Spectral Theory of Orthogonal Polynomials Floquet Solutions

Periodic Jacobi Periodic and Ergodic Spectral Problems Matrices Issac Newton Institute, January, 2015 The Discriminant

Gaps Spectrum Barry Simon Potential Theory IBM Professor of Mathematics and Theoretical Physics Quadratic Irrationalities California Institute of Technology Minimal Herglotz Pasadena, CA, U.S.A. Functions

Isospectral Tori Lecture 5: Isospectral Tori Spectral Theory of Orthogonal Polynomials

Lecture 1: Introduction and Overview Lecture 2: Szegő Theorem for OPUC Floquet Solutions

Periodic Jacobi Lecture 3: Three Kinds of Polynomial Asymptotics Matrices Lecture 4: Potential Theory The Discriminant

Gaps Lecture 5: Isospectral Tori Spectrum Lecture 6: Fuchsian Groups Potential Theory Lecture 7: Chebyshev Polynomials, I Quadratic Irrationalities Lecture 8: Chebyshev Polynomials, II Minimal Herglotz Functions

Isospectral Tori References

[OPUC] B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series Floquet Solutions 54.1, American Mathematical Society, Providence, RI, Periodic Jacobi Matrices 2005. The Discriminant [OPUC2] B. Simon, Orthogonal Polynomials on the Unit Gaps Circle, Part 2: Spectral Theory, AMS Colloquium Series, Spectrum 54.2, American Mathematical Society, Providence, RI, Potential Theory

Quadratic 2005. Irrationalities

Minimal Herglotz [SzThm] B. Simon, Szegő’s Theorem and Its Descendants: Functions Spectral Theory for L2 Perturbations of Orthogonal Isospectral Tori Polynomials, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011. Since you’ve seen some of these ideas already, I’ll be brief on some of the details. ∞ So {an, bn}n=−∞ are two-sided sequences with some p > 0 in Z so that an+p = an bn+p = bn ∞ For z ∈ C ﬁxed, we are interested in solutions {un}n=0 of

anun+1 + bnun + an−1un−1 = zun

Floquet Solutions

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori ∞ For z ∈ C ﬁxed, we are interested in solutions {un}n=0 of

anun+1 + bnun + an−1un−1 = zun

Floquet Solutions

We turn next to two sided periodic Jacobi matrices, their isospectral tori and use that as a jump oﬀ to the start of Floquet Solutions ﬁnite gap isospectral tori which will involve some almost Periodic Jacobi Matrices periodic examples. Since you’ve seen some of these ideas The Discriminant already, I’ll be brief on some of the details. Gaps ∞ So {an, bn} are two-sided sequences with some p > 0 Spectrum n=−∞ in so that Potential Theory Z

Quadratic an+p = an bn+p = bn Irrationalities

Minimal Herglotz Functions

Isospectral Tori Floquet Solutions

Quadratic an+p = an bn+p = bn Irrationalities ∞ Minimal Herglotz For z ∈ C ﬁxed, we are interested in solutions {un}n=0 of Functions

Isospectral Tori anun+1 + bnun + an−1un−1 = zun Such solutions are called Floquet solutions as they are analogs of solutions of ODE, especially Hill’s equation −u00 + V u = zu, V (x + p) = V (x). The analysis of such solutions is a delightful amalgam of three tools, the first of which is just the fact that the set of all solutions of the difference equation is two-dimensional. Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1.

Floquet Solutions

iθ that also obey for some λ ∈ C (λ = e , θ ∈ C)

Floquet Solutions un+p = λun Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The analysis of such solutions is a delightful amalgam of three tools, the first of which is just the fact that the set of all solutions of the difference equation is two-dimensional. Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1.

Floquet Solutions

iθ that also obey for some λ ∈ C (λ = e , θ ∈ C)

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori the first of which is just the fact that the set of all solutions of the difference equation is two-dimensional. Thus, there are, for z fixed, at most two different λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1.

Floquet Solutions

iθ that also obey for some λ ∈ C (λ = e , θ ∈ C)

Floquet Solutions un+p = λun Periodic Jacobi Matrices Such solutions are called Floquet solutions as they are The Discriminant analogs of solutions of ODE, especially Hill’s equation Gaps 00 Spectrum −u + V u = zu, V (x + p) = V (x). Potential Theory The analysis of such solutions is a delightful amalgam of Quadratic Irrationalities three tools,

Minimal Herglotz Functions

Isospectral Tori Thus, there are, for z ﬁxed, at most two diﬀerent λ’s for which there is a solution. If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1.

Floquet Solutions

iθ that also obey for some λ ∈ C (λ = e , θ ∈ C)

Minimal Herglotz all solutions of the diﬀerence equation is two-dimensional. Functions

Isospectral Tori If λ1, λ2 are two such λ’s, their Wronskian is non-zero so constancy of the Wronskian implies λ1λ2 = 1.

Floquet Solutions

iθ that also obey for some λ ∈ C (λ = e , θ ∈ C)

Minimal Herglotz all solutions of the diﬀerence equation is two-dimensional. Functions

Isospectral Tori Thus, there are, for z ﬁxed, at most two diﬀerent λ’s for which there is a solution. λ1λ2 = 1.

Floquet Solutions

iθ that also obey for some λ ∈ C (λ = e , θ ∈ C)

Minimal Herglotz all solutions of the diﬀerence equation is two-dimensional. Functions

iθ that also obey for some λ ∈ C (λ = e , θ ∈ C)

Minimal Herglotz all solutions of the diﬀerence equation is two-dimensional. Functions

Jjj = bj,Jj j+1 = aj,Jj j−1 = aj−1 −1 J1p = apλ ,Jp1 = apλ ∞ −1 If {un}n=−∞ is a Floquet solution, u0 = λ up, p per,λ up+1 = λu1 so ue = {un}n=1 has J ue = zue.

Conversely, if ue solves this, the unique u with un+p = λun ∞ and ue = {un}n=1 is a Floquet solution.

Periodic B.C. Jacobi Matrices

The (twisted) periodic boundary condition Jacobi matrix J per,λ is p × p. Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Jjj = bj,Jj j+1 = aj,Jj j−1 = aj−1 −1 J1p = apλ ,Jp1 = apλ ∞ −1 If {un}n=−∞ is a Floquet solution, u0 = λ up, p per,λ up+1 = λu1 so ue = {un}n=1 has J ue = zue.

Conversely, if ue solves this, the unique u with un+p = λun ∞ and ue = {un}n=1 is a Floquet solution.

Periodic B.C. Jacobi Matrices

The (twisted) periodic boundary condition Jacobi matrix J per,λ is p × p. It is the ﬁnite Jacobi matrix with 1p and p1 Floquet Solutions matrix elements added: Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori −1 J1p = apλ ,Jp1 = apλ ∞ −1 If {un}n=−∞ is a Floquet solution, u0 = λ up, p per,λ up+1 = λu1 so ue = {un}n=1 has J ue = zue.

Conversely, if ue solves this, the unique u with un+p = λun ∞ and ue = {un}n=1 is a Floquet solution.

Periodic B.C. Jacobi Matrices

The (twisted) periodic boundary condition Jacobi matrix J per,λ is p × p. It is the ﬁnite Jacobi matrix with 1p and p1 Floquet Solutions matrix elements added: Periodic Jacobi Matrices

The Discriminant Jjj = bj,Jj j+1 = aj,Jj j−1 = aj−1

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori ∞ −1 If {un}n=−∞ is a Floquet solution, u0 = λ up, p per,λ up+1 = λu1 so ue = {un}n=1 has J ue = zue.

Conversely, if ue solves this, the unique u with un+p = λun ∞ and ue = {un}n=1 is a Floquet solution.

Periodic B.C. Jacobi Matrices

The (twisted) periodic boundary condition Jacobi matrix J per,λ is p × p. It is the ﬁnite Jacobi matrix with 1p and p1 Floquet Solutions matrix elements added: Periodic Jacobi Matrices

The Discriminant Jjj = bj,Jj j+1 = aj,Jj j−1 = aj−1 Gaps −1 J1p = apλ ,Jp1 = apλ Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori p per,λ so ue = {un}n=1 has J ue = zue.

Conversely, if ue solves this, the unique u with un+p = λun ∞ and ue = {un}n=1 is a Floquet solution.

