Rotation Numbers, Boundary Forces and Gap Labelling

Johannes Kellendonk^{1}, Ioannis Zois^{2}

1 Institute Girard Desargues, Université Claude Bernard Lyon 1, F-69622 Villeurbanne e-mail: kellendonk@igd.univ-lyon1.fr 2 School of Mathematics, Cardif University, PO Box 926, Cardiff CF24 4YH, UK e-mail: zoisip@cf.ac.uk

July 9, 2004

Abstract

We review the Johnson-Moser rotation number and the K_{0}-theoretical gap labelling of Bellissard for one-dimensional Schrödinger operators. We compare them with two further gap-labels, one being related to the motion of Dirichlet eigenvalues, the other being a K_{1}-theoretical gap label. We argue that the latter provides a natural generalisation of the Johnson-Moser rotation number to higher dimensions.

1 Introduction

It is an interesting and very well known fact that the boundary of a domain plays a prominent role both in mathematics and in physics. A case that comes immediately into mind is for example the theory of differential equations where the boundary conditions determine quite a lot of the whole solution. Another class of prominent examples in physics are topological quantum field theories. In topological applications there are some even stronger statements which claim that what happens in the boundary determines the behaviour of the system in the bulk completely. Perhaps the most famous such statement is the holography principle in quantum gravity.

Following similar lines of thinking the first author and collaborators suggested in [KS04a, KS04b] that there should exist an equality between bulk and boundary topological invariants for certain physical systems arising in solid state physics. This suggestion was mathematically based on K-theoretic and cyclic cohomological properties of the Wiener-Hopf extension of the C^{∗}-algebra of observables. In most applications we have in mind in condensed matter physics this C^{∗}-algebra of observables is obtained by considering the Schrödinger operator and its translates describing the 1-particle approximation of the solid.

Here we consider a simple example, a Schrödinger operator on the real line. The K_{0}-theory gap labels introduced by Bellissard et al. [BLT85, Be92] are bulk invariants. It is known that these are equal to the Johnson-Moser rotation numbers [JM82] the existing proof being essentially a corollary of the Sturm-Liouville theorem by which they are identified with the integrated density of states on the gaps. Here we give a direct identification of the Johnson-Moser rotation number (for energies in gaps) with a boundary invariant, namely a Dirichlet rotation number. This number has an operator algebraic and physical interpretation, namely as boundary force per unit energy, and a K_{1}-theoretical interpretation. The equality between the even and the odd K-theoretic gap labels is an application of the correspondence between bulk and boundary invariants.

We organise this article as follows: after recalling some well-known facts for one-dimensional systems we discuss the Johnson-Moser rotation number in Section 3.1. In Section 3.2 we introduce the Dirichlet rotation number of a gap and show that it coincides with the Johnson-Moser rotation number for energies in that gap. We interprete the Dirchlet rotation number in Section 3.3 as an odd K gap-label and in Section 3.4 as boundary force per unit energy. All of Section 3 is based on a single operator, although its translates play a fundamental role. In Section 4 we place this into the framework of non-commutative topology which requires to work with the hull, a topological completion of the set of translates of the Schrödinger operator. This section is held quite briefly, since it is mainly based on [Kel].

2 Preliminaries

In this article we consider a one-dimensional Schrödinger operator H = −∂^{2} + V with (real) bounded differentiable potential and its translates H_{ξ} := −∂^{2} + V_{ξ}, V_{ξ}(x) = V (x + ξ). The differential equation HΨ = EΨ for complex valued functions Ψ over ℝ has for all E two linear independent solutions but not all E belong to the spectrum σ(H) of H as an operator acting on L^{2}(ℝ). We suppose that we are in the limit point case so that H is essentially self-adjoint and the following property of solutions holds [CL55].

Theorem 1 If E ∉ σ(H) there exist two real solutions Ψ_{+} and Ψ_{−} of (H − E)Ψ = 0 which are L^{2}-integrable at ∞ and −∞, resp.. These solutions are linear independent and unique up to multiplication by a factor.

If the potential is sufficiently regular L^{2}-integrability at ±∞ means that the solution vanishes at ±∞ and we will for simplicity assume that this is the case. Clearly σ(H_{ξ}) = σ(H) for all ξ.

