[PPT] - The characteristic function for infinite Jacobi matrices, its PowerPoint Presentation

SLIDE 1

The characteristic function for infinite Jacobi matrices, its logarithm, the spectral zeta function, and solvable examples

František Štampach1, Pavel Št’ovíˇ cek2

1Department of Applied Mathematics, Faculty of Information Technology

Czech Technical University in Prague, Czech Republic B frantisek.stampach@fit.cvut.cz

2Department of Mathematics, Faculty of Nuclear Science

Czech Technical University in Prague, Czech Republic B stovicek@kmlinux.fjfi.cvut.cz

3rd Najman Conference

ON SPECTRAL PROBLEMS FOR OPERATORS AND MATRICES

Biograd, Croatia September 16-20, 2013

SLIDE 2

The function F(x)

Define F : D → C, F(x) = 1 +

∞

m=1

(−1)m

∞

k1=1

∞

k2=k1+2

. . .

∞

km=km−1+2

× xk1xk1+1xk2xk2+1 . . . xkmxkm+1 where D =

{xk}∞

k=1 ⊂ C; ∞

k=1

|xkxk+1| < ∞

Note that

ℓ2(N) ⊂ D Put F(x1, x2, . . . , xn) = F(x1, x2, . . . , xn, 0, 0, 0, . . . ), F(∅) = 1 One has |F(x)| ≤ exp ∞

k=1

|xkxk+1|

, |F(x) − 1| ≤ exp

∞

k=1

|xkxk+1|

− 1

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 2/30

SLIDE 3

Basic properties of F(x)

A recurrence rule F({xn}∞

n=1) = F({xn+1}∞ n=1) − x1x2 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) xkxk+1 F({xk+n+1}∞

n=1)

For x ∈ D, F(x) = lim

n→∞ F(x1, x2, . . . , xn)

The function F is continuous on ℓ2(N)

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 3/30

SLIDE 4

F(x) and special functions

The Bessel functions of the first kind: for w, ν ∈ C, ν / ∈ −N, Jν(2w) = wν Γ(ν + 1) F

w

ν + k ∞

k=1

The basic hypergeometric series (q-hypergeometric series):

for t, w ∈ C, |t| < 1, F

tk−1w

∞

k=1

=

1 +

∞

m=1

(−1)m tm(2m−1)w2m (1 − t2)(1 − t4) . . . (1 − t2m) =

0φ1(; 0; t2, −t w2)

Here

0φ1(; b; q, z) = ∞

k=0

qk(k−1) (q; q)k(b; q)k zk , (a; q)k =

k−1

j=0
1 − aqj

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 4/30

SLIDE 5

A class of Jacobi matrices: a convergence condition

Consider a symmetric (in general complex) Jacobi (tridiagonal) matrix J =      λ1 w1 w1 λ2 w2 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 Cλ

0 := C \ {λn; n ∈ N}

The convergence condition

∞

n=1
w 2

n

(λn − z)(λn+1 − z)

< ∞

for some and hence any z ∈ Cλ Then ∞

n=1 w 2

n

(λn−z)(λn+1−z) converges locally uniformly on Cλ

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 5/30

SLIDE 6

The characteristic polynomial of a finite Jacobi matrix

One has F(x1, x2, . . . , xn) = det Xn where Xn =           1 x1 x2 1 x2 ... ... ... ... ... ... xn−1 1 xn−1 xn 1           Put γ2k−1 := k−1

j=1 w2j w2j−1 , γ2k := w1

k−1

j=1 w2j+1 w2j , k = 1, 2, 3, . . .

Then γkγk+1 = wk Let Jn be the n × n truncation of the Jacobi matrix. Then det(Jn − zIn) =

n
k=1

(λk − z)

F
γ 2

1

λ1 − z , γ 2

2

λ2 − z , . . . , γ 2

n

λn − z

František Štampach, Pavel Št’ovíˇ

cek The characteristic function for infinite Jacobi matrices ... 6/30

SLIDE 7

The characteristic function of J

Put FJ(z) := F

γ 2

n

λn − z ∞

n=1

and for z /

∈ C \ der(λ), r(z) :=

∞

k=1

δz,λk (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’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 7/30

SLIDE 8

The zero set of the characteristic function of J

Define the zero set Z(J) :=

z ∈ C \ der(λ); lim

u→z (u − z)r(z)FJ(u) = 0

and the functions ξk(z), k = 0, 1, 2,..., on C\ der(λ) (w0 = 1),

ξk(z) := lim

u→z (u − z)r(z)