Periodic B.C. Jacobi Matrices

The (twisted) periodic boundary condition Jacobi matrix J per,λ is p × p. It is the ﬁnite Jacobi matrix with 1p and p1 Floquet Solutions matrix elements added: Periodic Jacobi Matrices

The Discriminant Jjj = bj,Jj j+1 = aj,Jj j−1 = aj−1 Gaps −1 J1p = apλ ,Jp1 = apλ Spectrum ∞ −1 Potential Theory If {un}n=−∞ is a Floquet solution, u0 = λ up, Quadratic u = λu Irrationalities p+1 1

Minimal Herglotz Functions

Isospectral Tori Conversely, if ue solves this, the unique u with un+p = λun ∞ and ue = {un}n=1 is a Floquet solution.

Periodic B.C. Jacobi Matrices

The (twisted) periodic boundary condition Jacobi matrix J per,λ is p × p. It is the ﬁnite Jacobi matrix with 1p and p1 Floquet Solutions matrix elements added: Periodic Jacobi Matrices

Isospectral Tori Periodic B.C. Jacobi Matrices

The (twisted) periodic boundary condition Jacobi matrix J per,λ is p × p. It is the ﬁnite Jacobi matrix with 1p and p1 Floquet Solutions matrix elements added: Periodic Jacobi Matrices

The Discriminant Jjj = bj,Jj j+1 = aj,Jj j−1 = aj−1 Gaps −1 J1p = apλ ,Jp1 = apλ Spectrum ∞ −1 Potential Theory If {un}n=−∞ is a Floquet solution, u0 = λ up, p per,λ Quadratic up+1 = λu1 so u = {un} has J u = zu. Irrationalities e n=1 e e Minimal Herglotz Functions Conversely, if ue solves this, the unique u with un+p = λun ∞ Isospectral Tori and ue = {un}n=1 is a Floquet solution. (We’ll see soon that if λ 6= ±1, there are exactly p.) iθ If λ = e , θ ∈ R, λ 6= ±1, there are precisely p distinct z’s all real, for which there are Floquet solutions with that λ. The reality comes from hermicity of J per,λ. If λ 6= ±1, then λ¯ 6= λ. If u is a Floquet solution for λ, since z is real, u¯ is a Floquet solution for λ¯ so there is a per,λ unique solution for that z. Thus, for λ ∈ ∂D \ {±1}, J has p eigenvalues and each simple.

Periodic B.C. Jacobi Matrices

This implies

Floquet Solutions For any λ, there are at most p z’s which have a

Periodic Jacobi Floquet solution for that λ. Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori iθ If λ = e , θ ∈ R, λ 6= ±1, there are precisely p distinct z’s all real, for which there are Floquet solutions with that λ. The reality comes from hermicity of J per,λ. If λ 6= ±1, then λ¯ 6= λ. If u is a Floquet solution for λ, since z is real, u¯ is a Floquet solution for λ¯ so there is a per,λ unique solution for that z. Thus, for λ ∈ ∂D \ {±1}, J has p eigenvalues and each simple.

Periodic B.C. Jacobi Matrices

This implies

Floquet Solutions For any λ, there are at most p z’s which have a

Periodic Jacobi Floquet solution for that λ. (We’ll see soon that if Matrices λ 6= ±1, there are exactly p.) The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The reality comes from hermicity of J per,λ. If λ 6= ±1, then λ¯ 6= λ. If u is a Floquet solution for λ, since z is real, u¯ is a Floquet solution for λ¯ so there is a per,λ unique solution for that z. Thus, for λ ∈ ∂D \ {±1}, J has p eigenvalues and each simple.

Periodic B.C. Jacobi Matrices

This implies

Floquet Solutions For any λ, there are at most p z’s which have a

Periodic Jacobi Floquet solution for that λ. (We’ll see soon that if Matrices λ 6= ±1, there are exactly p.) The Discriminant iθ Gaps If λ = e , θ ∈ R, λ 6= ±1, there are precisely p distinct

Spectrum z’s all real, for which there are Floquet solutions with

Potential Theory that λ.

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori If λ 6= ±1, then λ¯ 6= λ. If u is a Floquet solution for λ, since z is real, u¯ is a Floquet solution for λ¯ so there is a per,λ unique solution for that z. Thus, for λ ∈ ∂D \ {±1}, J has p eigenvalues and each simple.

Periodic B.C. Jacobi Matrices

This implies

Floquet Solutions For any λ, there are at most p z’s which have a

Spectrum z’s all real, for which there are Floquet solutions with

Potential Theory that λ. Quadratic per,λ Irrationalities The reality comes from hermicity of J .

Minimal Herglotz Functions

Isospectral Tori per,λ Thus, for λ ∈ ∂D \ {±1}, J has p eigenvalues and each simple.

Periodic B.C. Jacobi Matrices

This implies

Floquet Solutions For any λ, there are at most p z’s which have a

Spectrum z’s all real, for which there are Floquet solutions with

Potential Theory that λ. Quadratic per,λ Irrationalities The reality comes from hermicity of J . Minimal Herglotz ¯ Functions If λ 6= ±1, then λ 6= λ. If u is a Floquet solution for λ,

Isospectral Tori since z is real, u¯ is a Floquet solution for λ¯ so there is a unique solution for that z. Periodic B.C. Jacobi Matrices

This implies

Floquet Solutions For any λ, there are at most p z’s which have a

Spectrum z’s all real, for which there are Floquet solutions with

Potential Theory that λ. Quadratic per,λ Irrationalities The reality comes from hermicity of J . Minimal Herglotz ¯ Functions If λ 6= ±1, then λ 6= λ. If u is a Floquet solution for λ,

Isospectral Tori since z is real, u¯ is a Floquet solution for λ¯ so there is a per,λ unique solution for that z. Thus, for λ ∈ ∂D \ {±1}, J has p eigenvalues and each simple. u1 u1 u1 Tp(z)( a0u0 ) = λ( a0u0 ) is equivalent to ( a0u0 ) generating a Floquet solution ! (Note: a0 may not be 1.) ∞ In terms of the OP’s for {an, bn}n=1, pp(z) −qp(z) Tp(z) = appp−1(z) −apqp−1(z)

The discriminant, ∆(z), is deﬁned by ∆(z) = Tr Tp(z) = pp(z) − apqp−1(z)

is a (real) polynomial of degree exactly p.

The Discriminant

The third tool concerns the p-step transfer matrix.

Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori (Note: a0 may not be 1.) ∞ In terms of the OP’s for {an, bn}n=1, pp(z) −qp(z) Tp(z) = appp−1(z) −apqp−1(z)

The discriminant, ∆(z), is deﬁned by ∆(z) = Tr Tp(z) = pp(z) − apqp−1(z)

is a (real) polynomial of degree exactly p.

The Discriminant

The third tool concerns the p-step transfer matrix.

u1 u1 u1 Floquet Solutions Tp(z)( a0u0 ) = λ( a0u0 ) is equivalent to ( a0u0 ) generating a Periodic Jacobi Floquet solution ! Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori ∞ In terms of the OP’s for {an, bn}n=1, pp(z) −qp(z) Tp(z) = appp−1(z) −apqp−1(z)

The discriminant, ∆(z), is deﬁned by ∆(z) = Tr Tp(z) = pp(z) − apqp−1(z)

is a (real) polynomial of degree exactly p.

The Discriminant

The third tool concerns the p-step transfer matrix.

u1 u1 u1 Floquet Solutions Tp(z)( a0u0 ) = λ( a0u0 ) is equivalent to ( a0u0 ) generating a Periodic Jacobi Floquet solution ! (Note: a0 may not be 1.) Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The discriminant, ∆(z), is deﬁned by ∆(z) = Tr Tp(z) = pp(z) − apqp−1(z)

is a (real) polynomial of degree exactly p.

The Discriminant

The third tool concerns the p-step transfer matrix.

u1 u1 u1 Floquet Solutions Tp(z)( a0u0 ) = λ( a0u0 ) is equivalent to ( a0u0 ) generating a Periodic Jacobi Floquet solution ! (Note: a0 may not be 1.) Matrices ∞ The Discriminant In terms of the OP’s for {an, bn}n=1, Gaps Spectrum pp(z) −qp(z) Tp(z) = Potential Theory appp−1(z) −apqp−1(z) Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori ∆(z) = Tr Tp(z) = pp(z) − apqp−1(z)

is a (real) polynomial of degree exactly p.

