J-theory in application to the spectral theory of periodic GMP - - PowerPoint PPT Presentation

J-theory in application to the spectral theory of periodic GMP matrices

Benjamin Eichinger

Institute of Analysis Johannes Kepler University Linz

OTIND , 17st-20th December

1 / 19

Jacobi matrices

Let dσ be a real compactly supported measure and Pn = Pn(dσ) the corresponding orthonormal polynomials. It is well known, that they satisfy a three-term recurrence relation. xPn(x) = anPn−1(x) + bnPn(x) + an+1Pn+1, an > 0. That is, multiplication by x in the basis {Pn}n≥0 has the one-sided Jacobi matrix J+ =       b0 a1 a1 b1 a2 ... ... ... ... ...       .

2 / 19

Jacobi matrices

Let dσ be a real compactly supported measure and Pn = Pn(dσ) the corresponding orthonormal polynomials. It is well known, that they satisfy a three-term recurrence relation. xPn(x) = anPn−1(x) + bnPn(x) + an+1Pn+1, an > 0. That is, multiplication by x in the basis {Pn}n≥0 has the one-sided Jacobi matrix J+ =       b0 a1 a1 b1 a2 ... ... ... ... ...       .

2 / 19

Shift on Jacobi matrices

Let us define the resolvent function r+(z) = (J+ − z)−1e0, e0, e0 =

• 1

. . .

• .

Let r(1)

+

be the resolvent function of J(1)

+ ,

J+ =   b0 a1 · · · · · · a1 J(1)

+

  . Then r+(z) = −1 z − b0 − a2

1r(1) + (z)

. That is, r+(z) 1

• ∼

−1/a1 a1 (z − b0)/a1 r(1)

+ (z)

1

• ,

where u v

• ∼

x y

• ⇐

⇒ ∃c u v

• = c

x y

• 3 / 19

Shift on Jacobi matrices

Let us define the resolvent function r+(z) = (J+ − z)−1e0, e0, e0 =

• 1

. . .

• .

Let r(1)

+

be the resolvent function of J(1)

+ ,

J+ =   b0 a1 · · · · · · a1 J(1)

+

  . Then r+(z) = −1 z − b0 − a2

1r(1) + (z)

. That is, r+(z) 1

• ∼

−1/a1 a1 (z − b0)/a1 r(1)

+ (z)

1

• ,

where u v

• ∼

x y

• ⇐

⇒ ∃c u v

• = c

x y

• 3 / 19

Shift on Jacobi matrices

Let us define the resolvent function r+(z) = (J+ − z)−1e0, e0, e0 =

• 1

. . .

• .

Let r(1)

+

be the resolvent function of J(1)

+ ,

J+ =   b0 a1 · · · · · · a1 J(1)

+

  . Then r+(z) = −1 z − b0 − a2

1r(1) + (z)

. That is, r+(z) 1

• ∼

−1/a1 a1 (z − b0)/a1 r(1)

+ (z)

1

• ,

where u v

• ∼

x y

• ⇐

⇒ ∃c u v

• = c

x y

• 3 / 19

Shift on Jacobi matrices

Let us define the resolvent function r+(z) = (J+ − z)−1e0, e0, e0 =

• 1

. . .

• .

Let r(1)

+

be the resolvent function of J(1)

+ ,

J+ =   b0 a1 · · · · · · a1 J(1)

+

  . Then r+(z) = −1 z − b0 − a2

1r(1) + (z)

. That is, r+(z) 1

• ∼

−1/a1 a1 (z − b0)/a1 r(1)

+ (z)

1

• ,

where u v

• ∼

x y

• ⇐

⇒ ∃c u v

• = c

x y

• 3 / 19

periodic Jacobi matrices

For periodic Jacobi matrices, i.e., ∃p ∀k ak+p = ak, bk+p = bk, it is convenient to extend the sequences periodically to Z and consider the two-sided Jacobi matrix            ... ... ... ap−1 ap−1 bp−1 a0 a0 b0 a1 a1 ... ... ...            =         J− a0 a0 J+         We define r−(z) = (J− − z)−1e−1, e−1.

