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Locality of the windowed local density of states

We introduce a generalization of local density of states which is"windowed"with respect to position and energy, called the windowed local density of states (wLDOS). This definition generalizes the usual LDOS in the sense that the usual LDOS is recovered in the limit where the position window captures individual sites and the energy window is a delta distribution. We prove that the wLDOS is local in the sense that it can be computed up to arbitrarily small error using spatial truncations of the system Hamiltonian. Using this result we prove that the wLDOS is well-defined and computable for infinite systems satisfying some natural assumptions. We finally present numerical computations of the wLDOS at the edge and in the bulk of a"Fibonacci SSH model", a one-dimensional non-periodic model with topological edge states.


Introduction
The density of states (DOS) is a fundamental concept in condensed matter physics which is crucial for understanding electronic conductivity properties of materials. Roughly speaking, the DOS is the density of electronic states available to be occupied by an electron as a function of energy (ignoring electron-electron interactions). Mathematically, the DOS is the density of eigenvalues of the single-particle electronic Hamiltonian viewed as a function of the spectral parameter. The local density of states (LDOS) is the contribution to the DOS from each point in space so that the average of the LDOS over all space equals the DOS at that energy. The LDOS has been used to clarify many phenomena in condensed matter physics. It is an especially important tool for studying systems without translational symmetry such as crystalline materials near defects or edges, disordered materials, and quasicrystals, where the Hamiltonian cannot be diagonalized using Bloch theory.
In this work we propose a generalization of the LDOS which is "windowed" with respect to position and energy, called the windowed local density of states (wLDOS). We start by defining the wLDOS for finite tight-binding systems, and in this context we show that the wLDOS reduces to the usual LDOS whenever the position window captures individual sites and the energy window is a delta distribution. We then prove that the wLDOS is local in the sense that it can be computed using a spatial truncation of the Hamiltonian to a neighborhood around each point of interest. Using locality of the wLDOS we then show that the wLDOS is well-defined and computable for a broad class of infinite tight-binding systems.
We finally present a numerical study of the wLDOS in the bulk and near the edge of a "quasi-crystalline" SSH model: a model of a one-dimensional material with no spatial periodicity which nonetheless supports a non-trivial bulk topological invariant and associated edge states. Note that there is no fundamental reason to restrict to a one-dimensional model, since the definition and locality property of the wLDOS are dimension-independent. However, in this work we restrict our numerical experiments to one spatial dimension because analogous numerical experiments in higher dimensions will be computationally more intensive and go beyond the scope of this work.
We are motivated to introduce the wLDOS as an alternative to the standard LDOS for several reasons. One is physical, in that experimental data extracted in spectroscopy is necessarily blurred with respect to energy and position because of the finite resolution of experimental probes. This can be seen clearly in Figures 1.1 and 1.2. We therefore expect numerical computations of the wLDOS to more closely resemble experimental data than numerical computations of the LDOS. This can be seen in Figure 1.3, where we present numerical computations of the wLDOS for a one-dimensional periodic SSH model and our quasicrystal variant. The energy and position windows used in these computations are shown in Figure 1.4. In our in-depth numerical study in Section 6 we use narrower position windows, but even the narrower windows, when placed over sites which are not uniformly spaced, can partially cover more than one site: see Figure 1.5.
Another reason to compute the wLDOS rather than the LDOS is numerical. When approximating the DOS one must smooth somewhat in energy to avoid implicitly attempting to compute all of the eigenvalues [15] and the same issue arises when approximating the LDOS. Finally, having the option of a variable spatial window may lead to more flexibility in how one parallelizes LDOS computations, and is convenient for plots of LDOS for systems with irregular positioning of sites and for systems with continuous degree of freedom.
1.1. Related literature. The derivation of the LDOS is standard in textbooks on condensed matter physics, see for example [13]. We do not attempt to summarize the physics literature on, or using, the LDOS, but recall some relevant mathematical works. Massatt, Luskin, and Ortner [19] (see also [20,4]) proposed a method for computing the DOS of an incommensurate bilayer system by averaging the LDOS over local atomic configurations. In the process they proved locality of the LDOS for that system using resolvent calculus. In contrast, we prove locality of the wLDOS (a generalization of the LDOS) for systems in arbitrary dimensions satisfying more general assumptions via a spectral flow argument. The idea to use locality of quantum mechanical models to develop schemes for computing quantum mechanical observables is now well-established, see [10,8,9,5,6]. The electronic properties of one-dimensional aperiodic models have been considered in the physics literature, see [26,7].