The Discriminant

The third tool concerns the p-step transfer matrix.

Isospectral Tori The Discriminant

The third tool concerns the p-step transfer matrix.

is a (real) polynomial of degree exactly p. ∆(z) = λ + λ−1; ∆(z) = 2 cos θ if λ = eiθ Floquet solutions correspond to geometric eigenvalues for Tp(z). If λ 6= ±1, it has multiplicity one, so is geometric. λ = ±1 has multiplicity 2, so there can be one or two Floquet solutions. An important consequence of the fact that ∆(z) ∈ (−2, 2) −1 implies all z’s are real is ∆ (−2, 2) ⊂ R.

The Discriminant

Since det Tp(z) = 1, it has algebraic eigenvalues λ and λ−1 where Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Floquet solutions correspond to geometric eigenvalues for Tp(z). If λ 6= ±1, it has multiplicity one, so is geometric. λ = ±1 has multiplicity 2, so there can be one or two Floquet solutions. An important consequence of the fact that ∆(z) ∈ (−2, 2) −1 implies all z’s are real is ∆ (−2, 2) ⊂ R.

The Discriminant

Since det Tp(z) = 1, it has algebraic eigenvalues λ and λ−1 where Floquet Solutions −1 iθ Periodic Jacobi ∆(z) = λ + λ ; ∆(z) = 2 cos θ if λ = e Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori An important consequence of the fact that ∆(z) ∈ (−2, 2) −1 implies all z’s are real is ∆ (−2, 2) ⊂ R.

The Discriminant

Since det Tp(z) = 1, it has algebraic eigenvalues λ and λ−1 where Floquet Solutions −1 iθ Periodic Jacobi ∆(z) = λ + λ ; ∆(z) = 2 cos θ if λ = e Matrices

Minimal Herglotz Functions

Isospectral Tori −1 ∆ (−2, 2) ⊂ R.

The Discriminant

Since det Tp(z) = 1, it has algebraic eigenvalues λ and λ−1 where Floquet Solutions −1 iθ Periodic Jacobi ∆(z) = λ + λ ; ∆(z) = 2 cos θ if λ = e Matrices

The Discriminant Floquet solutions correspond to geometric eigenvalues for Gaps Tp(z). If λ 6= ±1, it has multiplicity one, so is geometric. Spectrum λ = ±1 has multiplicity 2, so there can be one or two Potential Theory Floquet solutions. Quadratic Irrationalities An important consequence of the fact that ∆(z) ∈ (−2, 2) Minimal Herglotz implies all z’s are real is Functions

Isospectral Tori The Discriminant

Since det Tp(z) = 1, it has algebraic eigenvalues λ and λ−1 where Floquet Solutions −1 iθ Periodic Jacobi ∆(z) = λ + λ ; ∆(z) = 2 cos θ if λ = e Matrices

Isospectral Tori −1 0 Thus, ∆ (−2, 2) ⊂ R ⇒ ∆ (x0) 6= 0 if ∆(x0) ∈ (−2, 2). −1 Thus, ∆ (−2, 2) = (α1, β1) ∪ (α2, β2) ∪ ... ∪ (αp, βp)

where α1 < β1 ≤ α2 < β2 ≤ α3 < . . . < βp

with ∆ a smooth bijection of (αj, βj) to (−2, 2). Could be orientation reversing or not.

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori −1 Thus, ∆ (−2, 2) = (α1, β1) ∪ (α2, β2) ∪ ... ∪ (αp, βp)

where α1 < β1 ≤ α2 < β2 ≤ α3 < . . . < βp

with ∆ a smooth bijection of (αj, βj) to (−2, 2). Could be orientation reversing or not.

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori where α1 < β1 ≤ α2 < β2 ≤ α3 < . . . < βp

with ∆ a smooth bijection of (αj, βj) to (−2, 2). Could be orientation reversing or not.

The Discriminant

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori with ∆ a smooth bijection of (αj, βj) to (−2, 2). Could be orientation reversing or not.

The Discriminant

A basic fact of analytic functions is that if f(z) is real (i.e., 0 f(¯z) = f(z)), x0 ∈ R with f (x0) = 0, there are non-real Floquet Solutions z’s near x0 with f(z) real and near f(x0). Periodic Jacobi Matrices −1 0 Thus, ∆ (−2, 2) ⊂ R ⇒ ∆ (x0) 6= 0 if ∆(x0) ∈ (−2, 2). The Discriminant −1 Gaps Thus, ∆ (−2, 2) = (α1, β1) ∪ (α2, β2) ∪ ... ∪ (αp, βp) Spectrum where α1 < β1 ≤ α2 < β2 ≤ α3 < . . . < βp Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Could be orientation reversing or not.

The Discriminant

Minimal Herglotz Functions

Isospectral Tori The Discriminant

Isospectral Tori It follows that ∆(αp) = −2, ∆(βp−1) = −2, ∆(αp−1) = 2 ..., i.e., p−j p−j−1 ∆(βj) = (−1) 2, ∆(αj) = (−1) 2 If the α’s and β’s are all distinct, we have p points where ∆(x) = 2 and p where ∆(x) = −2. Since deg ∆ = p, these are all the points.

If βj−1 = αj, there is one less point where ∆(x) = p−j−1 0 p−j−1 (−1) 2, but ∆ (αj) = 0 since ∆ − (−1) 2 has the same sign on both sides of αj. It follows that

The Discriminant

Since ∆(x) → ∞ as x → ∞, we must have ∆(βp) = 2.

Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori p−j p−j−1 ∆(βj) = (−1) 2, ∆(αj) = (−1) 2 If the α’s and β’s are all distinct, we have p points where ∆(x) = 2 and p where ∆(x) = −2. Since deg ∆ = p, these are all the points.

If βj−1 = αj, there is one less point where ∆(x) = p−j−1 0 p−j−1 (−1) 2, but ∆ (αj) = 0 since ∆ − (−1) 2 has the same sign on both sides of αj. It follows that

The Discriminant

Since ∆(x) → ∞ as x → ∞, we must have ∆(βp) = 2.

Floquet Solutions It follows that ∆(αp) = −2, ∆(βp−1) = −2,

Periodic Jacobi ∆(αp−1) = 2 ..., i.e., Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori If the α’s and β’s are all distinct, we have p points where ∆(x) = 2 and p where ∆(x) = −2. Since deg ∆ = p, these are all the points.

If βj−1 = αj, there is one less point where ∆(x) = p−j−1 0 p−j−1 (−1) 2, but ∆ (αj) = 0 since ∆ − (−1) 2 has the same sign on both sides of αj. It follows that

The Discriminant

Since ∆(x) → ∞ as x → ∞, we must have ∆(βp) = 2.

Floquet Solutions It follows that ∆(αp) = −2, ∆(βp−1) = −2,

Periodic Jacobi ∆(αp−1) = 2 ..., i.e., Matrices p−j p−j−1 The Discriminant ∆(βj) = (−1) 2, ∆(αj) = (−1) 2

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Since deg ∆ = p, these are all the points.

If βj−1 = αj, there is one less point where ∆(x) = p−j−1 0 p−j−1 (−1) 2, but ∆ (αj) = 0 since ∆ − (−1) 2 has the same sign on both sides of αj. It follows that

The Discriminant

Since ∆(x) → ∞ as x → ∞, we must have ∆(βp) = 2.

Floquet Solutions It follows that ∆(αp) = −2, ∆(βp−1) = −2,

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori If βj−1 = αj, there is one less point where ∆(x) = p−j−1 0 p−j−1 (−1) 2, but ∆ (αj) = 0 since ∆ − (−1) 2 has the same sign on both sides of αj. It follows that

The Discriminant

Since ∆(x) → ∞ as x → ∞, we must have ∆(βp) = 2.

Floquet Solutions It follows that ∆(αp) = −2, ∆(βp−1) = −2,

Minimal Herglotz Functions

Isospectral Tori 0 p−j−1 but ∆ (αj) = 0 since ∆ − (−1) 2 has the same sign on both sides of αj. It follows that

The Discriminant

Since ∆(x) → ∞ as x → ∞, we must have ∆(βp) = 2.