We consider also the action of H_{ξ} on L^{2}(ℝ^{≤0}) with Dirichlet boundary conditions at the boundary. If we need to emphazise this we will also write Ĥ_{ξ} for the half-sided operator. The spectrum is then no longer the same. Whereas the essential part of the spectrum of Ĥ_{ξ} is contained in that of H_{ξ} the half sided operator may have isolated eigenvalues in the gaps in σ(H_{ξ}). Here a gap is a connected component of the complement of the spectrum, hence in particular an open set. E is an eigenvalue of Ĥ_{ξ} if (Ĥ_{ξ} −E)Ψ = Ψ for Ψ ∈ L^{2}(ℝ^{≤0}) which for E in a gap of σ(H_{ξ}) amounts to saying that the solution Ψ_{−} of (H_{ξ} − E)Ψ_{−} = 0 from Theorem 1 satisfies in addition Ψ_{−}(0) = 0.

Definition 1 We call E ∈ ℝ a right Dirichlet value of H_{ξ} if it is an eigenvalue of Ĥ_{ξ}.

We recall the important Sturm-Liouville theorem.

Theorem 2 Consider H := −∂^{2} + V with (real) bounded continuous potential acting on L^{2}([a, b]) with Dirichlet boundary conditions. The spectrum is discrete and bounded from below. A real eigenfunction to the nth eigenvalue (counted from below) has exactly n − 1 zeroes in the interior (a, b) of [a, b].

3 Rotation numbers

The winding number of a continuous function f : ℝ/ℤ → ℝ/ℤ is intuitively speaking the number of times its graph wraps around the circle ℝ/ℤ. This is counted relative to the orientations induced by the order on ℝ. Let Λ = {Λ_{n}}_{n} be an increasing chain of compact intervals Λ_{n} = [a_{n}, b_{n}] ⊂ Λ_{n+1} ⊂ ℝ whose union covers ℝ. The quantity

Λ(f) := lim_{n→∞} \frac{1}{b_{n} − a_{n}} ∫_{a_{n}}^{b_{n}} f(x)dx

is called the Λ-mean of the function f : ℝ → ℝ, existence of the limit assumed. Now let f : ℝ → ℝ/ℤ be continuous and choose a continuous extension f̃ : ℝ → ℝ. To define the rotation number of f we consider the expression

rot_{Λ}(f) = lim_{n→∞} \frac{f̃(b_{n}) − f̃(a_{n})}{b_{n} − a_{n}}

which becomes the rotation number of f if f is periodic of period 1. The limit does not exist in general but if it does it is independent of the extension f̃. If f is piecewise differentiable then rot_{Λ}(f) = Λ(f′). Moreover, if U : ℝ → ℂ is a nowhere vanishing continuous piecewise differentiable function then we can consider the rotation number of its argument function which becomes

rot_{Λ}(\frac{arg(U)}{2π}) = lim_{n→∞} \frac{1}{2πi(b_{n} − a_{n})} ∫_{a_{n}}^{b_{n}} \frac{\overline{U}}{|U|} (\frac{U}{|U|})′ dx  (1)

3.1 The Johnson-Moser rotation number

Johnson and Moser in [JM82] have defined rotation numbers for the Schrödinger operator H = −∂^{2} + V on the real line where V is a real quasi-periodic potential. They are defined as follows: Let Ψ(x) be the nonzero real solution of (H − E)Ψ = 0 which vanishes at −∞, then Ψ′ + iΨ : ℝ → ℂ is nowhere vanishing and

α_{Λ}(H, E) := 2 rot_{Λ}(\frac{arg(Ψ′ + iΨ)}{2π}).  (2)

(Our normalisation differs from that in [JM82] for later convenience.) For the class of potentials considered in [JM82] the limit is indeed defined and even independent on the choice of Λ. However we do not want to assume this special form for the potential. We will simply assume for now that V is a real bounded differentiable function such that the limit (2) exists provided E does not belong to the spectrum and the solution Ψ is taken which vanishes at −∞. In Section 4 we discuss the potentials for which these requirements can be fulfilled.

Note that α_{Λ}(H, E) has the following interpretations. If N(a, b; E) denotes the number of zeroes of the above solution Ψ in [a, b] then α_{Λ}(H, E) is the Λ-mean of the density of zeroes of Ψ, namely one has

α_{Λ}(H, E) = lim_{n→∞} \frac{N(a_{n}, b_{n}; E)}{b_{n} − a_{n}}.