1.2. Outline of paper. We recall usual definitions of the DOS and LDOS, and then define the wLDOS, in the relatively simple case of finite-dimensional tightbinding models in Section 2. We will then prove locality of the wLDOS for such systems in Section 3 before using this property to extend the definition to a class of infinite-dimensional tight-binding models in Section 4. We will then introduce the quasicrystalline SSH model in Section 5 and present results of our numerical experiments in Section 6.

Windowed local density of states
In this section we will recall standard definitions of the DOS and LDOS, and then introduce the windowed local density of states (wLDOS), in the simplest case of finite-dimensional tight-binding models.
2.1. Finite-dimensional tight-binding models and standard definitions of the DOS and LDOS. We start by establishing some notation for finite-dimensional tight-binding models.
We consider sets of N points (which we will refer to as sites) in R d , with coordinates x n = (x 1 , ..., x d ) for 1 ≤ n ≤ N . We consider an electron with M internal degrees of freedom hopping between these sites in the tight-binding approximation, so that the electronic wave-function is an element of the Hilbert space C N ⊗ C M ∼ = C N where N = M N . For the moment we allow the Hamiltonian H of the system to be an arbitrary N ×N Hermitian matrix. Let δ m n ∈ C N denote the vector equalling 1 at site n and internal degree of freedom m, and zero in every other entry. For 2. An example of the output of scanning tunneling microscopy/spectroscopy measurement of the LDOS performed on an encapsulation of europium nanowires encapsulated in a carbon nanotube. Red and blue correspond to high and low densities respectively. The horizontal yellow streak within the blue area corresponds to localized states with energy within the band gap. Reprinted with permission from [21] Remark 2.1. We expect that our results can be generalized to continuum PDE models, at the cost of some technical complications. One obvious difficulty is unboundedness of the Hamiltonian, since boundedness is essential at a few points in the present work, e.g. Lemma 3.1, Assumption 4.2. We expect this difficulty can be overcome in the same way that we overcome the restriction to finite systems below, by using the fact that the wLDOS depends only on an energy truncation of the Hamiltonian (restriction of the Hamiltonian to the subspace of eigenstates with energy close to the energy of interest).
The simplest definition of the DOS is directly as the distribution where λ j are the eigenvalues of H counted with multiplicity [13]. The LDOS is defined as follows. Let ψ j , j ∈ {1, ..., N }, denote the eigenvector of H corresponding to the eigenvalue λ j (if λ j is degenerate ψ j is not unique but any choice for ψ j will do) so that We can then define using the spectral representation of H    Remark 2.2. For the sake of clarity, we define the DOS and LDOS directly through the delta distribution. The DOS and LDOS can equivalently be defined via their action on test functions, see [19] for example.

2.2.
The windowed DOS and LDOS. We will shortly introduce the main object of study in this work, the windowed LDOS (wLDOS), which is a version of the LDOS which is windowed in both energy and position. For simplicity, we first restrict attention to one spatial dimension. Let f (ξ) be a positive function in L 1 (R). For example, we can take f to be a normalized Gaussian for some σ > 0. Let H be as in section 2.1. We now define the windowed DOS (wDOS).
Definition 2.1. We define the windowed DOS (wDOS) by Following the argument of the previous section we derive that We therefore define the windowed (in energy) LDOS by Definition 2.2. We define the windowed (in energy) LDOS by The wDOS and (2.11) clearly reduce to the standard DOS (2.2) and LDOS (2.7) in the limit where f (ξ) → δ(ξ).
We wish to consider a more general construction where the LDOS is windowed in position as well as energy. To this end, let g(ξ) ∈ L 1 (R) satisfy 0 ≤ g ≤ 1 and be compactly supported. For example, we can take g to be a "bump" function such as Let X be the one-dimensional position operator (recall (2.1)) where x n , 1 ≤ n ≤ N are the positions of each site. Then define for any real E, x (2.14) . We now define the one dimensional windowed LDOS (wLDOS) at energy E and position x with window functions f and g as follows.