Floquet Solutions It follows that ∆(αp) = −2, ∆(βp−1) = −2,

Since ∆(x) → ∞ as x → ∞, we must have ∆(βp) = 2.

Floquet Solutions It follows that ∆(αp) = −2, ∆(βp−1) = −2,

Periodic Jacobi ∆(αp−1) = 2 ..., i.e., Matrices p−j p−j−1 The Discriminant ∆(βj) = (−1) 2, ∆(αj) = (−1) 2 Gaps If the α’s and β’s are all distinct, we have p points where Spectrum ∆(x) = 2 and p where ∆(x) = −2. Potential Theory Quadratic Since deg ∆ = p, these are all the points. Irrationalities Minimal Herglotz If βj−1 = αj, there is one less point where ∆(x) = Functions p−j−1 0 p−j−1 (−1) 2, but ∆ (αj) = 0 since ∆ − (−1) 2 has the Isospectral Tori same sign on both sides of αj. It follows that −1 p ∆ {−2, 2} = {αj, βj}j=1 and 0 0 ∆ (αj) = 0 ⇔ αj = βj−1, ∆ (βj) = 0 ⇔ βj = αj+1 and in that case, ∆00 is not zero at that point.

The [αj, βj] are called the bands and (βj, αj+1) the gaps.

If βj < αj+1, we say that gap j is open.

If βj = αj+1, we say gap j is closed.

Open and Closed Gaps

−1 p Theorem. ∆ [−2, 2] = ∪j=1[αj, βj] and

Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori 0 0 ∆ (αj) = 0 ⇔ αj = βj−1, ∆ (βj) = 0 ⇔ βj = αj+1 and in that case, ∆00 is not zero at that point.

The [αj, βj] are called the bands and (βj, αj+1) the gaps.

If βj < αj+1, we say that gap j is open.

If βj = αj+1, we say gap j is closed.

Open and Closed Gaps

−1 p Theorem. ∆ [−2, 2] = ∪j=1[αj, βj] and −1 p Floquet Solutions ∆ {−2, 2} = {αj, βj}j=1 and Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The [αj, βj] are called the bands and (βj, αj+1) the gaps.

If βj < αj+1, we say that gap j is open.

If βj = αj+1, we say gap j is closed.

Open and Closed Gaps

−1 p Theorem. ∆ [−2, 2] = ∪j=1[αj, βj] and −1 p Floquet Solutions ∆ {−2, 2} = {αj, βj}j=1 and Periodic Jacobi 0 0 Matrices ∆ (αj) = 0 ⇔ αj = βj−1, ∆ (βj) = 0 ⇔ βj = αj+1 The Discriminant and in that case, ∆00 is not zero at that point. Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori If βj < αj+1, we say that gap j is open.

If βj = αj+1, we say gap j is closed.

Open and Closed Gaps

Spectrum The [αj, βj] are called the bands and (βj, αj+1) the gaps. Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori If βj = αj+1, we say gap j is closed.

Open and Closed Gaps

Spectrum The [αj, βj] are called the bands and (βj, αj+1) the gaps. Potential Theory Quadratic If βj < αj+1, we say that gap j is open. Irrationalities

Minimal Herglotz Functions

Isospectral Tori Open and Closed Gaps

Spectrum The [αj, βj] are called the bands and (βj, αj+1) the gaps. Potential Theory Quadratic If βj < αj+1, we say that gap j is open. Irrationalities Minimal Herglotz If βj = αj+1, we say gap j is closed. Functions

Isospectral Tori while at each of the edges of an open gap there is only one periodic (Floquet) solution. The transfer matrix has a 1 0 Jordan anomaly, i.e., det = 1, Tr = 2, but T 6= ( 0 1 ). Each of the gaps where ∆(x) ≥ 2 has two periodic solutions—either two at βj = αj+1 or one each at βj and αj+1 so there are p periodic Floquet solutions, as there must be from the J per analysis.

Open and Closed Gaps

Further analysis shows at a closed gap (with ∆(α) = 2 for simplicity) there are two periodic (Floquet) solutions, Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The transfer matrix has a 1 0 Jordan anomaly, i.e., det = 1, Tr = 2, but T 6= ( 0 1 ). Each of the gaps where ∆(x) ≥ 2 has two periodic solutions—either two at βj = αj+1 or one each at βj and αj+1 so there are p periodic Floquet solutions, as there must be from the J per analysis.

Open and Closed Gaps

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Each of the gaps where ∆(x) ≥ 2 has two periodic solutions—either two at βj = αj+1 or one each at βj and αj+1 so there are p periodic Floquet solutions, as there must be from the J per analysis.

Open and Closed Gaps

Further analysis shows at a closed gap (with ∆(α) = 2 for simplicity) there are two periodic (Floquet) solutions, while Floquet Solutions at each of the edges of an open gap there is only one Periodic Jacobi Matrices periodic (Floquet) solution. The transfer matrix has a 1 0 The Discriminant Jordan anomaly, i.e., det = 1, Tr = 2, but T 6= ( 0 1 ). Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori so there are p periodic Floquet solutions, as there must be from the J per analysis.

Open and Closed Gaps

Quadratic αj+1 Irrationalities

Minimal Herglotz Functions

Isospectral Tori Open and Closed Gaps

Quadratic αj+1 so there are p periodic Floquet solutions, as there Irrationalities must be from the J per analysis. Minimal Herglotz Functions

Isospectral Tori It follows that there are diﬀerent solutions u± decaying exponentially at ±∞ so their Wronskian is not zero. By (Schrödinger operator) Green’s function methods,

+ − Gnm(z) = umax(n,m)(z)umin(m.n)(z)/W (z)

is the matrix for J − z−1, i.e., z∈ / σ(J). If ∆(z) ∈ [−2, 2], there is a bounded Floquet solution (since |λ| = 1). Then k(J − z)[uχ[−N,N]]k is bounded, but since Pp 2 j=1|um+j| is constant, kuχ[−N,N]k → ∞ so z ∈ σ(J). Thus p Theorem. σ(J) = ∪j=1[αj, βj].

Spectrum and Spectral Types

If z is such that ∆(z) 6∈ [−2, 2], then the roots of λ + λ−1 = ∆(z) have |λ| > 1, |λ−1| < 1. Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori By (Schrödinger operator) Green’s function methods,

+ − Gnm(z) = umax(n,m)(z)umin(m.n)(z)/W (z)

Spectrum and Spectral Types

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori If ∆(z) ∈ [−2, 2], there is a bounded Floquet solution (since |λ| = 1). Then k(J − z)[uχ[−N,N]]k is bounded, but since Pp 2 j=1|um+j| is constant, kuχ[−N,N]k → ∞ so z ∈ σ(J). Thus p Theorem. σ(J) = ∪j=1[αj, βj].

Spectrum and Spectral Types

If z is such that ∆(z) 6∈ [−2, 2], then the roots of λ + λ−1 = ∆(z) have |λ| > 1, |λ−1| < 1. It follows that Floquet Solutions there are diﬀerent solutions u± decaying exponentially at Periodic Jacobi Matrices ±∞ so their Wronskian is not zero. By (Schrödinger The Discriminant operator) Green’s function methods, Gaps + − Spectrum Gnm(z) = umax(n,m)(z)umin(m.n)(z)/W (z) Potential Theory

Quadratic −1 Irrationalities is the matrix for J − z , i.e., z∈ / σ(J).

Minimal Herglotz Functions

Isospectral Tori but since Pp 2 j=1|um+j| is constant, kuχ[−N,N]k → ∞ so z ∈ σ(J). Thus p Theorem. σ(J) = ∪j=1[αj, βj].

Spectrum and Spectral Types

Quadratic −1 Irrationalities is the matrix for J − z , i.e., z∈ / σ(J).

Minimal Herglotz Functions If ∆(z) ∈ [−2, 2], there is a bounded Floquet solution (since

Isospectral Tori |λ| = 1). Then k(J − z)[uχ[−N,N]]k is bounded, p Theorem. σ(J) = ∪j=1[αj, βj].

Spectrum and Spectral Types

Quadratic −1 Irrationalities is the matrix for J − z , i.e., z∈ / σ(J).