The integrated density of states of H at E is

IDS_{Λ}(H, E) = lim_{n→∞} \frac{1}{|Λ_{n}|}Tr(P_{E}(H_{Λ_{n}}))  (3)

provided the limit exists. Here |Λ_{n}| = b_{n} − a_{n} is the volume of Λ_{n}, H_{Λ_{n}} the restriction of H to Λ_{n} with Dirichlet boundary conditions and, for self-adjoint A, P_{E}(A) is the spectral projection onto the spectral subspace of spectral values smaller or equal to E. It will be important that P (A) is a continuous function of A if E is not in the spectrum of A. Since Tr(P_{E}(H_{Λ_{n}}) is the number of eigenfunctions of H_{Λ_{n}} to eigenvalue smaller or equal E Theorem 2 implies

Corollary 1 α_{Λ}(H, E) = IDS_{Λ}(H, E).

In particular, like the integrated density of states α_{Λ}(H, E) is monotomically increasing in E and constant on the gaps of the spectrum of H. It is moreover the same for all H_{ξ}.

3.2 The Dirichlet rotation number

We now consider the continuous 1-parameter family of operators {H_{ξ}}_{ξ} with ξ ∈ ℝ and H_{ξ} = −∂^{2} + V_{ξ}, where V_{ξ}(x) = V (x + ξ). We shall prove that the Johnson-Moser rotation number is a rotation number which is defined by right Dirichlet values as a function of ξ.

Let Δ be a gap and define the set of right Dirichlet values in Δ

D_{ξ}(Δ) := {μ ∈ Δ|∃Ψ : (H_{ξ} − μ)Ψ = 0 and Ψ(0) = Ψ(−∞) = 0} .

Suppose D_{ξ}(Δ) ≠ ∅ for some ξ and let μ ∈ D_{ξ}(Δ) and Ψ the corresponding solution (H_{ξ}−μ)Ψ = 0 satisfying the required boundary conditions. Define the following two sets S(μ) := {η|μ ∈ D_{η}(Δ)} and Z(μ) := {x|Ψ(x − ξ) = 0}.

Lemma 1 S(μ) = Z(μ).

Proof: Define Ψ_{η}(x) = Ψ(x+(η −ξ)). Then (H_{η} −μ)Ψ_{η} = 0 and Ψ_{η}(−∞) = 0 for all η. Hence Z(μ) = {η|Ψ(η − ξ) = 0} = {η|Ψ_{η}(0) = 0} ⊂ S(μ).

For the opposite inclusion if μ ∈ D_{η}(Δ), then there exists Φ such that (H_{η} − μ)Φ = 0 with Φ(0) = Φ(−∞) = 0. Define Φ_{ξ}(x + (η − ξ)) = Φ(x). Then (H_{ξ} − μ)Φ_{ξ} = 0 with Φ_{ξ}(−∞) = 0. By Theorem 1, Ψ = λΦ_{ξ} for some λ ∈ ℂ^{∗}, which implies Ψ(η − ξ) = λΦ(0) = 0 and hence η ∈ Z(μ), thus S(μ) ⊆ Z(μ). □

Let ξ ∈ S(μ), μ ∈ Δ. Since the spectrum of Ĥ_{ξ} in the gap Δ consists of isolated eigenvalues which are non-degenerate by Theorem 1 we can use perturbation theory to find a neighbourhood (ξ − ϵ, ξ + ϵ) and a differentiable function ξ ↦ μ(ξ) on this neighbourhood which is uniquely defined by the property that μ(ξ) ∈ D_{ξ}(Δ). In fact, level-crossing of right Dirichlet values cannot occur in gaps, since it would lead to degeneracies. As in [Ke04] we see that its first derivative is strictly negative:

\frac{dμ(ξ)}{dξ} = ∫_{−∞}^{0} dx|Ψ_{ξ}(x)|^{2}V′_{ξ} = −|Ψ′_{ξ}(0)|^{2} < 0.

