SEMICLASSICAL QUANTIZATION CONDITIONS IN STRAINED MOIRÉ LATTICES

. In this article we generalize the Bohr-Sommerfeld rule for scalar symbols at a potential well to matrix-valued symbols having eigenvalues that may coalesce precisely at the bottom of the well. As an application, we study the existence of approximately flat bands in moiré heterostructures such as strained two-dimensional honeycomb lattices in a model recently introduced by Timmel and Mele.


Introduction
In a recent celebrated article by Tarnopolsky, Kruchkov, and Vishwanath [TKV19], it has been observed in a chiral model for twisted bilayer graphene, that a certain Dirac-type operator can exhibit flat bands at certain angles coined the magic angles. In [BEWZ21,BEWZ22] it has been shown that as the twisting angle is very small, essentially every band close to zero energy is flat. In this article, we study an analogue of the above-mentioned chiral model in one dimension, introduced by Timmel and Mele [TM20], where a moiré-type structure appears in one dimension through the application of physical strain. While this model does not exhibit perfectly flat bands, we show that there exist approximate eigenvalues of infinite multiplicity in the limit of very large moiré cells. Our main mathematical contribution is a generalization of the Bohr-Sommerfeld quantization condition at potential wells to fairly general matrixvalued symbols.
In Figure 1 we illustrate the superposition of two honeycomb lattices under strain. The left superposition is subjected to uniaxial strain in the x 1 direction, while the middle superposition is subjected to shear strain. Contrary to the case of twisted bilayer graphene in which the moiré pattern is a two-dimensional structure (see the right panel), the left and middle moiré patterns in Figure 1 are essentially one-dimensional. This is the setting that we will be concerned with in this article.
Apart from modeling twisted graphene sheets under mechanical strain, the model we analyze in this work has also been considered for low-energy electron diffraction (LEED) studies in surface reconstructions of metals. More precisely, metals such as Figure 1. Superposition of two honeycomb lattices with one of the lattices (with red/blue vertices) subject to uniaxial strain in x 1 direction (left) and shear strain (middle) creating 1D moiré patterns. For comparison, the right panel shows a 2D moiré pattern created by a superposition of twisted honeycomb lattices without strain.
iridium, platinum, and gold are known to exhibit columns of honeycomb lattice structures on their surface with pits in between them. This phenomenon where the crystal structure of the metal is broken up on the surface is known as surface reconstruction [Her12, VKS + 81]. In addition, the existence of one-dimensional flat bands in twisted one-dimensional Germanium selenide lattices has been recently discovered in [KXCR20].
In order to understand such emerging physical phenomena, we develop a new spectral analysis of operators with matrix-valued symbols exhibiting a potential well. For Schrödinger operators this has been studied by Barry Simon [Sim83] and has been generalized to pseudodifferential operators by Helffer-Robert [HR84] (see also [DS99]) to symbols that via a linear symplectic transformation can be locally reduced to a principal symbol with non-degenerate minimum at (x, ξ) = (0, 0) such that p 0 (x, ξ) = λ 2 (ξ 2 + x 2 ) + O((x, ξ) 3 ) with λ > 0. (1.1) The Bohr-Sommerfeld quantization condition near a potential well then allows for the following asymptotic spectral description, see also [CdV05,ILR18]. Indeed, we can parametrize periodic orbits p −1 0 (τ ) by the bicharacteristic flow I t → (x(t), ξ(t)) at the energy level τ. We shall then write p −1 0 (τ ) f dt := I f (x(t), ξ(t)) dt. The following theorem is discussed in [DS99, Theo.14.9], where we used the convention of [CdV05,ILR18] instead for the characterization of the approximate eigenvalues.
In F 1 the value 1 2 corresponds to the Maslov index, whereas γτ p 1 |dt| is associated with Berry's phase, see also [CU08]. Here the symbol class S(1) is the set of all smooth functions with bounded derivatives of any order, and p w (x, hD) is the semiclassical Weyl quantization, see Section 2. In the study of the Harper operator, this result has been generalized by Helffer-Sjöstrand [HS90, Corr. 3.1.2] to matrix-valued symbols that can be block-diagonalized to a scalar symbol exhibiting a potential well and a possibly matrix-valued remainder that is spectrally gapped from the well. In fact, let M ∈ S(1) be a self-adjoint matrix-valued symbol with M (x, ξ) ∈ C n×n and with one eigenvalue µ ∈ S(1), of algebraic multiplicity one, such that inf (x,ξ)∈T * R d(Spec(M (x, ξ))\µ(x, ξ), µ(x, ξ)) > 0. (1.2) Then there exists a unitary pseudodifferential operator U (x, hD) such that whereμ has principal symbol µ.
