The characteristic function for infinite Jacobi
matrices, its logarithm, the spectral zeta function,
and solvable examples
František Štampach1 , Pavel Št’ovíček2
1 Department
of Applied Mathematics, Faculty of Information Technology
Czech Technical University in Prague, Czech Republic
B [email protected]
2 Department
of Mathematics, Faculty of Nuclear Science
Czech Technical University in Prague, Czech Republic
B [email protected]
3rd Najman Conference
ON SPECTRAL PROBLEMS FOR OPERATORS AND MATRICES
Biograd, Croatia
September 16-20, 2013
The function F(x)
Define F : D → C,
F(x) = 1 +
∞
X
(−1)m
∞
X
∞
X
k1 =1 k2 =k1 +2
m=1
...
∞
X
km =km−1 +2
× xk1 xk1 +1 xk2 xk2 +1 . . . xkm xkm +1
where
(
D=
{xk }∞
k =1
⊂ C;
∞
X
)
|xk xk +1 | < ∞
k =1
Note that
`2 (N)
⊂D
Put F(x1 , x2 , . . . , xn ) = F(x1 , x2 , . . . , xn , 0, 0, 0, . . . ), F(∅) = 1
One has
!
!
∞
∞
X
X
|F(x)| ≤ exp
|xk xk +1 | , |F(x) − 1| ≤ exp
|xk xk +1 | − 1
k =1
František Štampach, Pavel Št’ovíček
k =1
The characteristic function for infinite Jacobi matrices ...
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Basic properties of F(x)
A recurrence rule
∞
∞
F({xn }∞
n=1 ) = F({xn+1 }n=1 ) − x1 x2 F({xn+2 }n=1 )
more generally, for any k ∈ N,
∞
F({xn }∞
n=1 ) = F(x1 , . . . , xk ) F({xk +n }n=1 )
− F(x1 , . . . , xk −1 ) xk xk +1 F({xk +n+1 }∞
n=1 )
For x ∈ D,
F(x) = lim F(x1 , x2 , . . . , xn )
n→∞
The function F is continuous on `2 (N)
František Štampach, Pavel Št’ovíček
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F(x) and special functions
The Bessel functions of the first kind:
for w, ν ∈ C, ν ∈
/ −N,
∞ wν
w
Jν (2w) =
F
Γ(ν + 1)
ν + k k =1
The basic hypergeometric series (q-hypergeometric series):
for t, w ∈ C, |t| < 1,
F
∞
n
o∞ X
= 1+
(−1)m
t k −1 w
k =1
m=1
2
t m(2m−1) w 2m
(1 − t 2 )(1 − t 4 ) . . . (1 − t 2m )
, −t w 2 )
=
0 φ1 (; 0; t
∞
X
kY
−1 q k (k −1)
k
z , (a; q)k =
1 − aq j
(q; q)k (b; q)k
Here
0 φ1 (; b; q, z)
=
k =0
František Štampach, Pavel Št’ovíček
j=0
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A class of Jacobi matrices: a convergence condition
Consider a symmetric (in general complex)
Jacobi (tridiagonal) matrix 
λ1 w1
w1 λ2 w2

J=
w2 λ3 w3

.. ..
.
.

..




.
∞
where λ = {λn }∞
n=1 ⊂ C and w = {wn }n=1 ⊂ C \ {0}
Denote der(λ) := the set of all finite accumulation points of λ
Cλ0 := C \ {λn ; n ∈ N}
The convergence condition
∞ X
wn2
< ∞ for some and hence any z ∈ Cλ
0
(λn − z)(λ
n+1 − z)
n=1
Then
wn2
n=1 (λn −z)(λn+1 −z)
P∞
František Štampach, Pavel Št’ovíček
converges locally uniformly on Cλ0
The characteristic function for infinite Jacobi matrices ...
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The characteristic polynomial of a finite Jacobi matrix
One has
Put
F(x1 , x2 , . . . , xn ) = det Xn where


1 x1

x2 1 x2




.. ..
..