Definition 2.3. Let H be the Hamiltonian of a finite-dimensional tight-binding model with d = 1, and let X be the position operator (2.13). Let E and x be real numbers, let f (ξ) and g(ξ) be positive L 1 functions such that g is compactly supported with 0 ≤ g(ξ) ≤ 1, and let F E,x (H, X) be as in (2.14). The windowed local density of states (wLDOS) at energy E and position x with window functions f and g is then defined by Note that the wLDOS can be defined at any x ∈ R, even if x is not the position of a site.
An alternative equivalent formulation which will be useful is as follows. For arbitrary Hermitian matrices A, we have the identity Since F E,x (H, X) is positive by construction, we have so that the right-hand side of (2.18) gives an alternative definition of the wLDOS. We now claim the following proposition, which establishes that the wLDOS defined by Definition 2.3, with a particular choice of window functions, reduces to the standard LDOS for finite-dimensional tight-binding models.
Proposition 2.1. Let f be a positive L 1 function. Let {g n } n∈I and {x n } n∈I denote sets of compactly supported functions each satisfying 0 ≤ g n ≤ 1, and real numbers respectively. For each n ∈ I, let W xn (E) denote the wLDOS at energy E and position x n using window functions f and g n . Then: (1) If the functions g n centered at x n form a partition of unity where W (E) is the wDOS defined by (2.10). (2) If each point x n is chosen as the co-ordinate of the nth site (so that I = {1, ..., N }), the functions g n are chosen such that exactly one site is in the support of g n (ξ − x n ) for all n ∈ I, and g n (0) = 1, then (2.20) holds and for each n ∈ I which is exactly the windowed (in energy) LDOS (2.11).
As an example of a set of functions satisfying (2.19), note that the bump function (2.12) satisfies Proof of Proposition 2.1. Using the cyclic property of the trace and the definition (2.14), we have that Now let {g n (ξ)} n∈I and {x n } n∈I be as in Proposition 2.1, and assume (2.19). Then which is nothing but (2.10), so (1) is proved. For (2), note that we can expand the trace in (2.23) in the basis of eigenvectors of X to derive Using the spectral representation of X we have that Part (2) of Proposition 2.1 is now clear from the assumptions that exactly one site is in the support of each g(x n − x) and g(0) = 1.
We now define the wLDOS in higher dimensions. In d dimensions, recall that we define position operators by ., x n,d ) are the co-ordinates of the nth site. We will use the obvious notation X := (X 1 , ..., X d ). Let f (ξ) be a positive L 1 function and g(ξ) denote a compactly supported function R d → R such that 0 ≤ g(ξ) ≤ 1. Then for arbitrary real E and x = ( Definition 2.4. Let H be the Hamiltonian of a finite-dimensional tight-binding model, and let X l , 1 ≤ l ≤ d denote the position operators (2.27). Let E be a real number and x = ( The windowed local density of states (wLDOS) at energy E and position x with window functions f and g is then defined by Since the proof is identical to that of Proposition 2.1, we state the following without proof.
Proposition 2.2. Let {g n (ξ)} n∈I and {x n } n∈I denote sets of compactly supported functions R d → R each satisfying 0 ≤ g n ≤ 1, and real numbers respectively. Let f (ξ) be a positive L 1 function. For each n ∈ I, let W xn (E) denote the wLDOS at energy E and position x n using window functions f and g n . Then: (1) If the functions g n centered at x n form a partition of unity (2) If each point x n is chosen as the co-ordinate of the nth site (so that I = {1, ..., N }) and the functions g n are chosen such that exactly one site is in the support of g n (ξ − x n ) for all n ∈ I, then (2.20) holds, and the wLDOS reduces to the windowed (in energy) LDOS: As an example of a set of functions satisfying (2.30) we can take products and translates of the bump function (2.12). For example in dimension d = 2 we have Remark 2.3. We briefly note some practical considerations which should be taken into account when numerically computing the wLDOS. First, note that a naïve computation of f (H − E) would require a potentially expensive diagonalization of H. Assuming f (ξ) is sufficiently smooth this can be avoided by approximating f (ξ) in L ∞ by a polynomial f p (ξ) so that f p (H − E) can be accurately computed by merely repeatedly applying H − E. This is known as the kernel polynomial method [25,15] and can be rigorously justified using Bernstein's theorem. Second, note that when F E,x (H, X) has large rank it may be costly to directly evaluate the trace defining the wLDOS. In this case it may be preferable to compute the trace using a randomized algorithm [12,15].