Our objective in this work is to study the case when the gap-condition (1.2) fails due to the existence of a degenerate potential well in the following sense: Definition 1.2. If P 0 (x, ξ) ∈ C 2×2 is positive semi-definite for all (x, ξ), and P 0 (x, ξ) = (a(ξ − ξ 0 ) 2 + b(x − x 0 ) 2 ) id 2 +O(|(x, ξ) − (x 0 , ξ 0 )| 3 ) (1.3) for some a, b > 0, then we say that P 0 has a potential well at (x 0 , ξ 0 ). If there is a neighborhood of (x 0 , ξ 0 ) in which P 0 has only one distinct eigenvalue of constant multiplicity 2 then we say that the well is non-degenerate, otherwise it is said to be degenerate. We say that the system P w (x, hD) has a (degenerate or non-degenerate) well at (x 0 , ξ 0 ) if the principal symbol σ 0 (P w (x, hD)) has a (degenerate or non-degenerate) well at (x 0 , ξ 0 ).
Note that if P 0 has a degenerate potential well at (x 0 , ξ 0 ) then the eigenvalues necessarily coalesce at (x 0 , ξ 0 ) since P 0 (x 0 , ξ 0 ) has only the eigenvalue λ(x 0 , ξ 0 ) = 0 with algebraic multiplicity 2. In particular, (1.2) is not satisfied. The main mathematical contribution of this article is then the construction of quasimodes for such symbols in Theorems 3.2 and 3.3, corresponding to operators on the real line and on the circle, respectively. After a conjugation in Proposition 3.1, the leading symbol of the matrix-valued operator is the direct sum of two harmonic oscillators T 0 (y, η) = diag(η 2 + ω 2 y 2 + µ 1 , η 2 + ω 2 y 2 + µ 2 ).
The subleading symbols then couple the two harmonic oscillators in a non-trivial way, preventing us from resorting to scalar methods.
In the case of µ 1 − µ 2 / ∈ (4Z + 2)ω we are then able to state an explicit construction of approximate eigenvalues and quasimodes in Theorem 3.5. For the case when this condition is not satisfied, we design a new phase space version of a quasimode construction used by Barry Simon [Sim83], to also obtain quasimodes and approximate eigenvalues in this case. This includes a phase space version of the IMS localization formula in Lemma 3.10 which may be of independent interest. To isolate the spectral contribution of the degenerate potential well at (x 0 , ξ 0 ) we use the notion of a massive Weyl operator by adding to P w (x, hD) the operator 1 − χ w (x, hD) where χ is a symbol having small support such that χ ≡ 1 near the well, see (3.14). Since the quasimodes have semiclassical wavefront set confined to {(x 0 , ξ 0 )}, the difference between P w (x, hD) and its massive counterpart acting on the quasimodes is O(h ∞ ) as h → 0.
In the second part of the paper we then apply these results to the models of Timmel and Mele [TM20] of one-dimensional moiré structures discussed above. To describe our results in more detail we first need to introduce the models. Before doing so we wish to mention that our study of quasimodes is inherently connected with flat bands, which appear for certain 2D twisted moiré materials. It is a classical result that a flat band corresponds to an eigenvalue of infinite multiplicity. Our construction of quasimodes shows that while there are no perfectly bands for the one-dimensional models considered in this article, there still exist almost flat bands in the corresponding chiral limits, while they are absent in the anti-chiral limits, see the discussion in §1.3 together with Figures 2 and 3. Spectral aspects of these models will be discussed in the forthcoming article [BGW22].
1.1. Model Hamiltonian. We start by introducing the tight-binding (i.e., discrete) model of one-dimensional moiré structures. Let ψ = (ψ n ) ∞ n=−∞ be a vector in 2 (Z). The Harper model for strained bilayer graphene [TM20] is defined as the action and k ⊥ ∈ R is the quasimomentum in the orthogonal periodic direction. Here, L is the length of a unit period of the moiré pattern.
