Ψj(λ, w)|2 = Im w + bn−1|Ψn−1(λ)|2 Im τn(λ, w).
From the last relation and relation (16) for the case w = −Qn−1(λ)
Pn−1(λ) , we see that the disc
Dn(λ) consists of w ∈ C such that
(26)
n−1∑
j=0
| Im aj − Im λ||Ψj(λ, w)|2 ≤ Im w.
From relation (26) it follows that
(27) Dn+1(λ) ⊆ Dn(λ), n ∈ N, λ ∈ Π+.
Hence, there exists a non-empty intersection D∞(λ) = ∩j∈NDj(λ). From relation (26) it
follows that D∞(λ) consists of w ∈ C such that
(28)
∞∑
j=0
| Im aj − Im λ||Ψj(λ, w)|2 ≤ Im w.
2) Choose an arbitrary λ ∈ Π−(r0) and n ∈ N. Reasoning similarly, we obtain that
(29) Im
(
Pn(λ)
Pn−1(λ)
)
< 0.
A pole of the map wn(λ, τ) lies in the lower half-plane C
′
−
= {λ ∈ C : Im λ < 0}. The
real line is mapped on a circle Cn(λ) and the upper half-plane C+ = {τ ∈ C : Im τ ≥ 0}
is mapped on a disc Dn(λ). For w ∈ C : Ψn−1(λ, w) �= 0, by virtue of relation (16) we
can write relation (24). In our case we have Im aj − Im λ > 0, j ∈ Z+, therefore
(30)
n−1∑
j=0
| Im aj − Im λ||Ψj(λ, w)|2 = − Imw − bn−1|Ψn−1(λ)|2 Im τn(λ, w).
From relation (15) and relation (16) for the case w = −Qn−1(λ)
Pn−1(λ) , we see that the disc
Dn(λ) consists of w ∈ C such that
(31)
n−1∑
j=0
| Im aj − Im λ||Ψj(λ, w)|2 ≤ − Imw.
INTEGRAL REPRESENTATIONS FOR SPECTRAL FUNCTIONS . . . 95
From relation (31) it follows that
(32) Dn+1(λ) ⊆ Dn(λ), n ∈ N, λ ∈ Π+.
Thus, there exists a non-empty intersection D∞(λ) = ∩j∈NDj(λ). From relation (31) it
follows that D∞(λ) consists of w ∈ C such that
(33)
∞∑
j=0
| Im aj − Im λ||Ψj(λ, w)|2 ≤ − Imw.
The radius of Cn(λ) is denoted by rn(λ), λ ∈ Π0. We will need an analytic expression
for rn(λ).
Proposition 2. Let λ ∈ Π0 and n ∈ N. The radius of the circle Dn(λ) is equal to
(34) rn(λ) =
1
bn−1|Pn(λ)Pn−1(λ) − Pn−1(λ)Pn(λ)|
=
1
2
∑n−1
j=0 | Im aj − Im λ||Pj(λ)|2
.
Proof. To obtain the first equality in (34), one repeats the standard arguments from the
proof of Theorem 1.2.3 in [1]. The second equality follows from relation (14). �
Consider a sequence of functions,
ŵn(λ) := wn(λ, 0) = −
Qn(λ)
Pn(λ)
, λ ∈ Π0(r0, r1), n ∈ N.
Notice that ŵn(λ) ∈ Dn(λ), n ∈ N, λ ∈ Π0. Hence, using relations (26), (31) we can
write
|ŵn(λ)|2 = |ŵn(λ)P0(λ) + Q0(λ)|2 ≤
n−1∑
j=0
| Im aj − Im λ|
| Im a0 − Im λ|
|ŵn(λ)Pj(λ) + Qj(λ)|2
≤
| Im ŵn(λ)|
| Im a0 − Im λ|
≤
|ŵn(λ)|
| Im a0 − Im λ|
.
Consequently, we obtain
(35) |ŵn(λ)| ≤
1
| Im a0 − Im λ|
, λ ∈ Π0, n ∈ N.
Thus, in any compact subset of Π0, the sequence of functions ŵn(λ) is uniformly bounded.
The functions ŵn(λ) are analytic in Π0 as it follows from Corollary 2. By virtue of
Montel’s theorem (see [7]) we can assert that there exists a subsequence ŵnk
(λ), k ∈ N,
which is uniformly convergent to a function m(λ) in Π0. The function m(λ) is analytic
by Weierstrass’s theorem. Passing to the limit in (35) with n = nk, k → ∞, we obtain
(36) |m(λ)| ≤
1
| Im a0 − Im λ|
, λ ∈ Π0.