Remark 2.4. We have seen that for finite systems the DOS can be recovered from the wLDOS by averaging over sites. Assuming the wLDOS can be computed efficiently, one can imagine a scheme for efficiently computing the DOS of a large system by averaging over local computations of the wLDOS. The efficiency of this approach would come from the fact that each local computation could be computed independently and hence could be parallelized.

Proof that the wLDOS is local
We now prove the wLDOS is local in the sense that it can be computed with a finite truncation of the system Hamiltonian nearby to the point of interest up to error which can be made arbitrarily small. We will prove this initially for finitedimensional models without aiming for optimal constants. In the next section we will introduce natural assumptions which will significantly improve these constants and allow us to exploit locality to define wLDOS for a class of infinite-dimensional tight-binding models. We will state and prove the result in one spatial dimension for clarity and then state the general result for models in R d without proof since the proof is similar to the one dimensional case.
Let f (ξ) and g(ξ) be as in Section 2.2, i.e., f is a positive L 1 function, and g ∈ L 1 compactly supported with 0 ≤ g ≤ 1. We now additionally assume that f is sufficiently smooth (specifically, we assume (3.4)). Now let k(ξ) ∈ L 1 be compactly supported with 0 ≤ k ≤ 1 equalling 1 on the support of g(ξ). For example, if g is given by (2.12), we can take k to be We think of k as modeling spatial truncation of the Hamiltonian. We first give an outline of our results before stating theorems. Detailed proofs will be postponed to an Appendix. Recall the definition of the wLDOS (Definition 2.3). The statement that the wLDOS is local is then the statement that for any real E and x, Note that the right-hand side only involves the "spatially truncated" Hamiltonian . The main step to prove (3.2) will be an estimate in the operator norm. Since (3.3) is equivalent to a statement about Frobenius norms using (2.18), we can pass to the estimate (3.2) using equivalence of finitedimensional norms (for large system sizes this step will give a large constant which can be avoided by making natural assumptions on H, see Section 4).
We now move to stating our results rigorously. We will establish (3.3) in two steps. The first and more difficult step is to prove the following lemma.
Lemma 3.1. Suppose H and X are finite-dimensional Hermitian operators. Let g and k be positive L 1 functions such that 0 ≤ g ≤ 1, 0 ≤ k ≤ 1, and kg = g. Let f be a positive L 1 function such that Condition (3.4) holds as long as f is twice differentiable with the Fourier transform of f in L 1 .
The second step is to prove that the right-hand side of (3.5) can be made arbitrarily small by an appropriate choice of k(ξ). Specifically, we will prove the following proposition.
where C > 0 is a constant independent of α. Note that the proof uses very naïve estimates that the resulting constant C will grow with system size. In the next section we will introduce assumptions that allow for estimates which are uniform in the system size.
For completeness we state the d-dimensional result without proof.
where C > 0 is a constant independent of α.

Defining and computing the wLDOS of infinite systems using locality
In this section we will introduce natural assumptions which will allow for locality estimates which are uniform in system size. Using these locality estimates we will then show that the wLDOS is well-defined and computable for a broad class of infinite-dimensional tight-binding systems. Suppose we fix a real number E and x ∈ R d . For any R > 0, we can define a finite-dimensional tight-binding model with Hamiltonian H R by restricting the infinite model to sites in the ball of radius R about x, and compute the wLDOS of this model at E and x. Since we recover the infinite-dimensional model in the limit R → ∞, it is natural to ask whether it makes sense to take the limit of the wLDOS of the sequence of finite models and define the wLDOS of the infinite model by this limit. In the previous section we proved that the wLDOS can be computed from a truncation of the Hamiltonian to a region nearby the point of interest. Formally then, the sequence of wLDOS values should converge as R → ∞. To make this rigorous, we have to control the constant in the estimate (3.9) as a function of R. The proof of Theorem 1 clearly does not provide this since, for example, we invoke equivalence of norms in C N , where N will increase as function of R. To pass to the limit R → ∞, we require three natural assumptions which we expect will be satisfied by any physically reasonable tight-binding model. We first assume that the Hamiltonian H ∞ is local in the following sense.