.
.
.


Xn = 

.
.
.
.
.
.


.
.
.



xn−1 1 xn−1 
xn
1
γ2k −1 :=
w2j
j=1 w2j−1
Qk −1
, γ2k := w1
Qk −1
j=1
w2j+1
w2j
, k = 1, 2, 3, . . .
Then γk γk +1 = wk
Let Jn be the n × n truncation of the Jacobi matrix. Then
det(Jn − zIn ) =
n
Y
k =1
František Štampach, Pavel Št’ovíček
!
γ12
γ22
γn2
(λk − z) F
,
,...,
λ1 − z λ2 − z
λn − z
The characteristic function for infinite Jacobi matrices ...
!
6/30
The characteristic function of J
Put
FJ (z) := F
and for z ∈
/ C \ der(λ),
r (z) :=
γn2
λn − z
∞
X
∞ !
n=1
δz,λk
k =1
(the number of occurrences of z in λ ;
r (z) = 0 for z ∈
/ {λn ; n ∈ N})
Lemma
Suppose J fulfills the convergence condition.
Then FJ (z) is a well defined analytic function on C \ {λn ; n ∈ N},
meromorphic on C \ der(λ) with poles at the points λn , n ∈ N,
(not belonging to der(λ), however) of order at most r (λn ).
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
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The zero set of the characteristic function of J
Define the zero set
n
o
Z(J) := z ∈ C \ der(λ); lim (u − z)r (z) FJ (u) = 0
u→z
and the functions ξk (z), k = 0, 1, 2,..., on C\ der(λ) (w0 = 1),

 (

)∞
k
2
Y
γ
w
j−1  
j

ξk (z) := lim (u − z)r (z) 
F
u→z
u − λj
λj − u
j=1
j=k +1
Remarks. (i) Notice that Z(J) ∩ C \ {λn ; n ∈ N} = FJ−1 (0)
(ii) z ∈ Z(J) iff ξ0 (z) = 0
(iii) For k sufficiently large,
)∞
!−1 (
k
k
Y
Y
γj 2
ξk (z) =
wj−1
F
(z − λj )
λj − z
j=1
František Štampach, Pavel Št’ovíček
j=1
λj 6=z
The characteristic function for infinite Jacobi matrices ...


j=k +1
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The spectrum of J in C\ der(λ)
Theorem
Suppose the convergence condition on J is fulfilled
and FJ (z) does not vanish identically on C \ {λn ; n ∈ N}.
Then J determines unambiguously a closed operator in `2 (N)
(denoted again by J),
spec(J) \ der(λ) = specp (J) \ der(λ) = Z(J)
The point spectrum of J is simple.
If z ∈ Z(J), i.e. ξ0 (z) = 0, then
ξ(z) = (ξ1 (z), ξ2 (z), ξ3 (z), . . .)
is a corresponding eigenvector.
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
9/30
The case J is real
Corollary
Suppose J is is real and fulfills the convergence condition.
Then J determines unambiguously a self-adjoint operator in `2 (N)
and
spec(J) ∩ (C \ der(λ)) = Z(J )
consists of simple real eigenvalues
which have no accumulation points in R \ der(λ).
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
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Example 1
J. Gard, E. Zakrajšek: J. Inst. Math. Appl. 11 (1973)
Y. Ikebe, Y. Kikuchi, I. Fujishiro: J. Comput. Appl. Math. 38 (1991)
λn = n and wn = w > 0 for all n ∈ N,


1 w
w 2 w



J=

w 3 w


.. .. ..
.
.
.
One has der(λ) = ∅, spec(J) = Z(J ), and for r ∈ Z+ ,
!
2 ∞
γn
F
= w −r +z Γ(1 + r − z) Jr −z (2w)
n − z n=r +1
Hence
spec(J) = {z ∈ C; J−z (2w) = 0}
Corresponding eigenvectors v (z) can be chosen as
vk (z) = (−1)k Jk −z (2w), k ∈ N
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
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Example 2
p
λn = 1/n, wn = β/ n(n + 1) , for β > 0 and all n ∈ N,
√