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periodic Jacobi matrices

For periodic Jacobi matrices, i.e., ∃p ∀k ak+p = ak, bk+p = bk, it is convenient to extend the sequences periodically to Z and consider the two-sided Jacobi matrix            ... ... ... ap−1 ap−1 bp−1 a0 a0 b0 a1 a1 ... ... ...            =         J− a0 a0 J+         We define r−(z) = (J− − z)−1e−1, e−1.

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We see that r+(z) 1

• ∼

−1/a1 a1 (z − b0)/a1

• ...

−1/ap ap (z − bp−1)/ap r(p)

+ (z)

1

• = Tp(z)

r+(z) 1

• .

Theorem Let us define the polynomial of degree p, ∆(z) = tr Tp(z). The spectrum of a periodic two-sided Jacobi matrix is purely absolutely continuous and it is given by E = ∆−1([−2, 2]). Moreover the resolvent functions are reflectionless on E, i.e., 1 r+(x + i0) = a2

0r−(x + i0).

Note that for this reason spectra of periodic Jacobi matrices are very special finite union of intervals.

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We see that r+(z) 1

• ∼

−1/a1 a1 (z − b0)/a1

• ...

−1/ap ap (z − bp−1)/ap r(p)

+ (z)

1

• = Tp(z)

r+(z) 1

• .

Theorem Let us define the polynomial of degree p, ∆(z) = tr Tp(z). The spectrum of a periodic two-sided Jacobi matrix is purely absolutely continuous and it is given by E = ∆−1([−2, 2]). Moreover the resolvent functions are reflectionless on E, i.e., 1 r+(x + i0) = a2

0r−(x + i0).

Note that for this reason spectra of periodic Jacobi matrices are very special finite union of intervals.

5 / 19

We see that r+(z) 1

• ∼

−1/a1 a1 (z − b0)/a1

• ...

−1/ap ap (z − bp−1)/ap r(p)

+ (z)

1

• = Tp(z)

r+(z) 1

• .

Theorem Let us define the polynomial of degree p, ∆(z) = tr Tp(z). The spectrum of a periodic two-sided Jacobi matrix is purely absolutely continuous and it is given by E = ∆−1([−2, 2]). Moreover the resolvent functions are reflectionless on E, i.e., 1 r+(x + i0) = a2

0r−(x + i0).

Note that for this reason spectra of periodic Jacobi matrices are very special finite union of intervals.

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Functional model

Let E be a finite union of intervals E = [a0, b0] \ g

j=1(aj, bj) and

Ω = C \ E. Let gΩ(z, z0) denote the Green’s function of Ω and define the complex Green’s function by Bz0(z) = e−gΩ(z,z0)−i

gΩ(z,z0).

Note that we have the properties: Bz0(z0) = 0, |Bz0| < 1 in Ω, |Bz0| = 1 on E, Bz0 ◦ ˜ γj = e2πiωΩ(Ej,z0)Bz0.

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Functional model

Let E be a finite union of intervals E = [a0, b0] \ g

j=1(aj, bj) and

Ω = C \ E. Let gΩ(z, z0) denote the Green’s function of Ω and define the complex Green’s function by Bz0(z) = e−gΩ(z,z0)−i

gΩ(z,z0).

Note that we have the properties: Bz0(z0) = 0, |Bz0| < 1 in Ω, |Bz0| = 1 on E, Bz0 ◦ ˜ γj = e2πiωΩ(Ej,z0)Bz0.

6 / 19

Functional model

Let E be a finite union of intervals E = [a0, b0] \ g

j=1(aj, bj) and

Ω = C \ E. Let gΩ(z, z0) denote the Green’s function of Ω and define the complex Green’s function by Bz0(z) = e−gΩ(z,z0)−i

gΩ(z,z0).

Note that we have the properties: Bz0(z0) = 0, |Bz0| < 1 in Ω, |Bz0| = 1 on E, Bz0 ◦ ˜ γj = e2πiωΩ(Ej,z0)Bz0.

6 / 19

Functional model

Let E be a finite union of intervals E = [a0, b0] \ g

j=1(aj, bj) and

Ω = C \ E. Let gΩ(z, z0) denote the Green’s function of Ω and define the complex Green’s function by Bz0(z) = e−gΩ(z,z0)−i

gΩ(z,z0).