Assumption 4.1. Let X l , 1 ≤ l ≤ d denote the position operators defined by (4.1), extended to all of R d . We assume that H ∞ is local in the sense that there exists a constant C loc > 0 such that Assumption 4.1 can be roughly stated as "H ∞ is a narrowly banded matrix in the position basis". We next assume that H ∞ is bounded.
Assumption 4.2. We assume that H ∞ is bounded in the sense that there exists a constant C norm > 0 such that The final assumption rules out some pathological situations where e.g. balls with finite radius can contain an unbounded number of sites.
Then we assume that the rank of the matrix is uniformly bounded above for all x ∈ R d by a positive integer M upper .
Note that Assumptions 4.1, 4.2, and 4.3 are trivial for Hamiltonians H of fixed finite-dimensional tight-binding models.
We now have the following. (1) The limit W x,∞ (E) := lim R→∞ W x,R (E) exists and equals (2) The limit W x,∞ (E) can be computed by the formula (4.6) as long as R > 0 is sufficiently large that Proof. Let R > 0 be arbitrary.
where C > 0 is independent of both R and α. Since k α (X − x) is a cutoff, by taking R sufficiently large we can ensure that We now claim that the sequence {W x,R (E)} is Cauchy. Let > 0 be arbitrary. For arbitrary R, R , (4.8) implies that (4.10) Now take α small enough such that the O(α) term is < , and then take R and R sufficiently large that (4.9) holds for both terms so that the other term in (4.10) vanishes.
We can now make the following definition.  That this quantity is well-defined and computable is guaranteed by Theorem 3.

The Fibonacci SSH model
In this section we introduce a one-dimensional model system which we refer to as the Fibonacci SSH model. We choose to study this model because it lacks any translational symmetry, making the wLDOS an important tool for understanding the electronic states of the system.
Before we can define the Fibonacci SSH model, we must first review the Fibonacci quasicrystal construction.
5.1. The Fibonacci Quasicrystal. The Fibonacci quasicrystal is a one dimensional chain made of two sorts of "links" which have lengths S and L with S < L. We take the lengths in a fixed ratio of the golden mean We can form an infinite quasiperiodic chain by starting with a series of links and then repeatedly applying the replacement rules If we start with the single letter S, we obtain the sequence and so on. It will be more convenient to work with a sequence which grows in two directions rather than one. To obtain such a sequence, instead of starting with S we start with LL. Applying the replacement rules we find (5.4)

L L LS LS LSL LSL LSLLS LSLLS LSLLSLSL LSLLSLSL
and so on, where indicates the center of the sequence. We will shortly want to identify the ends of each stage of the sequence. When we do this, we would like to guarantee that we do not create sequences of letters, known as words, which did not appear in the original sequence. We call such words invalid words.