1√ β/ 2
√
β/ 2 1/2 β/ 6



√
√
J=

β/ 6 1/3 β/ 12


..
..
..
.
.
.
Then, for r ∈ Z+ ,
! ∞
γn2
z r −1/z
1
2β
F
Γ r +1−
=
Jr −1/z
λn − z n=r +1
β
z
z
and
2β
spec(J) = z ∈ R \ {0}; J−1/z
= 0 ∪ {0}
z
Corresponding eigenvectors v (z) can be chose as
√
2β
vk (z) = k Jk −1/z
, k ∈N
z
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
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Example 3
λn = q n−1 and wn = βq (n−1)/2 , with 0 < q < 1, β > 0,


1
β
√
β

q
β q


√
J=
2

β
q
q
βq


..
.. ..
.
.
.
Then, for r ∈ Z+ ,
!
∞
r
γn2
q
qr β2
F
= 0 φ1 ; ; q, − 2
λn − z n=r +1
z
z
and so
spec(J) =
z ∈ R \ {0};
1
z
;q
∞
1
β2
= 0 ∪ {0}
0 φ1 ; ; q, − 2
z
z
A corresponding eigenvector v (z) can be written in the form
k −1 k k
q
q
qk β2
(k −1)(k −2)/4 β
;q
; q, − 2 , k ∈ N
vk (z) = q
0 φ1 ;
z
z
z
∞
z
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
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Example 4
A modification of Example 3 → an unbounded Jacobi operator
λn = q −n+1 and wn = βq −(n−1)/2 where again 0 < q < 1, β > 0


1
β
β

q −1
βq −1/2


J=
−1/2
−2
−1

βq
q
βq


..
..
..
.
.
.
Then, for r ∈ Z+ ,
!
∞
γn2
F
= 0 φ1 (; q r z; q, −q r +1 β 2 )
λn − z n=r +1
and so
n
o
2
spec(J) = z ∈ R; (z ; q)∞ 0 φ1 (; z; q, −qβ ) = 0
The k th entry of a corresponding eigenvector v (z)
vk (z) = q k (k +1)/4 (−β)k −1 (q k z ; q)∞ 0 φ1(; q k z; q, −q k +1 β 2 ), k ∈ N
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
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Example 5: Coulomb wave functions
Y. Ikebe: Math. Comp. 29 (1975)
For µ > 0, ν ∈ R, consider the Jacobi matrix J = J(µ, ν), with
1/2
ν
1
(µ + k )2 + ν 2
λk =
, wk =
, k ∈N
(µ + k − 1)(µ + k )
µ + k 4(µ + k )2 − 1
Then
∞ !
Γ
γk 2
F
=
−1
λk − ζ
k =1
1
2
+µ−
1
2
√
√
1 + 4νζ Γ 21 + µ + 12 1 + 4νζ
Γ(µ)Γ(µ + 1)
× e−iζ 1 F1 (µ + iν; 2µ; 2iζ)
This is expressible in terms of the regular Coulomb wave functions
|Γ(L + 1 + iη)| L+1 −iρ
ρ e 1 F1 (L+1−iη; 2L+2; 2iρ)
Γ(2L + 2)
n
o
spec(J(µ, ν))\{0} = ζ −1 ; 1 F1 (µ + iν; 2µ; 2iζ) = 0
FL (η, ρ) = 2L e−πη/2
Hence
The components of an eigenvector can be expressed explicitly, too
František Štampach, Pavel Št’ovíček
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Example 6: confluent hypergeometric functions
√
λk = γk , wk = α + βk , k ∈ N, where β > 0, γ > 0, α + β > 0
√