Minimal Herglotz Functions If ∆(z) ∈ [−2, 2], there is a bounded Floquet solution (since

Isospectral Tori |λ| = 1). Then k(J − z)[uχ[−N,N]]k is bounded, but since Pp 2 j=1|um+j| is constant, kuχ[−N,N]k → ∞ so z ∈ σ(J). Thus Spectrum and Spectral Types

Quadratic −1 Irrationalities is the matrix for J − z , i.e., z∈ / σ(J).

Minimal Herglotz Functions If ∆(z) ∈ [−2, 2], there is a bounded Floquet solution (since

Isospectral Tori |λ| = 1). Then k(J − z)[uχ[−N,N]]k is bounded, but since Pp 2 j=1|um+j| is constant, kuχ[−N,N]k → ∞ so z ∈ σ(J). Thus p Theorem. σ(J) = ∪j=1[αj, βj]. 2 2 and then by a Wronskian argument, |un| + |un+1| is bounded from below. So by a Carmona-type formula, one should expect purely a.c. spectrum. But this is whole line, not half line !

Here is a replacement: Away from the bands, Gnn = + − un un /W as we’ve seen. By continuity of eigenfunctions of ± transfer matrix in z, un has a limit at z = x + iε with ε ↓ 0 which are Floquet solutions. This is true at least at interiors of bands where the transfer matrix has distinct eigenvalues.

Spectrum and Spectral Types

If ∆(z) ∈ (−2, 2), we get that all solutions are bounded at ±∞ Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori So by a Carmona-type formula, one should expect purely a.c. spectrum. But this is whole line, not half line !

Spectrum and Spectral Types

If ∆(z) ∈ (−2, 2), we get that all solutions are bounded at 2 2 ±∞ and then by a Wronskian argument, |un| + |un+1| is Floquet Solutions bounded from below. Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori But this is whole line, not half line !

Spectrum and Spectral Types

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Here is a replacement: Away from the bands, Gnn = + − un un /W as we’ve seen. By continuity of eigenfunctions of ± transfer matrix in z, un has a limit at z = x + iε with ε ↓ 0 which are Floquet solutions. This is true at least at interiors of bands where the transfer matrix has distinct eigenvalues.

Spectrum and Spectral Types

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori By continuity of eigenfunctions of ± transfer matrix in z, un has a limit at z = x + iε with ε ↓ 0 which are Floquet solutions. This is true at least at interiors of bands where the transfer matrix has distinct eigenvalues.

Spectrum and Spectral Types

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori This is true at least at interiors of bands where the transfer matrix has distinct eigenvalues.

Spectrum and Spectral Types

If ∆(z) ∈ (−2, 2), we get that all solutions are bounded at 2 2 ±∞ and then by a Wronskian argument, |un| + |un+1| is Floquet Solutions bounded from below. So by a Carmona-type formula, one Periodic Jacobi Matrices should expect purely a.c. spectrum. But this is whole line, The Discriminant not half line ! Gaps Here is a replacement: Away from the bands, Gnn = Spectrum u+ u−/W as we’ve seen. By continuity of eigenfunctions of Potential Theory n n ± Quadratic transfer matrix in z, un has a limit at z = x + iε with ε ↓ 0 Irrationalities which are Floquet solutions. Minimal Herglotz Functions

Isospectral Tori Spectrum and Spectral Types

Isospectral Tori Thus, Gnn(z) is p continuous from C+ to C+ ∪ R \{αj, βj}j=1. (n) But if µ is the spectral measure of δn:

Z dµ(n)(x) G (z) = nn x − z

The continuity implies dµ(n) is purely a.c., so we have proven Theorem. A periodic two-sided Jacobi matrix has purely absolutely continuous spectrum. One can write out an explicit spectral representation with Floquet solutions with z ∈ (αj, βj) as continuum eigenfunctions.

Spectrum and Spectral Types

+ − W is non-vanishing on each (αj, βj) since u and u are distinct Floquet solutions (e±iθ). Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori (n) But if µ is the spectral measure of δn:

Z dµ(n)(x) G (z) = nn x − z

Spectrum and Spectral Types

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The continuity implies dµ(n) is purely a.c., so we have proven Theorem. A periodic two-sided Jacobi matrix has purely absolutely continuous spectrum. One can write out an explicit spectral representation with Floquet solutions with z ∈ (αj, βj) as continuum eigenfunctions.

Spectrum and Spectral Types

+ − W is non-vanishing on each (αj, βj) since u and u are ±iθ distinct Floquet solutions (e ). Thus, Gnn(z) is Floquet Solutions p continuous from C+ to C+ ∪ R \{αj, βj}j=1. Periodic Jacobi Matrices (n) But if µ is the spectral measure of δn: The Discriminant Gaps Z dµ(n)(x) Spectrum G (z) = nn x − z Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Spectrum and Spectral Types

Minimal Herglotz proven Functions

Isospectral Tori One can write out an explicit spectral representation with Floquet solutions with z ∈ (αj, βj) as continuum eigenfunctions.

Spectrum and Spectral Types

Minimal Herglotz proven Functions

Isospectral Tori Theorem. A periodic two-sided Jacobi matrix has purely absolutely continuous spectrum. Spectrum and Spectral Types

Minimal Herglotz proven Functions

Isospectral Tori Theorem. A periodic two-sided Jacobi matrix has purely absolutely continuous spectrum. One can write out an explicit spectral representation with Floquet solutions with z ∈ (αj, βj) as continuum eigenfunctions. Conversely, given βp, αp−1, βp−2,..., ∆ − 2 is determined up to a constant since we know its zeros.

That constant is determined by αp when ∆ is −2. Thus, βp, αp−1, βp−2 plus αp determine the remaining p − 1 α’s and β’s. Why this rigidity? Why can’t we have 2p arbitrary α’s and β’s? The answer will lie in potential theory.

Potential Theory

We start with a puzzle. ∆ determines 2 α1 < β1 ≤ α2 < β2 ≤ ... as the roots of ∆ − 4. Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori That constant is determined by αp when ∆ is −2. Thus, βp, αp−1, βp−2 plus αp determine the remaining p − 1 α’s and β’s. Why this rigidity? Why can’t we have 2p arbitrary α’s and β’s? The answer will lie in potential theory.

Potential Theory

We start with a puzzle. ∆ determines 2 α1 < β1 ≤ α2 < β2 ≤ ... as the roots of ∆ − 4. Floquet Solutions Conversely, given β , α , β ,..., ∆ − 2 is determined Periodic Jacobi p p−1 p−2 Matrices up to a constant since we know its zeros. The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Why this rigidity? Why can’t we have 2p arbitrary α’s and β’s? The answer will lie in potential theory.

Potential Theory

Gaps That constant is determined by αp when ∆ is −2. Thus,

Spectrum βp, αp−1, βp−2 plus αp determine the remaining p − 1 α’s

Potential Theory and β’s.

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The answer will lie in potential theory.

Potential Theory

Gaps That constant is determined by αp when ∆ is −2. Thus,

Spectrum βp, αp−1, βp−2 plus αp determine the remaining p − 1 α’s

Potential Theory and β’s. Why this rigidity? Why can’t we have 2p arbitrary

Quadratic α’s and β’s? Irrationalities

Minimal Herglotz Functions

Isospectral Tori Potential Theory

Gaps That constant is determined by αp when ∆ is −2. Thus,

Spectrum βp, αp−1, βp−2 plus αp determine the remaining p − 1 α’s

Potential Theory and β’s. Why this rigidity? Why can’t we have 2p arbitrary

Quadratic α’s and β’s? Irrationalities

Minimal Herglotz The answer will lie in potential theory. Functions

Isospectral Tori If |λ+| ≥ 1, we see that 1 1 γ(z) = lim log kTn(λ)k = log |λ+(z)| n→∞ n p Solving the quadratic equation for λ r 1 ∆(z) ∆(z)2 γ(z) = log + − 1 p 2 2

p On e = ∪j=1[αj, βj], |...| = 1, so γ(z) ≥ 0, γ(z) = 0 on e. γ(z) is harmonic on C \ e q ∆ ∆ 2 since 2 + 2 − 1 is analytic and non-vanishing there and γ(z) = log (|z|) + O(1) at ∞, since ∆(z) is a degree p polynomial.

