Singularly continuous spectrum of a self-similar Laplacian on the half-line

Several models of one-dimensional discrete Schrödinger operators have been proved to exhibit purely singular continuous spectrum; see for instance Refs. 7, 40, and 56. In this brief paper, we consider a particular family of self-similar Laplacians Δ_p on the integer half-line ℤ₊, parametrized by p ∈ (0, 1). The parameter p plays the role of the transition probability of a symmetrizable random walk. From the physical point of view, changing p corresponds to changing the contrast ratio of the fractal media. From the mathematical point of view, these Laplacians arise from the study of the unit interval endowed with a fractal measure and were first addressed by the second author in Ref. 63 in the context of spectral zeta functions; see also the related work.²⁷ In this context, the parameter p determines the resistance and measure scaling of the fractal space. In particular, we obtain a simple one-parameter family of models for which the spectral dimension d_s of Δ_p (see Remark 4) varies continuously,

It will be explained that when $p= 1 2$⁠, we recover the classical one-dimensional Laplacian with d_s = 1. We can also take the direct product of any number of these fractal intervals to construct a fractal of a higher dimension. For instance, a fractal with topological dimension 4 and spectral dimension 2 can be obtained by taking the direct product of 4 one-dimensional intervals, each equipped with a fractal Laplacian Δ_p with $p ( 1 − p ) = 1 40$⁠, or equivalently, $d s = 1 2$⁠.

The key question addressed in this paper concerns the spectral type of the fractal Laplacian Δ_p. Related questions about wave propagation on this fractal, viz., the modes of convergence of discrete wave solutions to the fractal wave solution, were studied in Refs. 6 and 16. For recent physics results, theoretical and experimental, see Refs. 1–3, 28, 44, and 61 and references therein. In general, weakly self-similar fractal systems are related to quasicrystals. Although we do not discuss this relation, the reader can find explanations in Refs. 12, 13, 19, 20, and 44 and references therein. Our long-term motivation comes from the fact that many problems of fractal nature appear in quantum gravity (e.g., Refs. 5, 30, 34, 43, and 51), but the related mathematical physics (e.g., Refs. 17, 31, 32, 36–39, 47, 49, 52, 53, 59, and 60) is not sufficiently developed to approach these problems, mainly because it is hard to tackle problems of fractal geometry and spectral analysis simultaneously. Hence analyzing a straightforward fractal model, such as the one described in this paper, may be of special interest.

Our result is relatively simple because of the minimality of our model, but it relies upon spectral decimation (or spectral similarity)^8,45 and its connection with the Julia set of a rational function. Parallel ideas have also appeared in the proofs of singularly continuous spectrum for Fibonacci Hamiltonians on ℤ (see Refs. 21–26 and 46 and references therein), as well as the relation between Julia sets and Jacobi matrices (see Refs. 10, 11, 14, and 15). We hope that the methods outlined in this paper can be generalized to more complicated settings.

The pq-model on ℤ₊ is defined as follows. Let p ∈ (0, 1) and q = 1 − p. For each x ∈ ℤ₊∖{0}, let m(x) be the largest natural number m such that 3^m divides x. Then for all functions f on ℤ₊, we set

Observe that the Laplacian Δ_p generates a nearest neighbor “pq random walk” on ℤ₊ as shown in Figure 1.

If $p= 1 2$⁠, we recover the symmetric simple random walk on the half-line with reflection at the origin, and $Δ 1 2$⁠, the classical one-dimensional Laplacian, is self-adjoint on ℓ²(ℤ₊). If $p≠ 1 2$⁠, Δ_p is not self-adjoint on ℓ²(ℤ₊). That said, we can identify the symmetrizing measure for Δ_p using the theory of Markov chains (see, e.g., Ref. 29, Ch. 1). One can readily verify that Δ_p generates an irreducible Markov chain on ℤ₊ whose transition probabilities satisfy p(x, y) = 0 whenever $x − y >1$⁠, i.e., it is a birth-and-death chain. As a result, one can explicitly compute the invariant measure π by iteratively solving the equation $π ( y ) = ∑ x ∈ Z + π ( x ) p ( x , y )$⁠. Moreover, the reversibility condition π(x) p(x, y) = π(y) p(y, x) holds for every x, y ∈ ℤ₊, which implies that Δ_p is symmetric with respect to π. In our example, π coincides with the discretization of the fractal measure described in Refs. 6 and 63 (π is a σ-finite but not finite measure on ℤ₊).