γ
α+β
√
√α + β

2γ
α + 2β


√
√


J=

α
+
2β
3γ
α
+
3β


..
..
..
.
.
.
By the Weyl theorem, the spectrum is discrete (and simple).
The convergence condition is violated.
Considering the limit Jn → J, where Jn is the principal n × n
submatrix of J, the characteristic function FJ (α, β, γ; z) equals
α
β
z
β
z β
β
z
− 2 − ;1 − 2 − ; 2
Γ 1− 2 −
1 F1 1 −
β
γ
γ γ
γ
γ
γ
γ
The components vk of a corresponding eigenvector v equal
1/2
(−1)k β k /2 γ −k Γ αβ + k
β
z
β
z
β
α
− 2 − ; 1 − 2 − + k; 2
1F1 1 −
β
γ
γ
γ
γ
γ
Γ 1 − γβ2 − γz + k
František Štampach, Pavel Št’ovíček
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The particular case for α = 0, β = δ 2 , γ = 1
Put α = 0, β = δ 2 , γ = 1, with δ > 0. Hence
√
λk = k , wk = δ k , k ∈ N,


1
δ
√
δ

2
δ 2


√
√

J=
 δ 2

3
δ 3


..
..
..
.
.
.
Then
FJ (0, δ , 1; z) = e
Γ 1 − δ2 − z
2
δ2
hence
spec J(0, δ 2 , 1) = −δ 2 + N
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
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Example 7: q-confluent hypergeometric functions
For σ ∈ R and γ > −1 and 0 < q < 1, J = J(σ, γ) is defined by
λn = q n−1 , wn =
p
1
sinh(σ)q (n−γ−1)/2 1 − q n+γ , n ∈ N
2
The characteristic function of J(σ, γ) on C \ {0} equals
cosh2 (σ/2)z −1 ; q ∞
× 1 φ1 q −γ cosh2 (σ/2)z −1 ; cosh2 (σ/2)z −1 ; q, − sinh2 (σ/2)z −1
where
∞
X
2
φ
a;
b;
q,
q
z
=
(−1)k q k (k −1)/2
1 1
k =0
(a; q)k
zk
(b; q)k (q; q)k
is the q-confluent hypergeometric function,
(a; q)k is the q-Pochhammer symbol.
The entries of an eigenvector corresponding to an eigenvalue
z 6= 0 can be expressed in terms of 1 φ1 as well.
František Štampach, Pavel Št’ovíček
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The particular case for γ = 0
Put γ = 0 (and still σ ∈ R) hence
λn = q n−1 , wn =
p
1
sinh(σ)q (n−1)/2 1 − q n , n ∈ N
2
The characteristic function of J(σ, 0) on C \ {0} equals
FJ (σ, 0; z) = cosh2 (σ/2)z −1 ; q ∞ − sinh2 (σ/2)z −1 ; q ∞
Hence
specJ(σ, 0) \ {0} =
František Štampach, Pavel Št’ovíček
n
o
q k cosh2 (σ/2); k = 0, 1, 2, . . .