Note that we have the properties: Bz0(z0) = 0, |Bz0| < 1 in Ω, |Bz0| = 1 on E, Bz0 ◦ ˜ γj = e2πiωΩ(Ej,z0)Bz0.

6 / 19

Functional model

Let E be a finite union of intervals E = [a0, b0] \ g

j=1(aj, bj) and

Ω = C \ E. Let gΩ(z, z0) denote the Green’s function of Ω and define the complex Green’s function by Bz0(z) = e−gΩ(z,z0)−i

gΩ(z,z0).

Note that we have the properties: Bz0(z0) = 0, |Bz0| < 1 in Ω, |Bz0| = 1 on E, Bz0 ◦ ˜ γj = e2πiωΩ(Ej,z0)Bz0.

6 / 19

Functional model

Let E be a finite union of intervals E = [a0, b0] \ g

j=1(aj, bj) and

Ω = C \ E. Let gΩ(z, z0) denote the Green’s function of Ω and define the complex Green’s function by Bz0(z) = e−gΩ(z,z0)−i

gΩ(z,z0).

Note that we have the properties: Bz0(z0) = 0, |Bz0| < 1 in Ω, |Bz0| = 1 on E, Bz0 ◦ ˜ γj = e2πiωΩ(Ej,z0)Bz0.

6 / 19

Functional Models

A uniformization of the domain Ω is a covering map z : D → Ω and a Fuchsian group Γ, i.e., a discrete subset of Möbius transformations on D, such that ∀z ∈ Ω ∃ζ ∈ D: z(ζ) = z and z(ζ1) = z(ζ2) ⇔ ζ1 = γ(ζ2). In particular, this implies that z ◦ γ = z. We assume that z(0) = ∞. Let Γ∗ be the group of characters of the discrete group Γ, Γ∗ = {α| α : Γ → R/Z such that α(γ1γ2) = α(γ1) + α(γ2)}. Note that Γ∗ ≃ Tg.

7 / 19

Functional Models

A uniformization of the domain Ω is a covering map z : D → Ω and a Fuchsian group Γ, i.e., a discrete subset of Möbius transformations on D, such that ∀z ∈ Ω ∃ζ ∈ D: z(ζ) = z and z(ζ1) = z(ζ2) ⇔ ζ1 = γ(ζ2). In particular, this implies that z ◦ γ = z. We assume that z(0) = ∞. Let Γ∗ be the group of characters of the discrete group Γ, Γ∗ = {α| α : Γ → R/Z such that α(γ1γ2) = α(γ1) + α(γ2)}. Note that Γ∗ ≃ Tg.

7 / 19

Functional Models

A uniformization of the domain Ω is a covering map z : D → Ω and a Fuchsian group Γ, i.e., a discrete subset of Möbius transformations on D, such that ∀z ∈ Ω ∃ζ ∈ D: z(ζ) = z and z(ζ1) = z(ζ2) ⇔ ζ1 = γ(ζ2). In particular, this implies that z ◦ γ = z. We assume that z(0) = ∞. Let Γ∗ be the group of characters of the discrete group Γ, Γ∗ = {α| α : Γ → R/Z such that α(γ1γ2) = α(γ1) + α(γ2)}. Note that Γ∗ ≃ Tg.

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Functional models

For α ∈ Γ∗ we define the Hardy space of character automorphic functions as H2(α) = H2

Ω(α) = {f ∈ H2 : f ◦ γ = e2πiα(γ)f , γ ∈ Γ},

where H2 denotes the standard Hardy class in D. Fix z0 ∈ Ω and ζ0 ∈ z−1(z0). By bz0 we denote the Blaschke product bz0(ζ) =

• γ∈Γ

−|γ(ζ0)| γ(ζ0) ζ − γ(ζ0) 1 − γ(ζ0)ζ We have bz0 = Bz0 ◦ z. It is character automorphic with character µz0.We define kα

ζ0(ζ) = kα(ζ, ζ0) as the reproducing kernels of the

space H2(α), i.e.,

• f , kα

ζ0

• = f (ζ0)

∀f ∈ H2(α).