The kinetic part is the discrete Dirac operator which, for γ 15 = diag(σ 1 , σ 1 ) and γ 25 = diag(σ 2 , σ 2 ), is defined as The honeycomb lattice consists of two types of atoms per fundamental domain, denoted A and B, respectively, corresponding to blue and red (alternatively green and black) nodes in Figure 1. The full potential V w is defined in terms of an anti-chiral potential (ac), describing the interaction between atoms A/A, B/B of the honeycomb lattice, and a chiral potential (c), describing the interaction between atoms A/B, B/A of the honeycomb lattice, of the form where U ac (x) = 1 + 2 cos(2πx) and U ± c (x) = 1 − cos(2πx) ± √ 3 sin(2πx). Then To use the general framework developed for operators with degenerate potential wells we note that the discrete operator (1.4) is unitarily equivalent to the pseudodifferential operator acting on L 2 (S), S = R/Z, see Lemma 2.1. Here, the semiclassical parameter is defined in terms of the moiré length via h = (2πL) −1 , i.e., we are concerned with the limit of large moiré lengths L 1. In the chiral limit (w 0 = 0), conjugating H ΨDO (w) by a unitary matrix leads to a system on off-diagonal block form, thus effectively reducing the spectral analysis to a 2 × 2 system, see Lemma 4.7. After locating the degenerate potential wells in Proposition 4.11 we then apply the results of Theorem 3.3 to obtain approximate eigenvalues and quasimodes to any order. We can do this for each of the degenerate wells, which appear periodically with period 1 in the fiber direction of phase space. In particular we show that near zero energy H ΨDO has approximate eigenvalues of infinite multiplicity given by see Corollary 4.12.
1.2. Effective Hamiltonian. We shall also consider an effective low-energy model for a moiré superlattice with layer-antisymmetric strain introduced in [TM20]. After a rescaling x/L → x, where L is the moiré length, the model is described by the semiclassical operator with semiclassical parameter h = 1/L, acting on L 2 (R). Here k ⊥ is the quasimomentum in the orthogonal periodic direction, and U (x) = U ac (x) = 1 + 2 cos(2πx) and 3 sin(2πx) as before. The kinetic differential operator is essentially a linearization in ξ of the symbol associated with the discrete model. We denote the symbol of the chiral Hamiltonian, when w 0 = 0, by H c and the symbol of the anti-chiral Hamiltonian, when w 1 = 0, by H ac , respectively.
Since the potential in H is 1-periodic in x, we can use the standard Bloch-Floquet transform to equivalently study the spectrum of H w (k x ) = H w (x, hD; k x ) on L 2 (S), (1.7) and Spec Spec(H w (k x )). (1.8) Then both H ΨDO and H(k x ) act on L 2 (S) which allows for a unified treatment of both models. Note that k x = O(h) so it does not contribute to the principal symbol of H w (k x ).
Similar to H ΨDO (w) we find in the chiral limit w 0 = 0 that conjugating H w (k x ) by a unitary matrix leads to a system on off-diagonal block form, see Lemma 4.1. We locate the degenerate potential wells in Proposition 4.3 and apply Theorem 3.3 to obtain approximate eigenvalues and quasimodes to any order. The approximate eigenvalues of H w (k x ) are independent of k x (see Remark 4.6), which by (1.8) leads to approximate eigenvalues of H w of infinite multiplicity. Near zero energy these are given by see Corollary 4.5.
1.3. Approximately flat bands. Existence of approximate eigenvalues for H w (k x ) in the chiral limit, as described in the previous section, results in the bands close to zero energy being almost flat, where we by a band mean an eigenvalue of H w (k x ) as a function of k x . This is illustrated in the left panel of Figure 2.
The corresponding notion of bands also exists for the discrete Hamiltonian (1.4). To see this, note that there is a version of Bloch-Floquet theory also for this model. Indeed, let L = p/q for positive integers p and q. then H TB commutes with translations τ q ψ n = ψ n−q . The Bloch transform (ψ n ) n∈Z → (φ n (k x )) n∈Z is then defined for k x ∈ R/(2πZ) by φ n (k x ) := m∈Z ψ n−qm e imkx with n ∈ {1, ..., q}, and the Floquet transformed Hamiltonian H TB (k x ) takes the form with matrices U ±,q = diag(U ± (ip/q)) 1≤i≤q , U q = diag(U (ip/q)) 1≤i≤q , and τ = 0 1 0 0 .
In particular, Spec(H TB (k x )). (1.9) In the left panel of Figure 3 the correpsonding bands close to zero energy are shown, some of which are indeed almost flat. Outline of the article. In Section 2, we briefly recall relevant background on semiclassical pseudodifferential operators. Section 3 contains our analysis of the spectral asymptotics for systems exhibiting a potential well. In Section 4, we then apply the spectral asymptotics derived in the previous section to the chiral Hamiltonian of the pseudodifferential Harper model (1.6) and of the low-energy model (1.7). The article also contains an appendix which consists of Section A where we prove auxiliary results used in the proofs of Section 3, and Section B where we for comparison discuss the anti-chiral limits of models (1.6) and (1.7). In the former model, there are various quasimodes at potential wells located at different energy levels, but not necessarily at zero, see Theorem B.2. The gap-condition (1.2) fails, but the operator is diagonalizable so the scalar results of Theorem 1.1 apply directly. In the latter model, there are no wells at all, see Remark B.3. The bands near zero energy of the anti-chiral limits of each corresponding model are shown in the right panels of Figures 2 and 3, none of which are almost flat.