Here Ψ_{ξ} is a normalised eigenfunction of Ĥ_{ξ}. Thus around each value ξ for which we find a right Dirichlet value in Δ we have locally defined curves μ(ξ) which are stricly monotonically decreasing and non-intersecting. Since {H_{ξ}}_{ξ} is a norm-continuous family, the spectra σ(H_{ξ}) are continuous in ξ (w.r.t. Hausdorff topology) so that the curves μ(ξ) can be continued until they reach the boundary of Δ or their limit at +∞ or −∞, if it exists. Let K be the circle of complex numbers of modulus 1 with anti-clockwise orientation. We define the function μ̃ : ℝ → K by

μ̃(ξ) = \{ \begin{matrix}exp 2πi\frac{μ(ξ)−E_{0}}{|Δ|} & in case D_{ξ}(Δ) ≠ ∅ \\ 1 & otherwise \end{matrix}

where E_{0} = inf Δ and |Δ| is the width of Δ. Then μ̃ is a continuous function which is differentiable at K\{1}.

Definition 2 The Dirichlet rotation number is

β_{Λ}(H, Δ) := −rot_{Λ}(\frac{arg μ̃}{2π}) ,

provided Δ contains right Dirichlet values. Otherwise we define it to be 0.

Theorem 3 α(H, E) = β(H, Δ).

Proof: Suppose Δ contains a right Dirichlet value E. By Lemma 1 α(H, E) is the Λ-mean of the density of S(μ). S(μ) can be identified with the set of intersection points between the constant curve ξ ↦ exp 2πi\frac{E−E_{0}}{|Δ|} and μ̃(ξ). Since μ′(ξ) < 0 the mean of the density of intersection points is minus the rotation number of \frac{arg μ̃}{2π}.

Now suppose that Δ does not contain right Dirichlet values and hence β(H, Δ) = 0. Let E ∈ Δ. By Theorem 1 there exists a solution (H − E)Ψ = 0 which vanishes at −∞. If α(H, E) were strictly greater than 0 this solution would have a zero, say at ξ. Set Ψ̃(x) = Ψ(x + ξ). Then (H_{ξ} − E)Ψ̃ = 0 and Ψ̃(0) = Ψ̃(−∞) = 0. Hence E ∈ D_{ξ}(Δ) which contradicts our assumption. Thus α(H, E) = 0. □

Remark 1 An even nicer geometric picture arrises if we take into account also the left Dirichlet values of H_{ξ}, values E for which exist Ψ solving (H_{ξ} − E)Ψ = 0 with Ψ(0) = Ψ(+∞) = 0. It can be shown that for fixed ξ each gap Δ contains at most one Dirichlet value of H_{ξ} (which might be a right or a left Dirichlet value). Redefine μ̃ : ℝ → K by

μ̃(ξ) = \{ \begin{matrix}exp πi\frac{μ(ξ)−E_{0}}{|Δ|} & in case D_{ξ}(Δ) ≠ ∅ \\ exp πi\frac{E_{0}−μ^{∗}(ξ)}{|Δ|} & in case D_{ξ}(Δ)^{∗} ≠ ∅ \\ 1 & otherwise \end{matrix}

where D_{ξ}(Δ)^{∗} is the set of left Dirichlet values of H_{ξ} in Δ and μ^{∗}(ξ) the corresponding locally defined curve. Then μ̃ is as well a continuous piecewise differentiable function and rot_{Λ}(\frac{arg μ̃}{2π}) is the same number as before except that it yields the Λ-mean of the winding per length of the Dirichlet values around a circle which is obtained from two copies of Δ by identification of their boundary points. For periodic systems, this circle can be identified with the homology cycle corresponding to a gap in the complex spectral curve of H [BBEIM] and so β(H, Δ) is the winding number of the Dirichlet values around it. This is similar to Hatsugai’s interpretation of the edge Hall conductivity as a winding number (see [Ha93]). There the role of the parameter ξ is played by the magnetic flux.

3.3 Odd K-gap labels and Dirichlet rotation numbers

We define another type of gap label which is formulated using operators traces and derivations instead of curves on topological spaces. It has its origin in an odd pairing between K-theory and cyclic cohomology.