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Functional models

For α ∈ Γ∗ we define the Hardy space of character automorphic functions as H2(α) = H2

Ω(α) = {f ∈ H2 : f ◦ γ = e2πiα(γ)f , γ ∈ Γ},

where H2 denotes the standard Hardy class in D. Fix z0 ∈ Ω and ζ0 ∈ z−1(z0). By bz0 we denote the Blaschke product bz0(ζ) =

• γ∈Γ

−|γ(ζ0)| γ(ζ0) ζ − γ(ζ0) 1 − γ(ζ0)ζ We have bz0 = Bz0 ◦ z. It is character automorphic with character µz0.We define kα

ζ0(ζ) = kα(ζ, ζ0) as the reproducing kernels of the

space H2(α), i.e.,

• f , kα

ζ0

• = f (ζ0)

∀f ∈ H2(α).

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Functional models

For α ∈ Γ∗ we define the Hardy space of character automorphic functions as H2(α) = H2

Ω(α) = {f ∈ H2 : f ◦ γ = e2πiα(γ)f , γ ∈ Γ},

where H2 denotes the standard Hardy class in D. Fix z0 ∈ Ω and ζ0 ∈ z−1(z0). By bz0 we denote the Blaschke product bz0(ζ) =

• γ∈Γ

−|γ(ζ0)| γ(ζ0) ζ − γ(ζ0) 1 − γ(ζ0)ζ We have bz0 = Bz0 ◦ z. It is character automorphic with character µz0.We define kα

ζ0(ζ) = kα(ζ, ζ0) as the reproducing kernels of the

space H2(α), i.e.,

• f , kα

ζ0

• = f (ζ0)

∀f ∈ H2(α).

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Functional models

We have an evident decomposition {kα

∞} = H2(α) ⊖ b∞H2(α − µ∞).

Theorem The system of functions eα

n (ζ) = bn ∞(ζ)kα−nµ∞ ∞

(ζ) kα−nµ∞

∞

• .

(i) forms an orthonormal basis in H2(α) for n ∈ N and (ii) forms an orthonormal basis in L2(α) for n ∈ Z, where L2(α) = {f ∈ L2 : f ◦ γ = e2πiα(γ)f , γ ∈ Γ}.

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Functional models

Let for a system of intervals E the isospectral torus of Jacobi matrices be defined by J (E) = {J : σ(J) = σac(J) = E, J is reflectionless on E}. Theorem The multiplication operator by z in L2(α) with respect to the basis {eα

n } is a Jacobi matrix J = J(α) ∈ J(E):

zeα

n = a(n; α)eα n−1 + b(n; α)eα n + a(n + 1; α)eα n+1,

where a(n; α) = A(α − nµ∞), b(n; α) = B(α − nµ∞).

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Functional models

a(n; α) = A(α − nµ∞), b(n; α) = B(α − nµ∞), This gives a one-to-one map between Tg ≃ Γ∗ and J(E). Note that since A, B are continuous on Γ∗, J(α) are almost periodic. S−1J(α)S = J(α − µ∞). J(E) consists of periodic Jacobi matrices iff there exists p ∈ N such that pµ∞ = 0Γ∗, i.e., ω(Ek, ∞; Ω) ∈ Q. This is the case, if there exists a polynomial of degree p such that T −1

p ([−2, 2]) = E + . . . .

μ α α-μ

.

11 / 19

Functional models

a(n; α) = A(α − nµ∞), b(n; α) = B(α − nµ∞), This gives a one-to-one map between Tg ≃ Γ∗ and J(E). Note that since A, B are continuous on Γ∗, J(α) are almost periodic. S−1J(α)S = J(α − µ∞). J(E) consists of periodic Jacobi matrices iff there exists p ∈ N such that pµ∞ = 0Γ∗, i.e., ω(Ek, ∞; Ω) ∈ Q. This is the case, if there exists a polynomial of degree p such that T −1

p ([−2, 2]) = E + . . . .

μ α α-μ

.

11 / 19

Functional models

a(n; α) = A(α − nµ∞), b(n; α) = B(α − nµ∞), This gives a one-to-one map between Tg ≃ Γ∗ and J(E). Note that since A, B are continuous on Γ∗, J(α) are almost periodic. S−1J(α)S = J(α − µ∞). J(E) consists of periodic Jacobi matrices iff there exists p ∈ N such that pµ∞ = 0Γ∗, i.e., ω(Ek, ∞; Ω) ∈ Q. This is the case, if there exists a polynomial of degree p such that T −1

p ([−2, 2]) = E + . . . .