n
o
∪ −q k sinh2 (σ/2); k = 0, 1, 2, . . .
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The q-Bessel function and the function F
Assume 0 < q < 1. The second definition of the q-Bessel function
(q ν+1 ; q)∞ x ν
q ν+1 x 2
(2)
ν+1
Jν (x; q) =
; q, −
0 φ1 ; q
(q; q)∞
2
4
where
0 φ1
is the basic hypergeometric series,
∞
X
q k (k −1)
φ
(;
b;
q,
z)
=
zk
0 1
(q; q)k (b; q)k
k =0
For 0 < q < 1, w, ν ∈ C,
F
q −ν
w
−(ν+k
)/2
q
− q (ν+k )/2
∈
/ q Z+ , one has
∞ = 0 φ1 (; q ν ; q, −q ν+1/2 w 2 )
k =0
One can prove that
∞ 2
X
(2)
(2)
2
J0 (2w; q) +
q k /2 + q −k /2 q k /2 Jk (2w; q)2 = (−w 2 ; q)∞
k =1
František Štampach, Pavel Št’ovíček
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Example 8: q-Bessel functions
Suppose 0 < q < 1, β ≥ 0; z ∈ C is a spectral parameter.
The bilateral difference equation
q (n−1)/2 βvn−1 + (q n − z) vn + q n/2 βvn+1 = 0, n ∈ Z
written in the matrix form (J − z)v = 0 where J = J(β, q)
is a Jacobi matrix operator in `2 (Z) with
λn = q n , wn = q n/2 β, n ∈ Z
Proposition
The spectrum of J(β, q) is pure point and simple,
specp J(β, q) = −β 2 q Z+ ∪ q Z
o
n
(+)
(+) ∞
Eigenvectors v m = vm,k
k =−∞
with the eigenvalues q m , m ∈ Z,
(+) (+)
(2)
vm,k = q (m−k )(m−k +1)/4 J−m+k (2q −m/2 β; q), v m 2 = (−q −m β 2 ; q)∞
František Štampach, Pavel Št’ovíček
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The logarithm of F(x)
For x = {xk }∞
k =1 ⊂ C such that
log F(x) = −
∞
X
X
P∞
k =1 |xk xk +1 |
α(m)
∞ d(m)
Y
X
< log 2 one has
xk +j−1 xk +j
mj
k =1 j=1
N=1 m∈M(N)
where, ∀m ∈ N` ,
`−1 `
X
1 Y mj + mj+1 − 1
α(m) :=
, d(m) := `, |m| :=
mj
m1
mj+1
j=1
j=1
and
(
M(N) :=
m∈
N
[
)
`
N ; |m| = N ,
∀N ∈ N
`=1
The logarithm formula can also be interpreted in the ring of formal
power series C[[x1 , x2 , x3 , . . .]].
František Štampach, Pavel Št’ovíček
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A Hilbert-Schmidt matrix operator on `2 (N)
Suppose x = {xk }∞
k =1 fulfills xk > 0, ∀k , and
∞
X
Put
xk xk +1 < ∞
k =1