We fix a gap Δ in the spectrum of H of length |Δ| and set E_{0} = inf(Δ). Let P_{Δ} = P_{Δ}(Ĥ_{ξ}) be the spectral projection of Ĥ_{ξ} onto the energy interval Δ. Then

U_{ξ} := P_{Δ}e^{2iπ \frac{Ĥ_{ξ}−E_{0}}{|Δ|}} + 1 − P_{Δ}

acts essentially as the unitary of time evolution by time \frac{1}{|Δ|} on the eigenfunctions of Ĥ_{ξ} in Δ. These eigenfunctions are all localised near the edge and therefore is the following expression a boundary quantity.

Definition 3 The odd K-gap label is

Π_{Λ}(H, Δ) = − lim_{n→∞} \frac{1}{2iπ|b_{n} − a_{n}|} ∫_{a_{n}}^{b_{n}} Tr[(U_{ξ}^{∗} − 1)∂_{ξ}U_{ξ}]dξ

Where Tr is the standard operator trace on L^{2}(ℝ).

Theorem 4 Π_{Λ}(H, Δ) = β_{Λ}(H, Δ).

Proof: Note that the rank of P_{Δ} is equal to |D_{ξ}(Δ)|, the number of elements in D_{ξ}(Δ), and

this is either 1 or 0. Since U_{ξ}^{∗} − 1 = P_{Δ}(e^{2iπ \frac{Ĥ_{ξ}−E_{0}}{|Δ|}} − 1) we can express the trace using the normalised eigenfunctions Ψ_{ξ} of Ĥ_{ξ} to μ(ξ), provided D_{ξ}(Δ) ≠ ∅,

Tr[(U_{ξ}^{∗} − 1)∂_{ξ}U_{ξ}] = ⟨Ψ_{ξ}|U_{ξ}^{∗} − 1|Ψ_{ξ}⟩⟨Ψ_{ξ}|∂_{ξ}U_{ξ}|Ψ_{ξ}⟩.

Substituting

⟨Ψ_{ξ}|∂_{ξ}U_{ξ}|Ψ_{ξ}⟩ = ∂_{ξ}⟨Ψ_{ξ}|U_{ξ}|Ψ_{ξ}⟩ = ∂_{ξ}e^{2iπ \frac{μ(ξ)−E_{0}}{|Δ|}}

in the previous expression we arrive at

Tr[(U_{ξ}^{∗} − 1)∂_{ξ}U_{ξ}] = (e^{−2iπ \frac{μ(ξ)−E_{0}}{|Δ|}} − 1)∂_{ξ}e^{2iπ \frac{μ(ξ)−E_{0}}{|Δ|}} .

Since U_{ξ}^{∗} − 1 = 0 if D_{ξ}(Δ) = ∅ we have

Π_{Λ}(H, Δ) = − lim_{n→∞} \frac{1}{2iπ|b_{n} − a_{n}|} ∫_{a_{n}}^{b_{n}}(\overline{μ̃(ξ)} − 1)μ̃′(ξ)dξ = −\frac{1}{2iπ} Λ(\overline{μ̃}μ̃′)  (4)

which is the expression for β_{Λ}(H, Δ). □

3.4 Interpretation as boundary force per unit energy

We obtain from (4)

Π_{Λ}(H, Δ) = − lim_{n→∞} \frac{1}{|b_{n} − a_{n}|} ∫_{a_{n}}^{b_{n}} μ′(ξ)\frac{|D_{ξ}(Δ)|}{|Δ|}dξ .

The r.h.s. is \frac{1}{|Δ|} times the Λ-mean of the expectation value of the gradient force w.r.t. the density matrix associated with the egde states in the gap. Since translating Ĥ_{ξ} in ξ is unitarily equivalent to translating the position of the boundary, Π can be seen as the force per unit energy the edge states in the gap of the system exhibit on the boundary [Kel].

4 Hulls and ergodic theorems

We have seen that, under the assumption that the limits exist, α_{Λ} = IDS_{Λ} and α_{Λ}(H, E) = Π_{Λ}(H, Δ) for E ∈ Δ, a gap in the spectrum of H. In this section we present some known results concerning classes of potentials for which the existence of the limit can be guaranteed and even shown to be independent of the sequence Λ. This is achieved by viewing the potential as an element of the topological space underlying a ℝ-dynamical system with invariant ergodic probability measure so that the result follows from an ergodic theorem. A second aim of the construction is perhaps more important. It allows for a K-theoretic description of the odd and even K-gap labels (justifying the terminology) and is the natural framework to prove their equality in any dimension.