Potential Theory

For any z ∈ C, there are two Floquet indices, λ±, solving λ + λ−1 = ∆(z). Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Solving the quadratic equation for λ r 1 ∆(z) ∆(z)2 γ(z) = log + − 1 p 2 2

Potential Theory

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori p On e = ∪j=1[αj, βj], |...| = 1, so γ(z) ≥ 0, γ(z) = 0 on e. γ(z) is harmonic on C \ e q ∆ ∆ 2 since 2 + 2 − 1 is analytic and non-vanishing there and γ(z) = log (|z|) + O(1) at ∞, since ∆(z) is a degree p polynomial.

Potential Theory

For any z ∈ C, there are two Floquet indices, λ±, solving −1 λ + λ = ∆(z). If |λ+| ≥ 1, we see that Floquet Solutions 1 1 Periodic Jacobi γ(z) = lim log kTn(λ)k = log |λ+(z)| Matrices n→∞ n p The Discriminant Solving the quadratic equation for λ Gaps Spectrum r 1 ∆(z) ∆(z)2 Potential Theory γ(z) = log + − 1 p 2 2 Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori γ(z) is harmonic on C \ e q ∆ ∆ 2 since 2 + 2 − 1 is analytic and non-vanishing there and γ(z) = log (|z|) + O(1) at ∞, since ∆(z) is a degree p polynomial.

Potential Theory

Isospectral Tori γ(z) = 0 on e. and γ(z) = log (|z|) + O(1) at ∞, since ∆(z) is a degree p polynomial.

Potential Theory

Isospectral Tori γ(z) = 0 on e. γ(z) is harmonic on C \ e q ∆ ∆ 2 since 2 + 2 − 1 is analytic and non-vanishing there Potential Theory

Isospectral Tori γ(z) = 0 on e. γ(z) is harmonic on C \ e q ∆ ∆ 2 since 2 + 2 − 1 is analytic and non-vanishing there and γ(z) = log (|z|) + O(1) at ∞, since ∆(z) is a degree p polynomial. Theorem. γ(z) as given above is the potential theorists’ Green’s function and periodic Jacobi parameters are associated to regular measures (in the Stahl–Totik sense). 1/p Corollary. C(e) = (a1 ··· ap) int By general principles, if Ge is smooth up to e on e , the equilibrium measure dρe(x) = fe(x)dx where 1 ∂ f (x) = G (x + iy) | e π ∂y e y=0 Thus, the equilibrium measure is

0 1 |∆ (x)| 1 d ∆(x) fe(z) = = arccos pπ p4 − ∆2(x) pπ dx 2

Potential Theory

Thus γ(z) = Ge(z) is the potential theorists’ Green’s function. Therefore, as we saw in Lecture 4, Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori int By general principles, if Ge is smooth up to e on e , the equilibrium measure dρe(x) = fe(x)dx where 1 ∂ f (x) = G (x + iy) | e π ∂y e y=0 Thus, the equilibrium measure is

0 1 |∆ (x)| 1 d ∆(x) fe(z) = = arccos pπ p4 − ∆2(x) pπ dx 2

Potential Theory

Thus γ(z) = Ge(z) is the potential theorists’ Green’s function. Therefore, as we saw in Lecture 4, Floquet Solutions

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Thus, the equilibrium measure is

0 1 |∆ (x)| 1 d ∆(x) fe(z) = = arccos pπ p4 − ∆2(x) pπ dx 2

Potential Theory

Thus γ(z) = Ge(z) is the potential theorists’ Green’s function. Therefore, as we saw in Lecture 4, Floquet Solutions

Periodic Jacobi Theorem. γ(z) as given above is the potential theorists’ Matrices Green’s function and periodic Jacobi parameters are The Discriminant associated to regular measures (in the Stahl–Totik sense). Gaps 1/p Spectrum Corollary. C(e) = (a1 ··· ap) int Potential Theory By general principles, if Ge is smooth up to e on e , the Quadratic equilibrium measure dρ (x) = f (x)dx where Irrationalities e e

Minimal Herglotz 1 ∂ Functions f (x) = G (x + iy) | e π ∂y e y=0 Isospectral Tori Potential Theory

Thus γ(z) = Ge(z) is the potential theorists’ Green’s function. Therefore, as we saw in Lecture 4, Floquet Solutions

Minimal Herglotz 1 ∂ Functions f (x) = G (x + iy) | e π ∂y e y=0 Isospectral Tori Thus, the equilibrium measure is

0 1 |∆ (x)| 1 d ∆(x) fe(z) = = arccos pπ p4 − ∆2(x) pπ dx 2 1 Theorem. ρe [αj, βj] = p .

This explains the puzzle mentioned earlier. This is also a density of zeros way of understanding why the above fe is the DOS. For the periodic eigenfunctions with a box of size kp are the Floquet solutions with λ = e2πij/k, j = 0, 1, 2, . . . , k − 1.

Potential Theory

∆ In each, band ∆(λ) goes from −2 to 2, so arccos( 2 ) from π to 0. Thus, Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori This explains the puzzle mentioned earlier. This is also a density of zeros way of understanding why the above fe is the DOS. For the periodic eigenfunctions with a box of size kp are the Floquet solutions with λ = e2πij/k, j = 0, 1, 2, . . . , k − 1.

Potential Theory

∆ In each, band ∆(λ) goes from −2 to 2, so arccos( 2 ) from π to 0. Thus, Floquet Solutions 1 Periodic Jacobi Theorem. ρe [αj, βj] = p . Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori This is also a density of zeros way of understanding why the above fe is the DOS. For the periodic eigenfunctions with a box of size kp are the Floquet solutions with λ = e2πij/k, j = 0, 1, 2, . . . , k − 1.

Potential Theory

∆ In each, band ∆(λ) goes from −2 to 2, so arccos( 2 ) from π to 0. Thus, Floquet Solutions 1 Periodic Jacobi Theorem. ρe [αj, βj] = p . Matrices The Discriminant This explains the puzzle mentioned earlier. Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori For the periodic eigenfunctions with a box of size kp are the Floquet solutions with λ = e2πij/k, j = 0, 1, 2, . . . , k − 1.

Potential Theory

Spectrum This is also a density of zeros way of understanding why the Potential Theory above fe is the DOS. Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Potential Theory

Spectrum This is also a density of zeros way of understanding why the Potential Theory above fe is the DOS. For the periodic eigenfunctions with a 2πij/k Quadratic box of size kp are the Floquet solutions with λ = e , Irrationalities j = 0, 1, 2, . . . , k − 1. Minimal Herglotz Functions

Isospectral Tori which one can prove by checking it solves the right diﬀerence equation with q−1 = −1.One deﬁnes Z dµ(x) m (z) = µ x − z −1 wn(x) ≡< pn, (• − z) >= qn(x) + m(x)pn(x) −1 2 the Weyl solution. Since (• − z) is in L (R, dµ) for 2 z∈ / supp(dµ), wn ∈ ` for such z.

If inf an > 0, constancy of the Wronskian shows this is the unique `2-solution.

Weyl Solutions

An important property of second kind OPRL, sometimes used as the deﬁnition is that for n ≥ 0, Floquet Solutions Z pn(x) − pn(y) Periodic Jacobi qn(x) = dµ(y) Matrices x − y The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori One deﬁnes Z dµ(x) m (z) = µ x − z −1 wn(x) ≡< pn, (• − z) >= qn(x) + m(x)pn(x) −1 2 the Weyl solution. Since (• − z) is in L (R, dµ) for 2 z∈ / supp(dµ), wn ∈ ` for such z.

If inf an > 0, constancy of the Wronskian shows this is the unique `2-solution.

Weyl Solutions

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori −1 2 Since (• − z) is in L (R, dµ) for 2 z∈ / supp(dµ), wn ∈ ` for such z.

If inf an > 0, constancy of the Wronskian shows this is the unique `2-solution.

Weyl Solutions

An important property of second kind OPRL, sometimes used as the definition is that for n ≥ 0, Floquet Solutions Z pn(x) − pn(y) Periodic Jacobi qn(x) = dµ(y) Matrices x − y The Discriminant which one can prove by checking it solves the right Gaps difference equation with q = −1.One defines Spectrum −1 Z Potential Theory dµ(x) mµ(z) = Quadratic x − z Irrationalities −1 Minimal Herglotz wn(x) ≡< pn, (• − z) >= qn(x) + m(x)pn(x) Functions Isospectral Tori the Weyl solution. 2 wn ∈ ` for such z.

If inf an > 0, constancy of the Wronskian shows this is the unique `2-solution.