 

k

j=1

wj−1 u − λj  F  

γ 2

j

λj − u ∞

j=k+1

 

Remarks. (i) Notice that

Z(J) ∩

C \ {λn; n ∈ N}
= F −1

J

(0) (ii) z ∈ Z(J) iff ξ0(z) = 0 (iii) For k sufficiently large, ξk(z) =

k

j=1

wj−1

k
j=1

λj=z

(z − λj)

−1

F  

γ 2

j

λj − z ∞

j=k+1

 

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 8/30

SLIDE 9

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 ℓ2(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’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 9/30

SLIDE 10

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 ℓ2(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’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 10/30

SLIDE 11

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, J =      1 w w 2 w w 3 w ... ... ...      One has der(λ) = ∅, spec(J) = Z(J ), and for r ∈ Z+, F

γ 2

n

n − z ∞

n=r+1

= w−r+z Γ(1 + r − z) Jr−z(2w)

Hence spec(J) = {z ∈ C; J−z(2w) = 0} Corresponding eigenvectors v(z) can be chosen as vk(z) = (−1)kJk−z(2w), k ∈ N

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 11/30

SLIDE 12

Example 2

λn = 1/n, wn = β/

n(n + 1) , for β > 0 and all n ∈ N,

J =      1 β/ √ 2 β/ √ 2 1/2 β/ √ 6 β/ √ 6 1/3 β/ √ 12 ... ... ...      Then, for r ∈ Z+, F

γ 2

n

λn − z ∞

n=r+1

=

z β

r−1/z

Γ

r + 1 − 1

z

Jr−1/z

2β z

and

spec(J) =

z ∈ R \ {0}; J−1/z

2β z

= 0
∪ {0}

Corresponding eigenvectors v(z) can be chose as vk(z) = √ k Jk−1/z 2β z

, k ∈ N

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 12/30

SLIDE 13

Example 3

λn = qn−1 and wn = βq(n−1)/2, with 0 < q < 1, β > 0, J =      1 β β q β√q β√q q2 βq ... ... ...      Then, for r ∈ Z+, F

γ 2

n

λn − z ∞

n=r+1

= 0φ1
; qr

z ; q, −qrβ2 z2

and so

spec(J) =

z ∈ R \ {0};

1 z ; q

∞ 0φ1
; 1

z ; q, −β2 z2

= 0
∪ {0}

A corresponding eigenvector v(z) can be written in the form vk(z) = q(k−1)(k−2)/4 β z

k−1 qk

z ; q

∞ 0φ1
; qk

z ; q, −qkβ2 z2

, k ∈ N

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 13/30

SLIDE 14

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 J =      1 β β q−1 βq−1/2 βq−1/2 q−2 βq−1 ... ... ...      Then, for r ∈ Z+, F

γ 2

n

λn − z ∞

n=r+1

= 0φ1(; qrz; q, −qr+1β2)

and so spec(J) =

z ∈ R; (z ; q)∞ 0φ1(; z; q, −qβ2) = 0
The kth entry of a corresponding eigenvector v(z)

vk(z) = qk(k+1)/4 (−β)k−1 (qkz ; q)∞ 0φ1(; qkz; q, −qk+1β2), k ∈ N

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 14/30

SLIDE 15

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 λk = ν (µ + k − 1)(µ + k), wk = 1 µ + k (µ + k)2 + ν2 4(µ + k)2 − 1

1/2

, k ∈ N Then F

γk 2

λk − ζ−1 ∞

k=1

= Γ

1

2 + µ − 1 2

√1 + 4νζ

Γ

1

2 + µ + 1 2

√1 + 4νζ

Γ(µ)Γ(µ + 1)

× e−iζ

1F1(µ + iν; 2µ; 2iζ)

This is expressible in terms of the regular Coulomb wave functions FL(η, ρ) = 2Le−πη/2 |Γ(L + 1 + iη)| Γ(2L + 2) ρL+1e−iρ

1F1(L+1−iη; 2L+2; 2iρ)

Hence spec(J(µ, ν))\{0} =

ζ−1; 1F1(µ + iν; 2µ; 2iζ) = 0
The components of an eigenvector can be expressed explicitly, too

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 15/30

SLIDE 16

Example 6: confluent hypergeometric functions

λk = γk, wk = √α + βk , k ∈ N, where β > 0, γ > 0, α + β > 0 J =       γ √α + β √α + β 2γ √α + 2β √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