Given a potential V consider its hull

Ω = \overline{\{V_{ξ}|ξ ∈ ℝ\}} ,

which is a compactification of the set of translates of V in the sense of [Jo86, Be92]. The action of ℝ by translation of the potential extends to an action on Ω by homeomorphisms which we denote by ω ↦ x·ω. The elements of Ω may be identified with those real functions (potentials) which may be obtained as limits of sequences of translates of V. We will write V_{ω} for the potential corresponding to ω ∈ Ω. If ω_{0} is the point of Ω corresponding to V then V_{ξ} = V_{−ξ·ω_{0}}. Then V_{y·ω}(x) = V_{ω}(x − y) and the family of Hamiltonians H_{ω} = −∂^{2} + V_{ω} is covariant in the sense that H_{x·ω} = U(x)H_{ω}U^{∗}(x) were U(x) is the operator of translation by x. The bulk spectrum is by definition the union of their spectra.

The assumption of the following theorem, namely that Ω carries an ℝ-invariant ergodic probability measure, can be verified for many situations and related to thermodynamical considerations [BHZ00].

Theorem 5 Suppose that (Ω, ℝ) carries an invariant ergodic probability measure P. Let Δ be a gap in the bulk spectrum and E ∈ Δ. Then almost surely (w.r.t. this measure) the limits to define α_{Λ}(H_{ω}, E) and Π_{Λ}(H_{ω}, Δ) exist and are independent of Λ and ω ∈ Ω. The almost sure value of Π_{Λ} is the P-average

Π(Δ) = \frac{1}{2iπ} ∫_{Ω} dP(ω)Tr((U_{ω}^{∗} − 1)δ^{⊥}U_{ω})

where (δ^{⊥}f)(ω) = \frac{df(t·ω)}{dt} |_{t=0}.

Proof: The crucial input is Birkhoff’s ergodic theorem which allows to replace

lim_{n→∞} \frac{1}{|Λ_{n}|} ∫_{Λ_{n}} F (x · ω)dx = ∫_{Ω} dPF (ω)

for almost all ω and any F ∈ L^{1}(Ω, P). The corresponding construction for the rotation number α has been carried out in [JM82] for quasi-periodic potentials and for the more general set up in [Jo86, Be92]. For Π_{Λ} the relevant function is F (ω) = Tr((U_{ω}^{∗} − 1)δ^{⊥}U_{ω}) which leads to the expression of the almost sure value of Π_{Λ}. □

4.1 K-theoretic interpretation

The dynamical system (Ω, ℝ) does not depend on the details of V, but only on its spatial structure (or what may be called its long range order). In fact, for systems whose atomic positions are described by Delone sets there are methods to construct the hull directly from this set, c.f. [BHZ00, FHK02]. The detailled form of the potential is rather encoded in a continuous function v : Ω → ℝ so that V_{ω}(x) = v(−x · ω) is the potential corresponding to ω. C(Ω) is thus the algebra of continuous potentials for a given spatial structure.

If one adds to this algebra the Weyl-algebra of rapidly decreasing functions of momentum operators one obtains the observable algebra which is in the continuous category the C^{∗}-crossed product C(Ω) ⋊_{φ} ℝ. It is the C^{∗}-closure of the convolution algebra of functions f : ℝ → C(Ω) with product f_{1}f_{2}(x) = ∫_{ℝ} dyf_{1}(y)φ_{y}f_{2}(x − y) and involution f^{∗}(x) = φ_{x}\overline{f(−x)}, where φ_{y}(f)(ω) = f(y · ω). It has a faithful family of representations {π_{ω}}_{ω∈Ω} on L^{2}(ℝ) by integral operators,

⟨x|π_{ω}(f)|y⟩ = f(y − x)(−x · ω).

It has the following important property. For each continuous function F : ℝ → ℂ vanishing at 0 and ∞ there exists an element F̃ ∈ C(Ω) ⋊_{φ} ℝ such that F (H_{ω}) = π_{ω}(F̃). Some of the topological properties of the family of Schrödinger operators {H_{ω}}_{ω∈Ω} are therefore captured by the topology of the C^{∗}-algebra. The invariant measure P over Ω gives rise to a trace T : C(Ω) ⋊_{φ} ℝ → ℂ, T (f) = ∫_{Ω} dPf(0).