Observe that m(λ) ∈ Dn(λ) for any n ∈ N, and therefore
(37) m(λ) ∈ D∞(λ), λ ∈ Π0.
For an arbitrary ε > 0 we set
Π0,ε = Π0,ε(r0, r1) = {λ ∈ C : Im λ ≤ r0 − ε} ∪ {λ ∈ C : Im λ ≥ r1 + ε}.
Proposition 3. For the function m(λ) the following relation holds true:
(38) m(λ) → 0, λ → ∞, λ ∈ Π0,ε, ε > 0.
96 S. M. ZAGORODNYUK
Proof. Since m(λ) ∈ Dn(λ), ŵn(λ) ∈ Dn(λ), n ∈ N, λ ∈ Π0, we can write
(39)
∣∣∣∣m(λ) +
Qn(λ)
Pn(λ)
∣∣∣∣ ≤ 2rn(λ), λ ∈ Π0,ε, ε > 0, n ∈ N.
From relation (34) we see that
|rn(λ)| ≤
1
2| Ima1 − Im λ||P1(λ)|2
≤
b2
0
2ε|λ − a0|2
, λ ∈ Π0,ε, n = 2, 3, . . .
Thus, for any fixed n, n = 2, 3, . . ., we obtain
rn(λ) → 0, λ → ∞, λ ∈ Π0,ε.
Passing to the limit in relation (39) we see that
m(λ) +
Qn(λ)
Pn(λ)
→ 0, λ → ∞, λ ∈ Π0,ε.
It remains to notice that
Qn(λ)
Pn(λ)
→ 0, λ → ∞,
since deg Qn = n − 1, deg Pn = n. �
The following theorem is valid.
Theorem 1. Difference equation (4) has a solution yn = m(λ)Pn(λ) + Qn(λ), n ∈ Z+,
which belongs to l2 for any λ ∈ Π0.
Proof. Since the function m(λ), λ ∈ Π0, belongs to the disc D∞(λ), from relations (28),
(33) it follows that
(40)
∞∑
j=0
| Im aj − Im λ||m(λ)Pj(λ) + Qj(λ)|2 < ∞.
Since | Im aj−Im λ| ≥ Im λ−r1 > 0, λ ∈ Π+, and | Im aj−Imλ| ≥ r0−Im λ > 0, λ ∈ Π−,
the result follows. �
4. The spectral function
Let J be the semi-infinite matrix from (1), (2). Observe that it is a matrix that is
complex symmetric (with respect to the transposition). Let {Pn(λ)}n∈Z+
, {Qn(λ)}n∈Z+
be the defined above solutions of the corresponding difference equations (3),(4). Recall
(see [4, p. 474]) that a linear with respect to the both arguments functional σ(u, v), u, v ∈
P, is called a spectral function of difference equations (3),(4) if it satisfies relations
(41) σ(Pn, Pm) = δn,m, n, m ∈ Z+.
For the given difference equations (3), (4) it is not hard to obtain the spectral function
using (41) as a definition and then extending this definition by the linearity. Namely, if
P (λ) =
∑
∞
j=0 ξjPj(λ), ξj ∈ C, and R(λ) =
∑
∞
j=0 νjPj(λ), νj ∈ C, we set
(42) σ(P, R) =
∞∑
j=0
ξjνj .
Here all sums are finite. However, representation (42) is not very convenient. It requires
the knowledge of all coefficients of resolutions of the polynomials P, R via the polyno-
mials {Pn(λ)}n∈Z+
. We are going to derive an analytic representation for the spectral
function σ.
INTEGRAL REPRESENTATIONS FOR SPECTRAL FUNCTIONS . . . 97
Note that according to Theorem 1 in [4] we have
(43) σ(P, R) = σ(PR, 1), P, R ∈ P.
That means that it is enough to obtain an analytic representation for σ(u, 1), u ∈ P. If
(44) u(λ) =
∞∑
j=0
ukPk(λ), uk ∈ C,
then by (41) we will get
(45) σ(u, 1) = u0.
Let Ψn(λ, w) and m(λ) be defined as in the previous Section. We set
(46) Ψn(λ) := m(λ)Pn(λ) + Qn(λ), λ ∈ Π0,
and
(47) Ψf (λ) :=
∞∑
j=0
Ψj(λ)fj , f = (f0, f1, f2, . . .) ∈ l2fin, λ ∈ Π0.
Proposition 4. Let f = (f0, f1, f2, . . .) ∈ l2fin and ε > 0. For the function Ψf (λ) the
following relation holds:
(48) Ψf (λ) = −
1
λ
(f0 + o(1)),
where o(1) → 0 as λ → ∞ in a strip Π0,ε.