1F1

1 − α

β − β γ2 − z γ ; 1 − β γ2 − z γ ; β γ2

Γ
1 − β

γ2 − z γ

The components vk of a corresponding eigenvector v equal

(−1)kβk/2γ−kΓ

α

β + k

1/2

Γ

1 − β

γ2 − z γ + k

1F1
1 − α

β − β γ2 − z γ ; 1 − β γ2 − z γ + k; β γ2

František Štampach, Pavel Št’ovíˇ

cek The characteristic function for infinite Jacobi matrices ... 16/30

SLIDE 17

The particular case for α = 0, β = δ2, γ = 1

Put α = 0, β = δ2, γ = 1, with δ > 0. Hence λk = k, wk = δ √ k , k ∈ N, J =       1 δ δ 2 δ √ 2 δ √ 2 3 δ √ 3 ... ... ...       Then FJ(0, δ2, 1; z) = eδ2 Γ

1 − δ2 − z
hence

spec J(0, δ2, 1) = −δ2 + N

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 17/30

SLIDE 18

Example 7: q-confluent hypergeometric functions

For σ ∈ R and γ > −1 and 0 < q < 1, J = J(σ, γ) is defined by λn = qn−1, wn = 1 2 sinh(σ)q(n−γ−1)/2 1 − qn+γ , n ∈ N The characteristic function of J(σ, γ) on C \ {0} equals

cosh2(σ/2)z−1; q
∞

× 1φ1

q−γ cosh2(σ/2)z−1; cosh2(σ/2)z−1; q, − sinh2(σ/2)z−1

where

1φ1

a; b; q, q2z
=

∞

k=0

(−1)kqk(k−1)/2 (a; q)k (b; q)k(q; q)k zk is the q-confluent hypergeometric function, (a; q)k is the q-Pochhammer symbol. The entries of an eigenvector corresponding to an eigenvalue z = 0 can be expressed in terms of 1φ1 as well.

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 18/30

SLIDE 19

The particular case for γ = 0

Put γ = 0 (and still σ ∈ R) hence λn = qn−1, wn = 1 2 sinh(σ)q(n−1)/2 1 − qn , n ∈ N 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} =

qk cosh2(σ/2); k = 0, 1, 2, . . .
∪
−qk sinh2(σ/2); k = 0, 1, 2, . . .
František Štampach, Pavel Št’ovíˇ

cek The characteristic function for infinite Jacobi matrices ... 19/30

SLIDE 20

The q-Bessel function and the function F

Assume 0 < q < 1. The second definition of the q-Bessel function J(2)

ν (x; q) = (qν+1; q)∞

(q; q)∞ x 2 ν

0φ1

; qν+1; q, −qν+1x2

4

where 0φ1 is the basic hypergeometric series,

0φ1(; b; q, z) = ∞

k=0

qk(k−1) (q; q)k(b; q)k zk For 0 < q < 1, w, ν ∈ C, q−ν / ∈ qZ+, one has F

w

q−(ν+k)/2 − q(ν+k)/2

∞

k=0

= 0φ1(; qν; q, −qν+1/2w2)

One can prove that J(2)

0 (2w; q)2 + ∞

k=1
qk/2 + q−k/2

q k2/2J(2)

k (2w; q)2 = (−w2; q)∞

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 20/30

SLIDE 21

Example 8: q-Bessel functions

Suppose 0 < q < 1, β ≥ 0; z ∈ C is a spectral parameter. The bilateral difference equation q(n−1)/2βvn−1 + (qn − z) vn + qn/2βvn+1 = 0, n ∈ Z written in the matrix form (J − z)v = 0 where J = J(β, q) is a Jacobi matrix operator in ℓ2(Z) with λn = qn, wn = qn/2β, n ∈ Z Proposition The spectrum of J(β, q) is pure point and simple, specp J(β, q) =

−β2qZ+

∪ qZ Eigenvectors v v v(+)

m

=

v(+)

m,k

∞

k=−∞ with the eigenvalues qm, m ∈ Z,

v(+)

m,k = q(m−k)(m−k+1)/4 J(2) −m+k(2q−m/2β; q),

v

v v(+)

m

2 = (−q−mβ2; q)∞

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 21/30

SLIDE 22

The logarithm of F(x)

For x = {xk}∞

k=1 ⊂ C such that ∞ k=1 |xkxk+1| < log 2 one has

log F(x) = −

∞

N=1
m∈M(N)

α(m)

∞

k=1

d(m)

j=1
xk+j−1xk+j

mj where, ∀m ∈ Nℓ, α(m) := 1 m1

ℓ−1

j=1

mj + mj+1 − 1 mj+1

, d(m) := ℓ, |m| :=

ℓ

j=1

mj and M(N) :=

m ∈

N

ℓ=1

Nℓ; |m| = N

,

∀N ∈ N The logarithm formula can also be interpreted in the ring of formal power series C[[x1, x2, x3, . . .]].