μ α α-μ

.

11 / 19

Functional models

a(n; α) = A(α − nµ∞), b(n; α) = B(α − nµ∞), This gives a one-to-one map between Tg ≃ Γ∗ and J(E). Note that since A, B are continuous on Γ∗, J(α) are almost periodic. S−1J(α)S = J(α − µ∞). J(E) consists of periodic Jacobi matrices iff there exists p ∈ N such that pµ∞ = 0Γ∗, i.e., ω(Ek, ∞; Ω) ∈ Q. This is the case, if there exists a polynomial of degree p such that T −1

p ([−2, 2]) = E + . . . .

μ α α-μ

.

11 / 19

Functional models

a(n; α) = A(α − nµ∞), b(n; α) = B(α − nµ∞), This gives a one-to-one map between Tg ≃ Γ∗ and J(E). Note that since A, B are continuous on Γ∗, J(α) are almost periodic. S−1J(α)S = J(α − µ∞). J(E) consists of periodic Jacobi matrices iff there exists p ∈ N such that pµ∞ = 0Γ∗, i.e., ω(Ek, ∞; Ω) ∈ Q. This is the case, if there exists a polynomial of degree p such that T −1

p ([−2, 2]) = E + . . . .

μ α α-μ

.

11 / 19

Functional models

a(n; α) = A(α − nµ∞), b(n; α) = B(α − nµ∞), This gives a one-to-one map between Tg ≃ Γ∗ and J(E). Note that since A, B are continuous on Γ∗, J(α) are almost periodic. S−1J(α)S = J(α − µ∞). J(E) consists of periodic Jacobi matrices iff there exists p ∈ N such that pµ∞ = 0Γ∗, i.e., ω(Ek, ∞; Ω) ∈ Q. This is the case, if there exists a polynomial of degree p such that T −1

p ([−2, 2]) = E + . . . .

μ α α-μ

.

11 / 19

Lemma For a system of intervals E there exists always a rational function ∆(z) = λ0z + c0 +

g

• j=1

λj cj − z , cj ∈ (aj, bj). such that E = ∆−1([−2, 2]), and Im ∆(z) > 0 for Im z > 0. Lemma There exists a function ψ(z) such that ∆(z) = ψ(z) + 1 ψ(z). Moreover ψ ◦ z = bc1 . . . bcg b∞.

12 / 19

Proof

Let f : D → C \ [−2, 2], f (ζ) = ζ + 1/ζ and g(u) = 1/2(u − √ u2 − 4) denote its inverse. Since ∆ : Ω → C \ [−2, 2], ψ(z) = 1 2(∆(z) −

• ∆(z)2 − 4),

is well defined. By definition we have λ0z + c0 +

g

• j=1

λj cj − z = ∆(z) = ψ(z) + 1 ψ(z) Moreover, |ψ| < 1 in Ω, |ψ| = 1 on E, ψ has simple zeros at c1, . . . , cg, ∞. Thus, ψ = Bbc1 . . . Bcg B∞, i.e., ψ ◦ z = bc1 . . . bcg b∞.

13 / 19

Proof

Let f : D → C \ [−2, 2], f (ζ) = ζ + 1/ζ and g(u) = 1/2(u − √ u2 − 4) denote its inverse. Since ∆ : Ω → C \ [−2, 2], ψ(z) = 1 2(∆(z) −

• ∆(z)2 − 4),

is well defined. By definition we have λ0z + c0 +

g

• j=1

λj cj − z = ∆(z) = ψ(z) + 1 ψ(z) Moreover, |ψ| < 1 in Ω, |ψ| = 1 on E, ψ has simple zeros at c1, . . . , cg, ∞. Thus, ψ = Bbc1 . . . Bcg B∞, i.e., ψ ◦ z = bc1 . . . bcg b∞.

13 / 19

Proof

Let f : D → C \ [−2, 2], f (ζ) = ζ + 1/ζ and g(u) = 1/2(u − √ u2 − 4) denote its inverse. Since ∆ : Ω → C \ [−2, 2], ψ(z) = 1 2(∆(z) −

• ∆(z)2 − 4),

is well defined. By definition we have λ0z + c0 +

g

• j=1

λj cj − z = ∆(z) = ψ(z) + 1 ψ(z) Moreover, |ψ| < 1 in Ω, |ψ| = 1 on E, ψ has simple zeros at c1, . . . , cg, ∞. Thus, ψ = Bbc1 . . . Bcg B∞, i.e., ψ ◦ z = bc1 . . . bcg b∞.