Semiclassical pseudodifferential operators
Notation. We denote by H m (R n ) the Sobolev space of order m. The Pauli matrices are denoted by σ i for i ∈ {1, 2, 3}. Recall that the Kohn-Nirenberg symbol class S m (R× By S m,k δ we denote the class of symbols p such that We let Ψ m (R) and Ψ m,k δ (R) denote the corresponding class of semiclassical operators, and recall that if a w ∈ Ψ m,k The class S m (T * S) is identified with the subset of S m (R × R) consisting of the functions which are 1-periodic in x. We shall also need the symbol classes S(m) where m is an order function of the type m(y, η) = (1 + |y| 2 + |η| 2 ) ν/2 for some ν ≥ 0, consisting of a ∈ C ∞ (T * R) such that |∂ α η ∂ β y a(y, η)| ≤ C αβ m(y, η) for all α, β ∈ N 0 . For such m we usually just write S( (y, η) ν ), where we use the notation t = (1 + |t| 2 ) 1 2 for t ∈ R. All these symbol classes generalize in the natural way to n × m systems a ∈ C ∞ (R × R; C n×m ) and we shall not emphasize the size C n×m in the notation.
2.1. Pseudodifferential calculus on S. In this subsection, we provide the relevant background on semiclassical operators on S (see [Zwo12,Section 5.3] for a detailed exposition). Let S = R/Z and identify S with the fundamental domain [0, 1). A function u ∈ L 2 (S) is identified with a periodic function on R with period 1. For a smooth function a(x, ξ) which is 1-periodic in x (and belongs to some appropriate symbol class) we can define a semiclassical operator A(h) = a(x, hD) through the standard quantization of a. (We write a(x, hD) instead of a(x, hD x ) for brevity if there is no risk of confusion.) Then A(h) acts on 1-periodic functions via interpreted in the weak sense. It follows that A(h)u(x) is 1-periodic, and if, say, a ∈ S(1), then A(h) : L 2 (S) → L 2 (S) is bounded.
Using the standard quantization above, we may express the action of A(h) in terms of Fourier coefficients: If u is 1-periodic write u(y) = n∈Z e 2πiny u n where u n = 1 0 e −2πiny u(y) dy. Inserting this into the definition of A(h)u(x) we obtain Remark. If we instead use the Weyl quantization, defined for a symbol a(x, ξ) as then a w (x, hD)u(x) is still 1-periodic, but formula (2.2) needs to be altered and becomes more involved in general. However, in the special case that a is a linear combination of functions that only depend on either x or ξ, so that a(x, ξ) = a 0 (x) + a 1 (ξ), it is easy to see that we similarly get for such operators. (Using the correspondence between different quantizations we have that a w (x, hD)u(x) = n∈Z e 2πinx (e i 2 hDxD ξ a)(x, 2πnh)u n in the general case.) We now show that H TB is unitarily equivalent to the semiclassical pseudodifferential operator H ΨDO (w) in (1.6).
Lemma 2.1. Let H ΨDO (w) be as in (1.6) with h = (2πL) −1 and set b(x, ξ) = 2t(k ⊥ ) cos(2πξ) + t 0 + V w (x). (2.4) Then the discrete operator (1.4) is unitarily equivalent to the pseudodifferential oper- Proof. Let ψ = (ψ n ) ∞ n=−∞ ∈ 2 (Z) and set Ψ(x) = n∈Z e 2πinx ψ n so that ψ n is the n:th Fourier coefficient of Ψ. With ψ = (ψ k ) k∈Z , (1.4) then gives rise to the action which we rewrite as where h = (2πL) −1 . In view of the remark above we may interpret this as the action of the semiclassical operator defined as the Weyl quantization A(w) = a w (x, hD) of the symbol a(x, ξ) = 2t(k ⊥ ) cos(2πx) where we have suppressed the dependence on w = (w 0 , w 1 ) for simplicity. Now let b be given by (2.4) and observe that a(x, [Zwo12,Theorem 4.9]. Hence, the discrete model (1.4) is unitarily equivalent to b w (x, hD). By the definition of V w we see that b is a bounded smooth function which implies that b w (x, hD) is bounded on L 2 (S), and the lemma follows.