Weyl Solutions

An important property of second kind OPRL, sometimes used as the definition is that for n ≥ 0, Floquet Solutions Z pn(x) − pn(y) Periodic Jacobi qn(x) = dµ(y) Matrices x − y The Discriminant which one can prove by checking it solves the right Gaps difference equation with q = −1.One defines Spectrum −1 Z Potential Theory dµ(x) mµ(z) = Quadratic x − z Irrationalities −1 Minimal Herglotz wn(x) ≡< pn, (• − z) >= qn(x) + m(x)pn(x) Functions −1 2 Isospectral Tori the Weyl solution. Since (• − z) is in L (R, dµ) for z∈ / supp(dµ), If inf an > 0, constancy of the Wronskian shows this is the unique `2-solution.

Weyl Solutions

An important property of second kind OPRL, sometimes used as the definition is that for n ≥ 0, Floquet Solutions Z pn(x) − pn(y) Periodic Jacobi qn(x) = dµ(y) Matrices x − y The Discriminant which one can prove by checking it solves the right Gaps difference equation with q = −1.One defines Spectrum −1 Z Potential Theory dµ(x) mµ(z) = Quadratic x − z Irrationalities −1 Minimal Herglotz wn(x) ≡< pn, (• − z) >= qn(x) + m(x)pn(x) Functions −1 2 Isospectral Tori the Weyl solution. Since (• − z) is in L (R, dµ) for 2 z∈ / supp(dµ), wn ∈ ` for such z. Weyl Solutions

An important property of second kind OPRL, sometimes used as the definition is that for n ≥ 0, Floquet Solutions Z pn(x) − pn(y) Periodic Jacobi qn(x) = dµ(y) Matrices x − y The Discriminant which one can prove by checking it solves the right Gaps difference equation with q = −1.One defines Spectrum −1 Z Potential Theory dµ(x) mµ(z) = Quadratic x − z Irrationalities −1 Minimal Herglotz wn(x) ≡< pn, (• − z) >= qn(x) + m(x)pn(x) Functions −1 2 Isospectral Tori the Weyl solution. Since (• − z) is in L (R, dµ) for 2 z∈ / supp(dµ), wn ∈ ` for such z.

If inf an > 0, constancy of the Wronskian shows this is the unique `2-solution. Thus if m1 is the m-function for ∞ {an+1, bn+1}n=1, we have that, for a constant, c(z), z − b1 −1 m(z) m1(z) 2 = c(z) a1 0 −1 −1 which leads to the recursion relation: 1 m(z) = 2 b1 − z − a1m1(z) which upon iterations yields the continued fraction of Jacobi, Markov and Stieltjes: 1 m(z) = 2 a1 b1 − z − 2 a2 b2−z− b3−z−...

Coeﬃcient Stripping and Continued Fractions

u1 In terms of initial data for ( a0u0 ), the Weyl solution has m(z) initial data ( −1 ). Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori z − b1 −1 m(z) m1(z) 2 = c(z) a1 0 −1 −1 which leads to the recursion relation: 1 m(z) = 2 b1 − z − a1m1(z) which upon iterations yields the continued fraction of Jacobi, Markov and Stieltjes: 1 m(z) = 2 a1 b1 − z − 2 a2 b2−z− b3−z−...

Coeﬃcient Stripping and Continued Fractions

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori which leads to the recursion relation: 1 m(z) = 2 b1 − z − a1m1(z) which upon iterations yields the continued fraction of Jacobi, Markov and Stieltjes: 1 m(z) = 2 a1 b1 − z − 2 a2 b2−z− b3−z−...

Coeﬃcient Stripping and Continued Fractions

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori which upon iterations yields the continued fraction of Jacobi, Markov and Stieltjes: 1 m(z) = 2 a1 b1 − z − 2 a2 b2−z− b3−z−...

Coeﬃcient Stripping and Continued Fractions

Isospectral Tori Coeﬃcient Stripping and Continued Fractions

u1 In terms of initial data for ( a0u0 ), the Weyl solution has m(z) initial data ( −1 ). Thus if m1 is the m-function for Floquet Solutions ∞ {an+1, bn+1}n=1, we have that, for a constant, c(z), Periodic Jacobi Matrices The Discriminant z − b1 −1 m(z) m1(z) 2 = c(z) Gaps a1 0 −1 −1 Spectrum which leads to the recursion relation: Potential Theory 1 Quadratic m(z) = Irrationalities 2 b1 − z − a1m1(z) Minimal Herglotz Functions which upon iterations yields the continued fraction of Isospectral Tori Jacobi, Markov and Stieltjes: 1 m(z) = 2 a1 b1 − z − 2 a2 b2−z− b3−z−... so m must obey: p (z) −q (z) m(z) m(z) p p = appp−1(z) −apqp−1 −1 −1 which leads to the quadratic equation

2 α(z)m(z) + β(z)m(z) + γ(z) = 0 α(z) = appp−1(z)

β(z) = pp(z) + apqp−1(z) γ(z) = qp(z) This is reminiscent of the results of Legendre and Galois on numeric continued fractions. By a direct calculation, the two discriminants are related by p β2 − 4αγ = ∆(z)2 − 4

Quadratic Equations

In the period p case, stripping p times leave J invariant

Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori which leads to the quadratic equation

2 α(z)m(z) + β(z)m(z) + γ(z) = 0 α(z) = appp−1(z)

Quadratic Equations

In the period p case, stripping p times leave J invariant so m must obey: Floquet Solutions pp(z) −qp(z) m(z) m(z) Periodic Jacobi = Matrices appp−1(z) −apqp−1 −1 −1 The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori α(z) = appp−1(z)

Quadratic Equations

In the period p case, stripping p times leave J invariant so m must obey: Floquet Solutions pp(z) −qp(z) m(z) m(z) Periodic Jacobi = Matrices appp−1(z) −apqp−1 −1 −1 The Discriminant

Gaps which leads to the quadratic equation

Spectrum α(z)m(z)2 + β(z)m(z) + γ(z) = 0 Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori This is reminiscent of the results of Legendre and Galois on numeric continued fractions. By a direct calculation, the two discriminants are related by p β2 − 4αγ = ∆(z)2 − 4

Quadratic Equations

In the period p case, stripping p times leave J invariant so m must obey: Floquet Solutions pp(z) −qp(z) m(z) m(z) Periodic Jacobi = Matrices appp−1(z) −apqp−1 −1 −1 The Discriminant

Gaps which leads to the quadratic equation

Spectrum α(z)m(z)2 + β(z)m(z) + γ(z) = 0 α(z) = a p (z) Potential Theory p p−1

Quadratic Irrationalities β(z) = pp(z) + apqp−1(z) γ(z) = qp(z) Minimal Herglotz Functions

Isospectral Tori By a direct calculation, the two discriminants are related by p β2 − 4αγ = ∆(z)2 − 4

Quadratic Equations

In the period p case, stripping p times leave J invariant so m must obey: Floquet Solutions pp(z) −qp(z) m(z) m(z) Periodic Jacobi = Matrices appp−1(z) −apqp−1 −1 −1 The Discriminant

Gaps which leads to the quadratic equation

Spectrum α(z)m(z)2 + β(z)m(z) + γ(z) = 0 α(z) = a p (z) Potential Theory p p−1

In the period p case, stripping p times leave J invariant so m must obey: Floquet Solutions pp(z) −qp(z) m(z) m(z) Periodic Jacobi = Matrices appp−1(z) −apqp−1 −1 −1 The Discriminant

Gaps which leads to the quadratic equation

Spectrum α(z)m(z)2 + β(z)m(z) + γ(z) = 0 α(z) = a p (z) Potential Theory p p−1

Quadratic Irrationalities β(z) = pp(z) + apqp−1(z) γ(z) = qp(z) Minimal Herglotz Functions This is reminiscent of the results of Legendre and Galois on Isospectral Tori numeric continued fractions. By a direct calculation, the two discriminants are related by p β2 − 4αγ = ∆(z)2 − 4 including points at inﬁnity. At closed gaps ∆(z)2 − 4 has double zeros so there are only branch points at the ends of open gaps and the surface is of genus ` where ` is the number of open gaps. Further analysis shows any meromorphic function which is diﬀerent on the two sheets has degree at least ` + 1. Moreover the m-function has a pole at ∞ on the “top” sheet and in order that Im m not change sign over a gap, on the two sheeted surface, m must have a pole in each of the ` gaps. Those points on the surface which “project down” to a gap, are a circle - two intervals glued together at the ends.