13 / 19

Proof

Let f : D → C \ [−2, 2], f (ζ) = ζ + 1/ζ and g(u) = 1/2(u − √ u2 − 4) denote its inverse. Since ∆ : Ω → C \ [−2, 2], ψ(z) = 1 2(∆(z) −

• ∆(z)2 − 4),

is well defined. By definition we have λ0z + c0 +

g

• j=1

λj cj − z = ∆(z) = ψ(z) + 1 ψ(z) Moreover, |ψ| < 1 in Ω, |ψ| = 1 on E, ψ has simple zeros at c1, . . . , cg, ∞. Thus, ψ = Bbc1 . . . Bcg B∞, i.e., ψ ◦ z = bc1 . . . bcg b∞.

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Definition of periodic GMP matrices

The relation ψ(z(ζ)) = bc1(ζ) . . . bcg (ζ)b∞(ζ) implies in particular that µ∞ + g

j=1 µcj = 0Γ∗. The vectors

f α

0 =

kα

c1

kα

c1, f α 1 = bc1k α−µc1 c2

k

α−µc1 c2

• , ..., f α

g =

g

j=1 bcjkα+µ∞ ∞

kα+µ∞

∞

• ,

form a basis of the g + 1 dimensional subspace K α

ψ = H2(α) ⊖ ψH2(α).

14 / 19

Definition of periodic GMP matrices

The relation ψ(z(ζ)) = bc1(ζ) . . . bcg (ζ)b∞(ζ) implies in particular that µ∞ + g

j=1 µcj = 0Γ∗. The vectors

f α

0 =

kα

c1

kα

c1, f α 1 = bc1k α−µc1 c2

k

α−µc1 c2

• , ..., f α

g =

g

j=1 bcjkα+µ∞ ∞

kα+µ∞

∞

• ,

form a basis of the g + 1 dimensional subspace K α

ψ = H2(α) ⊖ ψH2(α).

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Definition of periodic GMP matrices

H2(α) = K α

ψ ⊕ ψK α ψ ⊕ ψ2K α ψ ⊕ . . .

Theorem The system of functions f α

n = ψmf α j ,

n = (g + 1)m + j, j ∈ [0, . . . , g] (i) forms an orthonormal basis in H2(α) for n ∈ N and (ii) forms an orthonormal basis in L2(α) for n ∈ Z. Definition We call the multiplication operator by z with respect to the basis {f α

n } a GMP matrix. For a given α ∈ Γ∗ we denote the

corresponding GMP matrix by A(α). The isospectral torus of GMP matrices is defined by A(E) = {A(α) : α ∈ Γ∗}.

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Definition of periodic GMP matrices

H2(α) = K α

ψ ⊕ ψK α ψ ⊕ ψ2K α ψ ⊕ . . .

Theorem The system of functions f α

n = ψmf α j ,

n = (g + 1)m + j, j ∈ [0, . . . , g] (i) forms an orthonormal basis in H2(α) for n ∈ N and (ii) forms an orthonormal basis in L2(α) for n ∈ Z. Definition We call the multiplication operator by z with respect to the basis {f α

n } a GMP matrix. For a given α ∈ Γ∗ we denote the

corresponding GMP matrix by A(α). The isospectral torus of GMP matrices is defined by A(E) = {A(α) : α ∈ Γ∗}.

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Definition of periodic GMP matrices

H2(α) = K α

ψ ⊕ ψK α ψ ⊕ ψ2K α ψ ⊕ . . .

Theorem The system of functions f α

n = ψmf α j ,

n = (g + 1)m + j, j ∈ [0, . . . , g] (i) forms an orthonormal basis in H2(α) for n ∈ N and (ii) forms an orthonormal basis in L2(α) for n ∈ Z. Definition We call the multiplication operator by z with respect to the basis {f α

n } a GMP matrix. For a given α ∈ Γ∗ we denote the

corresponding GMP matrix by A(α). The isospectral torus of GMP matrices is defined by A(E) = {A(α) : α ∈ Γ∗}.