Theorem 6 ([Be92]) Let E be in a gap of the bulk spectrum and suppose that the potential is smooth. The almost sure value of IDS_{Λ}(H, E) is IDS(E) := T (P̃_{E}).

We mention that this result is more subtle then just an application of Birkhoff’s theorem and interpretating the result in C^{∗}-algebraic terms as it needs a Shubin type argument which holds for smooth potentials, namely

lim_{n→∞} \frac{1}{|Λ_{n}|}(Tr(P_{E}(H_{Λ_{n}}) − Tr(χ_{Λ_{n}}P_{E}(H))) = 0.

The element P̃_{E} is a projection. As any trace on a C^{∗}-algebra, T depends only on the homotopy class of P̃_{E} in the set of projections of C(Ω) ⋊_{φ} ℝ. The even K-group K_{0}(C(Ω ⋊_{φ} ℝ) is constructed from homotopy classes of projections and the map on projections P ↦ T (P) induces a functional on this group, or stated differently, the elements of the K_{0}-group pair with T . It is therefore that we refer to T (P̃_{E}) as an even K-gap label of the gap.

There is a similar identification of the odd gap label as the result of a functional applied to the odd K-group of a C^{∗}-algebra. This C^{∗}-algebra is the C^{∗}-algebra of observables on the half space near 0, the position of the boundary. It turns out convenient to consider also the cases in which the boundary is at s ≠ 0. We therefore consider the space Ω × ℝ with product topology. This topological space, whose second component denotes the position of the boundary, carries an action of ℝ by translation of the potential and the boundary (so that their relative position remains the same). The relevant C^{∗}-algebra is then the crossed product (constructed as above) C_{0}(Ω×ℝ)⋊φ̃ℝ with φ̃_{y}(f)(ω, s) = f(y·ω, s+y). It has a family of representations {π_{ω,s}}_{ω∈Ω,s∈ℝ} on L^{2}(ℝ) by integral operators,

⟨x|π_{ω,s}(f)|y⟩ = f(y − x)(−x · ω, s − x).

It has the following important property. For each continuous function F : ℝ → ℂ vanishing at 0 and ∞ and such that F (H_{ω}) = 0 for all ω there exists an element F̂ ∈ C_{0}(Ω × ℝ) ⋊φ̃ ℝ such that F (H_{ω,s}) = π_{ω,s}(F̂) where H_{ω,s} is the restriction of H_{ω} to ℝ^{≤s} with Dirichlet boundary conditions at s.

The product measure of P with the Lebesgue measure is an ℝ-invariant measure on Ω × ℝ and defines a trace T̂(f) = ∫_{Ω} ∫_{ℝ} dPdsf(0).

Theorem 7 ([Kel]) The almost sure value of Π(Δ) is

Π_{Λ}(H, Δ) = Π(Δ) := \frac{1}{2iπ} T̂({ \atop U^{∗} − 1}δ^{⊥}{ \atop U − 1}).

The expression of the theorem depends only on the homotopy class of { \atop U − 1} + 1 in the set of unitaries of (the unitization of) C_{0}(Ω × ℝ) ⋊φ̃ ℝ. The odd K-group K_{1}(C_{0}(Ω × ℝ) ⋊φ̃ ℝ) is constructed from homotopy classes of unitaries and the map on unitaries U ↦ T̂((U^{∗} − 1)δ^{⊥}U) induces a functional on this group. It is therefore that we refer to \frac{1}{2iπ} T̂({ \atop U^{∗} − 1}δ^{⊥}{ \atop U − 1}) as an odd gap label of the gap.

The proof of the following theorem is based on the topology of the above C^{∗}-algebras.

Theorem 8 ([Kel]) T (P̃_{E}) = \frac{1}{2iπ} T̂({ \atop U^{∗} − 1}δ^{⊥}{ \atop U − 1}). In other words, IDS(E) = Π(Δ).