Our main result is as follows.

It is well-known that the spectrum of $Δ 1 2$⁠, the classical one-dimensional Laplacian on ℤ₊, is the interval [0, 2] and is absolutely continuous. So in a sense, there is a “phase transition” in the spectral type of Δ_p as p varies through $1 2$⁠, going from a singular spectrum to an absolutely continuous spectrum and back to a singular spectrum.

Throughout the section, ρ(A) and σ(A) stands for the resolvent set and the spectrum of an operator A, respectively.

We briefly review the necessary ingredients from spectral decimation that will be used in the proof. Spectral decimation originated from Refs. 12 and 50 and was implemented on the Sierpinski gasket in Refs. 33, 54, and 62 and on post-critically finite fractals in Ref. 55. Here we follow Ref. 45, Definition 2.1 (see also Ref. 8 for more information). Let $H$ and $H 0$ be Hilbert spaces, and H (respectively, H₀) be operators on $H$ (respectively, $H 0$⁠). We say that H is spectrally similar to H₀ with functions φ₀, φ₁ : ρ(H) → ℂ if there exists a (partial) isometry $U: H 0 →H$ such that

whenever both sides are defined.

For concreteness, we will specialize spectral similarity to the case where $H 0$ is a closed subspace of $H$⁠, and U^∗ ≕ P₀ is the orthogonal projection from $H$ to $H 0$⁠. Let $H 1$ be the orthogonal complement of $H 0$ in $H$⁠, and P₁ = I − P₀ be the orthogonal projection from $H$ to $H 1$⁠. Define $I 0 : H 0 → H 0$⁠, $X: H 0 → H 1$⁠, $X ̄ : H 1 → H 0$⁠, and $Q: H 1 → H 1$ by $I 0 = P 0 H P 0 ∗$⁠, $X= P 1 H P 0 ∗$⁠, $X ̄ = P 0 H P 1 ∗$⁠, and $Q= P 1 H P 1 ∗$⁠. In other words, H has the following block structure with respect to the representation $H= H 0 ⊕ H 1$⁠:

According to Ref. 45, Corollary 3.4, without loss of generality, we may assume that φ₀ and φ₁ are defined on ρ(Q). Then by Ref. 45, Definition 3.5, we introduce the exceptional set of the spectrally similar operators H and H₀ as follows:

Let R(z) = φ₁(z)/φ₀(z) whenever φ₀(z) ≠ 0.

The key result we need is

We now apply the above framework to the operator Δ_p on ℓ²(ℤ₊). Here we put ℋ = ℓ²(ℤ₊) and ℋ₀ = ℓ²(3ℤ₊). Then

is the orthogonal projection defined by

Moreover, following the idea of Bellissard [Ref. 12, p. 125], we can define a dilation operator, D : ℓ²(3ℤ₊) → ℓ²(ℤ₊),

and it co-isometric adjoint, D : ℓ²(ℤ₊) → ℓ²(3ℤ₊),

L²(ℤ₊, π). Then we define the operator $Δ p +$ on ℓ²(3ℤ₊) to be

Note that, by definition, $Δ p +$ is isometrically equivalent to Δ_p.

In what follows, one of the key observations is that the invariant measure π satisfies the relation π(x) = π(3x) for all x ∈ ℤ₊, which allows us to use the same definitions in L²(ℤ₊, π).

Proposition 5 was essentially proved in Refs. 6 and 63. It follows from taking the Schur complement of Δ_p with respect to the block corresponding to projection of functions onto ℤ₊∖(3ℤ₊). For the reader’s convenience, we give a self-contained proof in the  Appendix.

Next, we identify the exceptional set of Δ_p and $Δ p +$⁠. Note that φ₀(z) ≠ 0 for all z ∈ ℂ. As for the operator $Q: H 1 → H 1$⁠, (2.1) yields

for each x ∈ ℤ₊. This means that Q, as a matrix with respect to the natural basis of delta functions on ℤ₊∖3ℤ₊, is a block diagonal matrix consisting of 2 × 2 blocks,

From this it is easy to deduce that σ(Q) = {1 + p, 1 − p}. Thus $E ( Δ p , Δ p + ) = { 1 + p , 1 − p }$⁠.