Proof. Let f and ε be from the statement of the Proposition. Set g = (g0, g1, g2, . . .),
where
g0 = a0f0 + b0f1,
gn = bn−1fn−1 + anfn + bnfn+1, n ∈ N.
Observe that g ∈ l2fin. We can write
Ψg(λ) =
∞∑
j=0
Ψj(λ)gj = Ψ0(λ)(a0f0 + b0f1) +
∞∑
j=1
Ψj(λ)(bj−1fj−1 + ajfj + bjfj+1)
= Ψ0(λ)(a0f0 + b0f1) +
∞∑
k=0
Ψk+1(λ)bkfk +
∞∑
j=1
Ψj(λ)ajfj +
∞∑
l=2
Ψl−1(λ)bl−1fl
= Ψ0(λ)a0f0 + Ψ1(λ)b0f0 +
∞∑
j=1
(bj−1Ψj−1(λ) + ajΨj(λ) + bjΨj+1(λ))fj
= Ψ0(λ)a0f0 + Ψ1(λ)b0f0 + λ
∞∑
j=1
Ψj(λ)fj , λ ∈ Π0,
where we have used the fact that Ψj(λ) is a solution of difference equation (4).
Since Ψ0(λ) = m(λ), and b0Ψ1(λ) = λm(λ) − a0m(λ) + 1, we get
Ψg(λ) = λm(λ)f0 + f0 + λ
∞∑
j=1
Ψj(λ)fj = f0 + λ
∞∑
j=0
Ψj(λ)fj = f0 + λΨf (λ).
Therefore
(49) Ψf(λ) =
1
λ
(−f0 + Ψg(λ)), λ ∈ Π0.
98 S. M. ZAGORODNYUK
By virtue of the Cauchy-Buniakovskiy inequality we can write
(50) |Ψg(λ)| ≤
( ∞∑
j=0
|Ψj(λ)|2
) 1
2
( ∞∑
j=0
|gj(λ)|2
) 1
2
, λ ∈ Π0.
If λ ∈ Π0,ε then | Imaj − Im λ| > ε. Since the function m(λ), λ ∈ Π0,ε, belongs to the
disc D∞(λ), by virtue of relations (28), (33) we can write
(51) ε
∞∑
j=0
|m(λ)Pj(λ)+Qj(λ)|2 ≤
∞∑
j=0
| Im aj − Imλ||m(λ)Pj(λ)+Qj(λ)|2 ≤ | Im m(λ)|.
Hence, we get
(52) |Ψg(λ)| ≤
|m(λ)|
ε
( ∞∑
j=0
|gj(λ)|2
) 1
2
, λ ∈ Π0,ε.
Applying Proposition 3 we complete the proof. �
Theorem 2. The spectral function σ of difference equations (3),(4) has the following
representation:
(53)
σ(P, R) =
1
2πi
lim
δ→0
{∫
∞+i(r1+ε)
−∞+i(r1+ε)
P (λ)R(λ)e−δλ2
m(λ) dλ
+
∫
−∞+i(r0−ε)
∞+i(r0−ε)
P (λ)R(λ)e−δλ2
m(λ) dλ
}
, P, R ∈ P,
where
(54) ε > 0 : ε > −r1, ε > r0.
Proof. We first note that the function Ψf (λ) from (47) is analytic in Π0. Choose an
arbitrary ε > 0 which satisfies (54) and consider points a+
N = −N+i(r1+ε), c+ = i(r1+ε),
b+
N = N + i(r1 + ε), and a−
N = −N + i(r0 − ε), c− = i(r0 − ε), b−N = N + i(r0 − ε) in the
complex λ-plane. We also denote
C+
N = {λ ∈ C : |λ − c+| = N, Im λ ≥ r1 + ε},
C−
N = {λ ∈ C : |λ − c−| = N, Im λ ≤ r0 − ε}.
Condition (54) ensures that the points a+
N , c+, b+
N and the half of the circle, C+
N , lie in
the open upper half-plane C′
+. The points a−
N , c−, b−N and the half of the circle, C−
N , lie
in the open lower half-plane C′
−
. Using the analyticity we can write
(55)
∫ b+
N
a
+
N
Ψf(λ) dλ +
∫
C
+
N
Ψf (λ) dλ = 0,
(56)
∫ a
−
N
b
−
N
Ψf (λ) dλ +
∫
C
−
N
Ψf(λ) dλ = 0.