A=


0 a1 0 0
a1 0 a2 0
0 a2 0 a3
0 0 a3 0
..
..
..
..
.
.
.
.
···
···
···
···
..
.



√

, where ak = xk xk +1 , k ∈ N


A is a Hilbert-Schmidt operator on `2 (N), its Hilbert-Schmidt norm
kAk22 = 2
∞
X
k =1
|ak |2 = 2
∞
X
xk xk +1
k =1
The characteristic function of A is analytic on C \ {0},
n x o∞ k
FA (z) = F
z k =1
František Štampach, Pavel Št’ovíček
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The spectral zeta function of A−2
If
P∞
k =1 1/x2k −1
= ∞ then 0 is not an eigenvalue of A. Put
f (z) = FA (z −1 ) = F({z xk }∞
k =1 )
f (z) is the characteristic function of A−1 ;
it is entire, even, with simple real roots.
Let 0 < ζ1 < ζ2 < ζ3 < . . . be its positive roots; limk →∞ ζk = ∞.
The spectral zeta function of A−2 is, at the same time,
the Rayleigh-like function for f (z),
ZA−2 (s) := Tr A2s = 2
∞
X
ζk−2s ,
Re s ≥ 1
k =1
Using the logarithm formula for F one derives that, ∀N ∈ N,
ZA−2 (N) = 2N
X
m∈M(N)
František Štampach, Pavel Št’ovíček
α(m)
∞ d(m)
X
Y
xk +j−1 xk +j
mj
k =1 j=1
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Several first values of the spectral zeta function
For example, the first three values on N of the spectral zeta
function ZA−2 are
∞
X
1
xk xk +1
ZA−2 (1) =
2
k =1
∞
∞
X
X
1
2 2
Z −2 (2) =
xk xk +1 + 2
xk xk2+1 xk +2
2 A
k =1
1
Z −2 (3) =
2 A
∞
X
k =1
xk3 xk3+1
k =1
+3
+3
∞
X
xk xk3+1 xk2+2
k =1
∞
X
+3
∞
X
xk2 xk3+1 xk +2
k =1
xk xk2+1 xk2+2 xk +3
k =1
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Example: Bessel functions and the Rayleigh function
Recall that, for ν > −1 and w ∈ C,
∞ wν
w
Jν (2w) =
F
Γ(ν + 1)
ν + k k =1
Let 0 < jν,1 < jν,2 < jν,3 < . . . be the positive roots of Jν (z).
For values on 2N of the Rayleigh function (as originally introduced
for Bessel functions) we get, ∀ ∈ N,
∞
X
k =1
1
jν,k 2N
=2
−2N
N
∞
X
X
d(m) α(m)
k =1 m∈M(N)
j=1
For example,
∞
X
1
1
=
,
2
4(ν + 1)
jν,k
k =1
∞
X
k =1
1
jν,k
6
František Štampach, Pavel Št’ovíček
=
Y
1
(j + k + ν − 1)(j + k + ν)
∞
X
1
1
= 4
4
jν,k
2 (ν + 1)2 (ν + 2)
k =1
1
25 (ν
+
mj
1)3 (ν
+ 2)(ν + 3)
,
etc.
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Example: the q-Airy function
The Ramanujan function, also interpreted as the q-Airy function
2
∞
X
qn
Aq (z) := 0 φ1 ( ; 0; q, −qz) =
(−z)n
(q; q)n
n=1
We still suppose 0 < q < 1 and z ∈ C. Recall that, for w ∈ C,
n
o∞ k −1
= 0 φ1 (; 0; q 2 , −qw 2 )
F q
w
k =1
Hence
Aq (w 2 ) = F
n
o∞ wq (2k −1)/4
k =1
The zeros of Aq (z) are all positive and simple; denote them
0 < ι1 (q) < ι2 (q) < ι3 (q) < . . ..
A formula for integer values of the Rayleigh-like function, ∀N ∈ N,
∞
X
k =1
Nq N
1
=
ιk (q)N
1 − qN
where ∀m ∈ N` , 1 (m) =
František Štampach, Pavel Št’ovíček
P`
j=1 (j
X
α(m) q 1 (m)
m∈M(N)
− 1) mj .
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Several first values of the Rayleigh-like function for Aq
Several first instances of the Rayleigh-like function for Aq
∞
X
1
ιk (q)
=
q
1−q
1
ιk (q)2
=
q 2 (1 + 2q)
1 − q2
1
ιk (q)3
=
q 3 (1 + 3q + 3q 2 + 3q 3 )
1 − q3
1
ιk (q)4
=
q 4 (1 + 4q + 6q 2 + 8q 3 + 8q 4 + 4q 5 + 4q 6 )
1 − q4
k =1
∞
X
k =1
∞
X
k =1
∞
X
k =1
František Štampach, Pavel Št’ovíček
The characteristic function for infinite Jacobi matrices ...
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Bibliography
• F. Štampach, P. Št’ovíček: On the eigenvalue problem for a
particular class of finite Jacobi matrices,
Linear Alg. Appl. 434 (2011) 1336-1353
• F. Štampach, P. Št’ovíček: The characteristic function for Jacobi
matrices with applications,
Linear Alg. Appl. 438 (2013) 4130-4155
• F. Štampach, P. Št’ovíček: Special functions and spectrum of
Jacobi matrices,
Linear Alg. Appl. (2013) (in press),
http://dx.doi.org/10.1016/j.laa.2013.06.024
František Štampach, Pavel Št’ovíček
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THANK YOU FOR YOUR ATTENTION!