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Structure of GMP matrices

We say A ∈ A if there exist

• pj = [p(j)

0 , . . . , p(j) g ]∗,

qj = [q(j)

0 , . . . , q(j) g ]∗ and C = {c1, . . . , cg}

such that A =            ... ... ... A( p−1) A∗( p−1) B( p−1, q−1) A( p0) A∗( p0) B( p0, q0) A( p1) A∗( p1) ... ... ...            , where A( p) = δg p ∗ and B( p, q) = diag[c1, . . . , cg, 0] + ( q p ∗)− + ( p q ∗)+ .

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The Jacobi flow on GMP matrices

Corollary The map F(A(α)) = J(α) gives a parametrization of the isospectral torus of almost periodic Jacobi matrices by periodic GMP matrices. Recall, S−1J(α)S = J(α − µ∞). Definition The Jacobi flow in A(E) is defined by J (A(α)) = A(α − µ∞). A

F

• J
• J

S

• A

F

J

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The Jacobi flow on GMP matrices

Corollary The map F(A(α)) = J(α) gives a parametrization of the isospectral torus of almost periodic Jacobi matrices by periodic GMP matrices. Recall, S−1J(α)S = J(α − µ∞). Definition The Jacobi flow in A(E) is defined by J (A(α)) = A(α − µ∞). A

F

• J
• J

S

• A

F

J

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Spectral theory of periodic GMP matrices

Let a(z; c, p, q) = I − 1 c − z p q p q

• j,

j = −1 1

• ,

a(z; ∞, p, q) = −p

1 p z−pq p

• and let us define the transfer matrix of A ∈ A(E) by

A(z, A) = a(z, c1, p0, q0), a(z, c2, p1, q1) . . . a(z, ∞, pg, qg). Let r+(z) be the resolvent function of A+ and r(1)

+

be the resolvent function of (S∗

+)g+1A+Sg+1 +

, then r+(z) 1

• ∼ a(z, c1, p0, q0), a(z, c2, p1, q1) . . . a(z, ∞, pg, qg)
• r(1)

+ (z)

1

• 18 / 19

Spectral theory of periodic GMP matrices

Let a(z; c, p, q) = I − 1 c − z p q p q

• j,

j = −1 1

• ,

a(z; ∞, p, q) = −p

1 p z−pq p

• and let us define the transfer matrix of A ∈ A(E) by

A(z, A) = a(z, c1, p0, q0), a(z, c2, p1, q1) . . . a(z, ∞, pg, qg). Let r+(z) be the resolvent function of A+ and r(1)

+

be the resolvent function of (S∗

+)g+1A+Sg+1 +

, then r+(z) 1

• ∼ a(z, c1, p0, q0), a(z, c2, p1, q1) . . . a(z, ∞, pg, qg)
• r(1)

+ (z)

1

• 18 / 19

Spectral theory of periodic GMP matrices

Let a(z; c, p, q) = I − 1 c − z p q p q

• j,

j = −1 1

• ,

a(z; ∞, p, q) = −p

1 p z−pq p

• and let us define the transfer matrix of A ∈ A(E) by

A(z, A) = a(z, c1, p0, q0), a(z, c2, p1, q1) . . . a(z, ∞, pg, qg). Let r+(z) be the resolvent function of A+ and r(1)

+

be the resolvent function of (S∗

+)g+1A+Sg+1 +

, then r+(z) 1

• ∼ a(z, c1, p0, q0), a(z, c2, p1, q1) . . . a(z, ∞, pg, qg)
• r(1)

+ (z)

1

• 18 / 19

Spectral theory of periodic GMP matrices

Theorem Let A ∈ A(E, C) and let ∆(z) := tr A(z; A). Then the spectrum E

• f A is given by

E = ∆−1([−2, 2]) = {x : ∆(x) ∈ [−2, 2]}.

Thank you for your attention!

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Spectral theory of periodic GMP matrices

Theorem Let A ∈ A(E, C) and let ∆(z) := tr A(z; A). Then the spectrum E

• f A is given by

E = ∆−1([−2, 2]) = {x : ∆(x) ∈ [−2, 2]}.

Thank you for your attention!

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