By virtue of Proposition 4 we can write
(57)
∫
C
+
N
Ψf(λ) dλ = −f0
∫
C
+
N
1
λ
dλ −
∫
C
+
N
1
λ
o(1) dλ,
where o(1) = −Ψg(λ) (see (49)) is an analytic function in Π0. Since |λ| ≥ N − |r1 + ε|
in C+
N , we get
|
o(1)
λ
| ≤
|o(1)|
N − |r1 + ε|
,
INTEGRAL REPRESENTATIONS FOR SPECTRAL FUNCTIONS . . . 99
and the second term in the right-hand side of (57) tends to zero as N → ∞. For the first
term in the right-hand side of (57), we can write
−f0(ln a+
N − ln b+
N ) = −f0i(arg a+
N − arg b+
N) → −πif0,
as N → ∞. Here we have used an arbitrary analytic branch of the logarithm in
C\[0, +∞). Calculating arguments we used that points a+
N , b+
N lie in C′
+.
Passing to the limit in (55) we get
(58) lim
N→∞
∫ b
+
N
a
+
N
Ψf (λ) dλ = πif0.
Proceeding in an analogous manner with relation (56) we obtain
(59) lim
N→∞
∫ a
−
N
b
−
N
Ψf (λ) dλ = πif0.
Summing up relations (58) and (59) we get
(60) lim
N→∞
{∫ b
+
N
a
+
N
Ψf (λ) dλ +
∫ a
−
N
b
−
N
Ψf (λ) dλ
}
= 2πif0.
Let us show that
(61) lim
N→∞
∫ b
+
N
a
+
N
Ψf (λ) dλ = lim
δ→0
∫
∞+i(r1+ε)
−∞+i(r1+ε)
e−δλ2
Ψf (λ) dλ.
We first note that the integral in the right-hand side of (61) exists, since Ψf (λ) is bounded
(see (48)). For an arbitrary ε̂ > 0 we can write
(62)
∣∣∣∣
∫
∞+i(r1+ε)
−∞+i(r1+ε)
e−δλ2
Ψf (λ) dλ − lim
N→∞
∫ b
+
N
a
+
N
Ψf (λ) dλ
∣∣∣∣
=
∣∣∣∣ lim
N→∞
∫ b
+
N
a
+
N
(e−δλ2
− 1)Ψf (λ) dλ
∣∣∣∣ ≤
∣∣∣∣
∫ b
+
N
a
+
N
(e−δλ2
− 1)Ψf (λ) dλ
∣∣∣∣ +
ε̂
2
,
for N ≥ N0, N0 ∈ N. On the finite segment [a+
N0
, b+
N0
], the function (e−δλ2
− 1)Ψf(λ)
uniformly tends to zero as δ → 0. Therefore,
∫ b
+
N0
a+
N0
(e−δλ2
− 1)Ψf(λ) dλ → 0, δ → 0.
Hence, we can choose δ̂ > 0 such that |δ| < δ̂0 implies
(63)
∣∣∣∣
∫ b
+
N0
a
+
N0
(e−δλ2
− 1)Ψf (λ) dλ
∣∣∣∣ ≤
ε̂
2
.
From relations (62), (63) it follows that (61) holds. In an analogous manner we obtain
(64) lim
N→∞
∫ a
−
N
b
−
N
Ψf(λ) dλ = lim
δ→0
∫
−∞+i(r0−ε)
∞+i(r0−ε)
e−δλ2
Ψf(λ) dλ.
From (60), (61), (64) we obtain
(65) lim
δ→0
{∫
∞+i(r1+ε)
−∞+i(r1+ε)
e−δλ2
Ψf(λ) dλ +
∫
−∞+i(r0−ε)
∞+i(r0−ε)
e−δλ2
Ψf (λ) dλ
}
= 2πif0.
100 S. M. ZAGORODNYUK
Let u(λ) ∈ P be an arbitrary complex polynomial which has resolution (44). A vector of
coefficients u = (u0, u1, u2, . . .) belongs to l2fin. For λ ∈ Π0 we can write
(66) Ψu(λ) =
∞∑
j=0
Ψj(λ)uj =
∞∑
j=0
(m(λ)Pj(λ) + Qj(λ))uj = m(λ)u(λ) +
∞∑
j=0
Qj(λ)uj .
Let us show that
(67) lim
δ→0
{∫
∞+i(r1+ε)
−∞+i(r1+ε)
e−δλ2
Qj(λ) dλ +
∫
−∞+i(r0−ε)
∞+i(r0−ε)
e−δλ2
Qj(λ) dλ
}
= 0, j ∈ Z+.