The next result is a direct consequence of Proposition 5.

Actually we can say more. Due to the self-similarity of the Laplacian Δ_p, $Δ p +$ has the same spectrum as Δ_p, and in fact they are isomorphic as bounded symmetrizable operators. This observation combined with Proposition 7 leads to

It remains to resolve the status of the exceptional points.

We now recall some facts from complex dynamics (see, e.g., Ref. 48, Sec. 4). The Fatou set $F ( g )$ of a nonconstant holomorphic function g on the Riemann sphere $C ˆ$ is the domain in which the family of iterates {g^∘n}_n converges uniformly on compact subsets. The complement of the Fatou set in $C ˆ$ is the Julia set $J ( g )$⁠. Both $F ( g )$ and $J ( g )$ are fully invariant under g: that is, $g − 1 ( F ( g ) ) =F ( g )$ and $g − 1 ( J ( g ) ) =J ( g )$⁠. Moreover, $J ( g )$ is a closed subset of $C ˆ$⁠.

For the spectral decimation function R in (3.4), we have the following characterization of the Julia set $J ( R )$⁠, which is standard in complex dynamics (see Ref. 48).

By Ref. 48, Lemma 4.6, ${ 0 , 1 , 2 } ⊂J ( R )$ because they are repulsive fixed points of R.

We now have all the ingredients to prove Theorem 1.

A substantial part of this work was completed and presented at the workshop “Spectral Properties of Quasicrystals via Analysis, Dynamics, and Geometric Measure Theory” at the Casa Matemática Oaxaca (CMO). We thank the Banff International Research Station for Mathematical Innovation and Discovery, and the organizers and participants for their support. We are especially grateful to D. Damanik and A. Gorodetski for many insightful remarks and suggestions. Research partially supported by NSF No. DMS-1262929.

Let us divide ℤ₊ into two disjoint subspaces 3ℤ₊ and ℤ₊∖3ℤ₊. Then for z ∈ ℂ, the operator Δ_p − z acting on functions on ℤ₊ can be represented in block matrix form,

where

The Schur complement S(z) of Δ_p − z with respect to the block corresponding to functions on ℤ₊∖3ℤ₊ is then given by

We claim that this equals $φ 0 ( z ) ( Δ p + − R ( z ) )$ as an operator acting on functions on 3ℤ₊. More formally, we consider the matrices of operators with respect to the natural basis of delta functions on ℤ₊.

To compute S(z), let us observe that I₀ − z is a diagonal matrix with all diagonal elements equal to 1 − z; $X ̄$ has nonzero matrix elements $X ̄ ( 0 , 1 ) =−1$⁠, $X ̄ ( 3 x , 3 x − 1 ) =−q$ (respectively, −p) and $X ̄ ( 3 x , 3 x + 1 ) =−p$ (respectively, −q) if 3^−m(3x)(3x) ≡ 1(mod3) (respectively, if 3^−m(3x)(3x) ≡ 2(mod3) ); X has nonzero matrix elements X(3x, 3x ± 1) = − q for all x ∈ ℤ₊; and Q − z is a block diagonal matrix consisting of 2 × 2 blocks,

Since Q − z is block diagonal, it is easy to see that it has an inverse (Q − z)⁻¹ whenever z ∉ {1 − p, 1 + p}. (Q − z)⁻¹ is a block diagonal matrix consisting of 2 × 2 blocks,

After some algebra, we verify that $X ̄ ( Q − z ) − 1 X$ has all diagonal elements equal to $q ( 1 − z ) ( 1 − z ) 2 − p 2$ and off-diagonal elements,

Therefore $( I 0 − z ) − X ̄ ( Q − z ) − 1 X$ has all diagonal elements equal to φ₀(z)[1 − R(z)] and off-diagonal elements $φ 0 ( z ) Δ p + ( x , y )$ in the (3x, 3y)-entry. This proves the claim. Since Δ_p − z is invertible if and only if both Q − z and the Schur complement $( I 0 − z ) − X ̄ ( Q − z ) − 1 X$ are invertible, the claim implies Proposition 5.