Riemann Surface of the m-function

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori At closed gaps ∆(z)2 − 4 has double zeros so there are only branch points at the ends of open gaps and the surface is of genus ` where ` is the number of open gaps. Further analysis shows any meromorphic function which is diﬀerent on the two sheets has degree at least ` + 1. Moreover the m-function has a pole at ∞ on the “top” sheet and in order that Im m not change sign over a gap, on the two sheeted surface, m must have a pole in each of the ` gaps. Those points on the surface which “project down” to a gap, are a circle - two intervals glued together at the ends.

Riemann Surface of the m-function

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Further analysis shows any meromorphic function which is diﬀerent on the two sheets has degree at least ` + 1. Moreover the m-function has a pole at ∞ on the “top” sheet and in order that Im m not change sign over a gap, on the two sheeted surface, m must have a pole in each of the ` gaps. Those points on the surface which “project down” to a gap, are a circle - two intervals glued together at the ends.

Riemann Surface of the m-function

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Moreover the m-function has a pole at ∞ on the “top” sheet and in order that Im m not change sign over a gap, on the two sheeted surface, m must have a pole in each of the ` gaps. Those points on the surface which “project down” to a gap, are a circle - two intervals glued together at the ends.

Riemann Surface of the m-function

Isospectral Tori in order that Im m not change sign over a gap, on the two sheeted surface, m must have a pole in each of the ` gaps. Those points on the surface which “project down” to a gap, are a circle - two intervals glued together at the ends.

Riemann Surface of the m-function

We have thus proven that the m-function of a periodic Jacobi matrix has a continuation as a meromorphic function Floquet Solutions on the two sheeted Riemann surface of p∆(z)2 − 4, S, Periodic Jacobi 2 Matrices including points at infinity. At closed gaps ∆(z) − 4 has The Discriminant double zeros so there are only branch points at the ends of Gaps open gaps and the surface is of genus ` where ` is the Spectrum number of open gaps. Potential Theory Further analysis shows any meromorphic function which is Quadratic Irrationalities different on the two sheets has degree at least ` + 1. Minimal Herglotz Moreover the m-function has a pole at ∞ on the “top” Functions sheet and in order that Im m not change sign over a gap, on Isospectral Tori the two sheeted surface, m must have a pole in each of the ` gaps. Those points on the surface which “project down” to a gap, are a circle - two intervals glued together at the ends. Moreover, if we consider the map as defined on all periodic Jacobi matrices with a given discriminant, ∆, one can prove with some effort that this map is a bijection. Thus, this set is a torus of dimensions `, called the isospectral torus. The possible m-functions are thus precisely the Herglotz functions (i.e. analytic functions in the upper half plane with positive imaginary part) that have an meromorphic continuation to the Riemann surface of p∆(z)2 − 4 that have minimal degree and which are normalized to look like −/z near ∞. Christiansen, Simon and Zinchenko call these minimal Herglotz functions.

Dirichlet Data

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Thus, this set is a torus of dimensions `, called the isospectral torus. The possible m-functions are thus precisely the Herglotz functions (i.e. analytic functions in the upper half plane with positive imaginary part) that have an meromorphic continuation to the Riemann surface of p∆(z)2 − 4 that have minimal degree and which are normalized to look like −/z near ∞. Christiansen, Simon and Zinchenko call these minimal Herglotz functions.

Dirichlet Data

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The possible m-functions are thus precisely the Herglotz functions (i.e. analytic functions in the upper half plane with positive imaginary part) that have an meromorphic continuation to the Riemann surface of p∆(z)2 − 4 that have minimal degree and which are normalized to look like −/z near ∞. Christiansen, Simon and Zinchenko call these minimal Herglotz functions.

Dirichlet Data

There is thus a map from the m-function to the set of points on the product over the gaps of the points on the Floquet Solutions surface that project down to that gap, i.e. onto a Periodic Jacobi Matrices `-dimensional torus. Moreover, if we consider the map as The Discriminant deﬁned on all periodic Jacobi matrices with a given Gaps discriminant, ∆, one can prove with some eﬀort that this Spectrum map is a bijection. Thus, this set is a torus of dimensions `, Potential Theory called the isospectral torus. Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Dirichlet Data

Isospectral Tori with positive imaginary part) that have an meromorphic continuation to the Riemann surface of p∆(z)2 − 4 that have minimal degree and which are normalized to look like −/z near ∞. Christiansen, Simon and Zinchenko call these minimal Herglotz functions. Thus

e = [α1, β1] ∪ [α2, β2] ∪ ... ∪ [α`+1, β`+1] where the meaning of the α’s and β’s has changed subtly from the prior notation when we have a perioidc problem with closed gaps. The same method which we didn’t describe that constructs minimal Herglotz functions in the periodic case lets us do the same for the Reimann surface of v u`+1 uY t (z − αj)(z − βj) j=1

Finite Gap Sets

Now consider a general compact subset e ⊂ R which has ` + 1 components so its complement in R has ` gaps. Floquet Solutions

Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori where the meaning of the α’s and β’s has changed subtly from the prior notation when we have a perioidc problem with closed gaps. The same method which we didn’t describe that constructs minimal Herglotz functions in the periodic case lets us do the same for the Reimann surface of v u`+1 uY t (z − αj)(z − βj) j=1

Finite Gap Sets

Now consider a general compact subset e ⊂ R which has ` + 1 components so its complement in R has ` gaps. Thus Floquet Solutions Periodic Jacobi e = [α1, β1] ∪ [α2, β2] ∪ ... ∪ [α`+1, β`+1] Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The same method which we didn’t describe that constructs minimal Herglotz functions in the periodic case lets us do the same for the Reimann surface of v u`+1 uY t (z − αj)(z − βj) j=1

Finite Gap Sets

The Discriminant where the meaning of the α’s and β’s has changed subtly Gaps from the prior notation when we have a perioidc problem Spectrum with closed gaps. Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Finite Gap Sets

The Discriminant where the meaning of the α’s and β’s has changed subtly Gaps from the prior notation when we have a perioidc problem Spectrum with closed gaps. Potential Theory The same method which we didn’t describe that constructs Quadratic Irrationalities minimal Herglotz functions in the periodic case lets us do Minimal Herglotz the same for the Reimann surface of Functions Isospectral Tori v u`+1 uY t (z − αj)(z − βj) j=1 The frequency spectrum of the almost periodic function is generated by the harmonic measures of the intervals, i.e. ρe([αj, βj]). The result is thus Theorem Every ﬁnite gap set, e has an isospectral torus of almost periodic Jacobi matrices associated to it. These are all periodic with period p if and only if each band has j harmonic measure p for j ∈ {1, ..., p − 1}.

Almost Periodic Isospectral Torus

There is again an ` dimensional torus of half line Jacobi matrices, each of them almost periodic with essential Floquet Solutions spectrum exactly equal to e. Periodic Jacobi Matrices

The Discriminant

Gaps

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori The result is thus Theorem Every ﬁnite gap set, e has an isospectral torus of almost periodic Jacobi matrices associated to it. These are all periodic with period p if and only if each band has j harmonic measure p for j ∈ {1, ..., p − 1}.

Almost Periodic Isospectral Torus

Spectrum

Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori These are all periodic with period p if and only if each band has j harmonic measure p for j ∈ {1, ..., p − 1}.

Almost Periodic Isospectral Torus

There is again an ` dimensional torus of half line Jacobi matrices, each of them almost periodic with essential Floquet Solutions spectrum exactly equal to e. The frequency spectrum of the Periodic Jacobi Matrices almost periodic function is generated by the harmonic The Discriminant measures of the intervals, i.e. ρe([αj, βj]). The result is thus Gaps Theorem Every ﬁnite gap set, e has an isospectral torus of Spectrum almost periodic Jacobi matrices associated to it. Potential Theory

Quadratic Irrationalities

Minimal Herglotz Functions

Isospectral Tori Almost Periodic Isospectral Torus

Quadratic all periodic with period p if and only if each band has Irrationalities j harmonic measure p for j ∈ {1, ..., p − 1}. Minimal Herglotz Functions

Isospectral Tori