Since the function e−δλ2
Qj(λ) is analytic in C, we have
(68)
∫ N+i(r1+ε)
−N+i(r1+ε)
e−δλ2
Qj(λ) dλ +
∫
−N+i(r0−ε)
N+i(r0−ε)
e−δλ2
Qj(λ) dλ
+
∫ N+i(r0−ε)
N+i(r1+ε)
e−δλ2
Qj(λ) dλ +
∫
−N+i(r1+ε)
−N+i(r0−ε)
e−δλ2
Qj(λ) dλ = 0.
The last two terms in the left-hand side of (68) tend to zero as N → ∞. In fact, the
length of the path of integration is constant and the function under the integral tends to
zero as N → ∞, in the both cases. So, proceeding to the limit in (68) we obtain (67).
If we write relation (65) for the function Ψu(λ) from (66) and use (67), we will get
(69)
lim
δ→0
{∫
∞+i(r1+ε)
−∞+i(r1+ε)
e−δλ2
m(λ)u(λ) dλ +
∫
−∞+i(r0−ε)
∞+i(r0−ε)
e−δλ2
m(λ)u(λ) dλ
}
= 2πiu0 = 2πiσ(u(λ), 1).
If we take into account relation (43), we will obtain relation (53). The proof is complete.
�
References
1. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis,
Hafner, New York, 1965. (Russian edition: Fizmatgiz, Moscow, 1961)
2. Yu. M. Berezanskii, Expansion in Eigenfunction of Self-Adjoint Operators, Amer. Math. Soc.,
Providence, R. I., 1968. (Russian edition: Naukova Dumka, Kiev, 1965)
3. F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York–
London, 1964. (Russian edition: Mir, Moscow, 1968)
4. S. M. Zagorodnyuk, Direct and inverse spectral problems for (2N +1)-diagonal, complex, sym-
metric, non-Hermitian matrices, Serdica Math. J. 30 (2004), no. 4, 471–482.
5. S. M. Zagorodnyuk, The direct and inverse spectral problems for (2N + 1)-diagonal complex
antisymmetric (with respect to the transposition) matrices, Methods Funct. Anal. Topology 14
(2008), no. 2, 124–131.
6. V. A. Marchenko, A resolution by eigenfunctions of nonself-adjoint singular differential oper-
ators of the second order, Mat. Sb. 52(94) (1960), no. 2, 739–788. (Russian)
7. A. I. Markushevich, The Theory of Analytic Functions, Vol. 1, Nauka, Moscow, 1967. (Russian)
School of Mathematics and Mechanics, Karazin Kharkiv National University, 4 Svobody
sq., Kharkiv, 61077, Ukraine
E-mail address: Sergey.M.Zagorodnyuk@univer.kharkov.ua
Received 28/08/2008
|
| id | nasplib_isofts_kiev_ua-123456789-5696 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1029-3531 |
| language | English |
| last_indexed | 2025-11-29T12:47:28Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zagorodnyuk, S.M. 2010-02-02T12:55:55Z 2010-02-02T12:55:55Z 2009 Integral representations for spectral functions of some nonself-adjoint Jacobi matrices / S.M. Zagorodnyuk // Methods of Functional Analysis and Topology. — 2009. — Т. 15, № 1. — С. 91-100. — Бібліогр.: 7 назв. — англ. 1029-3531 https://nasplib.isofts.kiev.ua/handle/123456789/5696 We study a Jacobi matrix J with complex numbers an, n belongs Z+, in the main diagonal such that r0 ≤ Im an ≤ r1, r0, r1 belongs Z+. We obtain an integral representation for the (generalized) spectral function of the matrix J. Themethod of our study is similar to Marchenko’s method for nonself-adjoint differential operators. en Інститут математики НАН України Integral Representations for Spectral Functions of Some Nonself-Adjoint Jacobi Matrices Article published earlier |
| spellingShingle | Integral Representations for Spectral Functions of Some Nonself-Adjoint Jacobi Matrices Zagorodnyuk, S.M. |
| title | Integral Representations for Spectral Functions of Some Nonself-Adjoint Jacobi Matrices |
| title_full | Integral Representations for Spectral Functions of Some Nonself-Adjoint Jacobi Matrices |
| title_fullStr | Integral Representations for Spectral Functions of Some Nonself-Adjoint Jacobi Matrices |
| title_full_unstemmed | Integral Representations for Spectral Functions of Some Nonself-Adjoint Jacobi Matrices |
| title_short | Integral Representations for Spectral Functions of Some Nonself-Adjoint Jacobi Matrices |
| title_sort | integral representations for spectral functions of some nonself-adjoint jacobi matrices |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/5696 |
| work_keys_str_mv | AT zagorodnyuksm integralrepresentationsforspectralfunctionsofsomenonselfadjointjacobimatrices |