To obtain a sequence such that identifying the ends of each stage does not create invalid words, we start with LLS instead of LL. In this case, invalid words are not created by identifying ends because LLSL and LSLL and SLLS all are valid words, appearing by stage 3. The first five sequences obtained from the quasicrystal construction are then stage 1: Notice that stage 1 appears at the center of stage 3, stage 3 appears at the center of stage 5, and so on. We can therefore consider the sequence of odd stages as a sequence of chains which grow at their ends. The infinite quasicrystal is defined as the infinite limit of this sequence. Note that if we took instead the even stages, we would have something locally indistinguishable from what we are using (see Example 4.6 of [3]). Our final task in this section is to compute values for the lengths of S and L so that the average distance between vertices (points between links) is 1. The number of symbols at stage n is a Fibonacci number 1 , Suppose we replace S by an edge of length 1 and L by an edge of length φ where (Thus φ 2 = φ + 1, φ 3 = 2φ + 1, etc.). Since the total length represented by L is φ and the total length represented by LS is φ + 1, the length of the resulting finite quasilattice will be growing at each stage by the factor φ. At stage 1 the total length of the quasilattice is 2φ + 1 so the total length of the quasilattice at stage n is φ n+2 . This means the average distance between vertices is To get the average distance between vertices to be 1, we rescale, and so use The Fibonacci SSH model. The original SSH model [23] describes hopping on a one-dimensional lattice where the hopping amplitudes alternate between two values, known as the inter-site and onsite hopping amplitudes. In the Fibonacci SSH model, the onsite hopping amplitude is held fixed while the inter-site hopping amplitude takes on one of two values, with the choice determined by the Fibonacci quasicrystal construction constructed in the previous section. 1 The Fibonacci numbers are defined by F 0 = 0, F 1 = 1 and Fn = F n−1 + F n−2 for all n > 1.
A straightforward induction proves the stronger statement that the numbers of copies of S and L at stage n are F n+1 and F n+2 respectively.
The model is defined on the Hilbert space 2 (V )⊗C 2 with V the set of vertices of the infinite Fibonacci quasicrystal. As usual we define the one-dimensional position operator by where x n is the co-ordinate of each vertex. Let σ x denote the Pauli matrix The Hamiltonian is where h.c. denotes the Hermitian conjugate. Here t o is a real constant defining the onsite hopping amplitude, and t i (n, n + 1) is the inter-site hopping amplitude, which depends on whether the link between vertices n and n + 1 is S or L. We will take t o = 2.15 and (5.14) t i (n, n + 1) = 3.04 if link between sites n and n + 1 is S 2.73 if link between sites n and n + 1 is L.
With these choices, the inter-site hopping term is, on average, 2.85. This is comparable to the original SSH model [24], where the onsite and inter-site hopping strengths are 2.15 and 2.85 respectively. The quasicrystal SSH model retains the chiral symmetry of the original periodic SSH model, i.e. where S = I ⊗ σ z . We can therefore consider the model as belonging to class BDI of the Altland-Zirnbauer classification of topological insulators [1,14]. It is straightforward to compute the Bloch eigenvalue bands of the periodic SSH model [24,2] as The bulk winding number is 1 whenever |t o | < |t i |. When t 0 = 2.15 and t i = 2.85 the bulk spectrum of the periodic SSH model is therefore (5.17) [−5, −0.7] ∪ [0.7, 5], and the bulk winding number is 1. The infinite Fibonacci SSH Hamiltonian with t i chosen according to (5.14) is a perturbation whose size in the operator norm is bounded by 0.19 of the standard SSH Hamiltonian which has gap 0.7 and topological index 1. By standard arguments the Fibonacci SSH Hamiltonian has a gap of at least 0.51 and must also have topological index 1. When the model is truncated and Dirichlet boundary conditions are imposed at both ends, we expect edge states, eigenvectors of the Hamiltonian supported near to the physical edge of the model, to occur.
We can compute an approximation of the spectrum and integrated DOS for the Fibonacci SSH model by imposing periodic boundary conditions on a finite

Investigation of the wLDOS for the Fibonacci SSH model
In this section we show computations of the wLDOS for the Fibonacci SSH model introduced in the previous section. For real positive η > 0, we define a Gaussian energy window with standard deviation η by To avoid diagonalization of the Hamiltonian, we approximate f η by a 14th order polynomial. We define a position window by where g(ξ) is as in (2.12), so that g 2 (ξ) is supported on the interval [−1, 1]. This window function is plotted in Figure 1.5. With these window functions, we compute the wLDOS in the bulk (Figure 6.1) and at the edge ( Figure 6.3) of finite truncations of the model. By our theoretical results we know that when the truncation is sufficiently far from the point of interest that the results will be identical to those obtained if we were able to compute with the fully infinite model (half-infinite when we look at the edge). We can computationally test locality of the wLDOS by comparing the computed wLDOS as we increase the system size (Figures 6.2 and 6.4). We find that in practice the computed wLDOS converges quickly as system size is increased, suggesting the wLDOS is more local than our theoretical guarantees.