5 Conclusion and final remarks

We have discussed four quantities which serve as gap-labels for one-dimensional Schrödinger operators. They are all equal but their definition relies on different concepts. The Johnson-Moser rotation number α measures the mean oscillation of a single solution. The Dirichlet rotation number β counts the mean winding of the eigenvalues of the halfsided operators around a circle compactification of the gap. Π and IDS are operator algebraic expressions with concrete physical interpretations, the boundary force per energy and the integrated density of states. Whereas the identities α = −β = Π are rather elementary, their identity with IDS is based on a fundamental theorem, the Sturm-Liouville theorem. We tend to think therefore of Π as the natural operator algebraic formulation of the Johnson-Moser rotation number and of Theorem 8 as an operator analog of the Sturm-Liouville theorem. The advantage is that Π, IDS and Theorem 8 generalise naturally to higher dimensions [Kel]. In fact, the expression for IDS is the same as in (3) if one uses Føllner sequences {Λ_{n}}_{n} for ℝ^{d}. The expression of Π_{Λ} in ℝ^{d} requires a choice of a d − 1-dimensional subspace, the boundary, and so Ĥ_{ξ} is the restriction of the Schrödinger operator H_{ξ} = −Σ_{j}∂_{j}^{2} +V_{ξ}, V_{ξ}(x) = V (x+ξe_{d}), to the half space ℝ^{d−1} ×ℝ^{≤0} with Dirichlet boundary conditions. Then, for a single operator

Π_{Λ} = − lim_{n→∞} \frac{1}{|Σ_{n}|(b_{n} − a_{n})} ∫_{a_{n}}^{b_{n}} Tr((U_{ξ,Σ_{n}}^{∗} − 1)∂_{ξ}U_{ξ,Σ_{n}})dξ .

Here Σ_{n} is a Føllner sequence for the boundary and U_{ξ,Σ_{n}} = e^{−2πiG(Ĥ_{ξ,Σn})} where Ĥ_{ξ,Σ_{n}} is the restriction of H_{ξ} to Σ_{n}×ℝ^{≤0} with Dirichlet boundary conditions. We do not know of a direct link between this expression and the generalisation proposed by Johnson [Jo91] for odd-dimensional systems.

References

[BBEIM] E.D. Belokolos, A.I. Bobenko, V.Z. Enol’skii, A.R. Its, V.B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer-Verlag 1995.

[BLT85] J. Bellissard, R. Lima, D. Testard, Almost periodic Schrödinger operators, 1–64 in Mathematics + Physics. Vol. 1, World Scientific Publishing, Singapore 1985.

[Be92] J. Bellissard, Gap labelling theorems for Schrödinger operators, 538–630, in From Number Theory to Physics, (Springer, Berlin, 1993).

[BHZ00] J. Bellissard, D.J.L. Herrmann, M. Zarrouati, Hulls of aperiodic solids and gaplabelling theorems, 217–258, in Directions in Mathematical Quasicrystals, M. Baake and R.V. Moody, editors, Amer. Math. Soc. Providence, RI, 2000.

[CL55] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill 1955.

[FHK02] A.H. Forrest, J. Hunton, and J. Kellendonk, Cohomology of canonical projection tilings, Comm. Math. Phys. 226 (2002), no. 2, 289–322.

[Ha93] Y. Hatsugai, Edge States in the integer quantum Hall effect and the Riemann surface of the Bloch function, Phys. Rev. B 48, 11851-11862 (1993). The Chern Number and Edge States in the Integer Quantum Hall Effect, Phys. Rev. Lett. 71, 3697-3700 (1993).

[JM82] R. Johnson, J. Moser, The rotation number for almost periodic potentials, Commun. Math. Phys. 84, 403–438 (1982).

[Jo86] R. Johnson, Exponential Dichotomy, Rotation Number, and Linear Differential Operators with Bounded Coefficients, J. Differential Equations 61, 54–78 (1986).

[Jo91] R. Johnson, Oscillation theory for the odd-dimensional Schrödinger operator, J. Differential Equations 92, 145–162 (1991).

[KS04a] J. Kellendonk, H. Schulz-Baldes, Quantization of edge currents for continuous magnetic operators, J. Funct. Anal. 209, 388–413 (2004).

[KS04b] J. Kellendonk, H. Schulz-Baldes, Boundary maps for C^{∗}-crossed products with ℝ with an application to the Quantum Hall Effect, to appear in Commun. Math. Phys..

[Ke04] J. Kellendonk, Topological quantization of boundary forces and the integrated density of states, J. Phys. A. 37 L161–L166 (2004).

[Kel] J. Kellendonk, Gap Labelling and the pressure on the boundary, Preprint.