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 22/30

SLIDE 23

A Hilbert-Schmidt matrix operator on ℓ2(N)

Suppose x = {xk}∞

k=1 fulfills

xk > 0, ∀k, and

∞

k=1

xkxk+1 < ∞ Put A =        a1 · · · a1 a2 · · · a2 a3 · · · a3 · · · . . . . . . . . . . . . ...        , where ak = √xkxk+1 , k ∈ N A is a Hilbert-Schmidt operator on ℓ2(N), its Hilbert-Schmidt norm A 2

2 = 2 ∞

k=1

|ak|2 = 2

∞

k=1

xkxk+1 The characteristic function of A is analytic on C \ {0}, FA(z) = F xk z ∞

k=1

František Štampach, Pavel Št’ovíˇ

cek The characteristic function for infinite Jacobi matrices ... 23/30

SLIDE 24

The spectral zeta function of A−2

If ∞

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

∞

k=1

ζ −2s

k

, Re s ≥ 1 Using the logarithm formula for F one derives that, ∀N ∈ N, ZA−2(N) = 2N

m∈M(N)

α(m)

∞

k=1

d(m)

j=1
xk+j−1xk+j

mj

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 24/30

SLIDE 25

Several first values of the spectral zeta function

For example, the first three values on N of the spectral zeta function ZA−2 are 1 2 ZA−2(1) =

∞

k=1

xkxk+1 1 2 ZA−2(2) =

∞

k=1

x 2

k x 2 k+1 + 2 ∞

k=1

xkx 2

k+1xk+2

1 2 ZA−2(3) =

∞

k=1

x 3

k x 3 k+1 + 3 ∞

k=1

xkx 3

k+1x 2 k+2 + 3 ∞

k=1

x 2

k x 3 k+1xk+2

+ 3

∞

k=1

xkx 2

k+1x 2 k+2xk+3

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 25/30

SLIDE 26

Example: Bessel functions and the Rayleigh function

Recall that, for ν > −1 and w ∈ C, Jν(2w) = wν Γ(ν + 1) F

w

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, For example,

∞

k=1

1 jν,k 2 = 1 4(ν + 1) ,

∞

k=1

1 jν,k 4 = 1 24(ν + 1)2(ν + 2)

∞

k=1

1 jν,k 6 = 1 25(ν + 1)3(ν + 2)(ν + 3) , etc.

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 26/30

∞

k=1

1 jν,k 2N = 2−2NN

∞

k=1
m∈M(N)

α(m)

d(m)

j=1
1

(j + k + ν − 1)(j + k + ν)

mj

SLIDE 27

Example: the q-Airy function

The Ramanujan function, also interpreted as the q-Airy function Aq(z) := 0φ1( ; 0; q, −qz) =

∞

n=1

qn2 (q; q)n (−z)n We still suppose 0 < q < 1 and z ∈ C. Recall that, for w ∈ C, F

qk−1w

∞

k=1

= 0φ1(; 0; q2, −qw2)

Hence Aq(w2) = F

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,

∞

k=1

1 ιk(q)N = NqN 1 − qN

m∈M(N)

α(m) qǫ1(m) where ∀m ∈ Nℓ, ǫ1(m) = ℓ

j=1(j − 1) mj .

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 27/30

SLIDE 28

Several first values of the Rayleigh-like function for Aq

Several first instances of the Rayleigh-like function for Aq

∞

k=1

1 ιk(q) = q 1 − q

∞

k=1

1 ιk(q)2 = q2 (1 + 2q) 1 − q2

∞

k=1

1 ιk(q)3 = q3 (1 + 3q + 3q2 + 3q3) 1 − q3

∞

k=1

1 ιk(q)4 = q4 (1 + 4q + 6q2 + 8q3 + 8q4 + 4q5 + 4q6) 1 − q4

František Štampach, Pavel Št’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 28/30

SLIDE 29

Bibliography

F

. Štampach, P . Št’ovíˇ cek: On the eigenvalue problem for a particular class of finite Jacobi matrices, Linear Alg. Appl. 434 (2011) 1336-1353

F

. Štampach, P . Št’ovíˇ cek: The characteristic function for Jacobi matrices with applications, Linear Alg. Appl. 438 (2013) 4130-4155

F

. Štampach, P . Št’ovíˇ cek: 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’ovíˇ cek The characteristic function for infinite Jacobi matrices ... 29/30

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