Conclusion
In this work we have proposed a variant of the usual LDOS called the windowed local density of states or wLDOS where the LDOS is "windowed" with respect to energy and position. We proved that for finite systems the wLDOS generalizes the usual LDOS in the sense that for narrow window functions the wLDOS reduces to the usual LDOS. We proved that the wLDOS is local in the sense that it can be computed accurately from a finite truncation of the Hamiltonian about the point of interest. We used this property to show that under natural locality assumptions on the Hamiltonian the wLDOS is well-defined and computable even for tight-binding models on infinite domains. We finally investigated the wLDOS for the "Fibonacci SSH" model, a one dimensional aperiodic model which nonetheless has a non-trivial bulk index and associated topological edge states. Our computations show that the wLDOS is considerably more local than our theory guarantees, suggesting our locality estimates might be further improved.
These results demonstrate that the wLDOS can be a useful tool for computing and visualizing material properties. Since our theory allows for computing the wLDOS for both finite and infinite models, it may be useful for predicting when finite size effects will be important in computations and/or experiments.
For future directions, we mention two potential applications of our results. First, to compute the DOS of a large finite system, instead of diagonalizing the Hamiltonian, one could average the wLDOS over the system. This computation would , the sequence shows rather quick convergence to machine precision, as should be expected from the uncertainty principle.
require many computations of the wLDOS, but importantly, using locality, these computations could be performed in parallel. This scheme could be applied, for example, to compute the DOS of a large finite quasicrystal as studied by one of the authors in [18]. Similar schemes have been employed in the study of incommensurate layered materials, see [19,20,4]. The second potential application involves an apparent link between the wLDOS and the spectral localizer [11,17,16]. If one uses a broad window g in position and a modest window f in energy then g(X) and f (H) become almost commuting matrices. Given the known connection [17] between almost commuting matrices, K-theory, and the spectral localizer, it should be interesting to look at the wLDOS on topological systems. Moreover, it should be useful to work with versions of the wLDOS using other norms, c.f. (2.18).
Appendix A. Proofs of Lemma 3.1 and Proposition 3.2 In this section we give the proofs of Lemma 3.1 and Proposition 3.2.
A.1. Proof of Lemma 3.1. Before we can give the proof of Lemma 3.1 we require two preliminary Lemmas. Note that we take x = E = 0 for simplicity.  Re-arranging the right-hand side of (A.8) and using kg = k gives d dt e −itH e itk(X)Hk(X) g as required.
We can now give the proof of Lemma 3.1.
proof of Lemma 3.1. Let be the inverse Fourier transform of f , so For bounded Hermitian operators K, we have by functional calculus that It then follows that (A.14) g Comparing this identity with K = H and K = k(X)Hk(X) we find  Since (t) = it f (t) the statement follows.
A.2. Proof of Proposition 3.2. We now prove Proposition 3.2. We again start with a preliminary Lemma. The result now follows by Lemma A.1.
We are now in a position to give an explicit construction of k α (ξ) equalling 1 for all ξ ∈ [−L, L] and satisfying (3.7).
Proof of Proposition 3.2. Fix L > 0, and define M L to be the smallest integer such that 2M L > L. Define k(ξ) as in (2.12), i.e.
(A. 21) k It is clear that and more generally, Using the fact that 2M > L by assumption, we have that k 1 (ξ) acts by 1 over the whole interval [−L, L]. For positive α, we now define (A.24) k α (ξ) := k 1 (αξ).
It is easy to see that k α (ξ) acts as 1 over the interval − 2M α , 2M α and hence acts as 1 over the interval − L α , L α , which contains [−L, L] for 0 < α < 1. The support of k α (ξ) is clearly confined to − 2M +2 α , 2M +2 α . Using the definition of M as the smallest integer such that 2M > L we see that L ≤ 2M ≤ L + 2 ≤ 2M + 2 ≤ L + 4 and hence the support of k α (ξ) is confined to − 2M +4 α , 2M +4 α . We will now prove that k α (ξ) satisfies the bound (3.7) using Lemma A.3. Our strategy is to build up to a bound on the Fourier transform of k α (ξ) from a bound on the Fourier transform of k (ξ).