Quasimodes near degenerate wells
Let P w (x, hD) be a 2 × 2 system of self-adjoint semiclassical operators with matrix valued symbol P ∈ C ∞ (T * S). As usual we identify functions on T * S with functions on T * R that are 1-periodic in the base variable. We shall assume that P has an expansion P ∼ ∞ k=0 h k P k where either P k ∈ S(1) for k ≥ 0 or P k ∈ S 2−k (T * S) for k ≥ 0. (The former implies that P ∈ S(1) and the latter that P ∈ S 2 (T * S).) In both The purpose of this section is to study quasimodes of P w when P w has a degenerate potential well in the sense of Definition 1.2. By multiplying P 0 by a scalar if necessary we may assume that a = 1 in (1.3). We will also assume that x 0 = 0, but keep ξ 0 for now to illustrate its effects. We thus assume that P w has a degenerate potential well at (0, ξ 0 ). In addition we shall also assume that if P ∈ S 2 (T * S) then where V is a real-valued function such that V (x) = Cx 2 + O(x 3 ) for some C > 0.
Note that if P 0 ∈ S(1) is positive semi-definite then the sharp Gårding inequality for systems (see Hörmander [Hör79, Theorem 6.8] gives (P w 0 (x, hD)u, u) ≥ −Ch u 2 which implies that also If u is microlocally small in a neighborhood of the well at (0, ξ 0 ), this will lead to a sufficiently good positive lower bound for our applications. Assumption (3.1) will allow us to argue in a similar way when P ∈ S 2 (T * S).
3.1. Normal form. If P 0 has a degenerate potential well then the gap condition (1.2) is clearly violated. We shall therefore have to study the spectrum of P w by another approach, the first step of which is to obtain a suitable normal form.
We begin with a general discussion and first recall the standard rescaling, so suppose that p ∈ S m (T * S) and make the change of variables y = h − 1 2 x. Then p w (x, hD x )u(x) = p w h (y, D y )v(y) where u(x) = v(y) and the Weyl quantization of p h (y, η) = p(h 1 2 y, h 1 2 η) is understood to be non-semiclassical, i.e., In other words, if and if m ≥ 0 then standard calculus shows that the right-hand side is bounded by which cannot vanish since this would imply that j/m = −h 1 2 |η|. Since the expression in parenthesis tends to j > m ≥ 0 as h → 0 we conclude that f is increasing and therefore f (h) ≤ f (1) = (1 + |η|) m−j when 0 < h < 1.
If p ∈ S(1) then the same calculations show that p h ∈ S(1) uniformly for 0 < h < 1. We then have the following normal form.
Note in particular that T 0 is just the symbol of a direct sum of harmonic oscillators which are perturbed and coupled to one another through terms appearing in T j , R k .
Proof. First note that for a symbol p(x, ξ) we have after the symplectic change of variables ζ = ξ − ξ 0 that Applying this to P w (x, hD) we find that . Now Taylor expand T (y, η) near y = 0, η = 0. In view of Definition 1.2 we get, both when P ∈ S(1) and when P ∈ S 2 (T * S), Taylor's formula also on P 1 ∈ S(1) gives We then continue in this way to Taylor expand h 2 P 2 , . . . , h k+1 P k+1 , and since , the result follows by combining the expansions.

3.2.
Quasimodes. From now on we work with assumptions and notation as in Proposition 3.1. Since T is periodic in y with period h − 1 2 (although the terms T j , R k are not), we shall identify R/h − 1 2 Z with the fundamental domain I h = [− 1 2 h − 1 2 , 1 2 h − 1 2 ) and study approximate eigenvalues of T , viewed as a densely defined operator on since e iξ 0 •/ √ h is not periodic with period h − 1 2 in general, but T preserves periodicity: where the second identity follows from . The main objective of this section is to obtain approximate eigenvalues and quasimodes for P w (x, hD) on L 2 (S), which we will do by reducing it to the task of obtaining corresponding ones for T w (y, D) on L 2 (R).
Theorem 3.2. Let P and T be as in Proposition 3.1 and assume The proof of Theorem 3.2 makes up the bulk of Section 3. We note that the existence of approximate eigenvalues and quasimodes for P w (x, hD) on L 2 (S) is an immediate consequence.
where the remainder r (j) = r (j) 2 (h) ∈ S (R) has seminorms in S bounded uniformly in 0 < h < 1. Let us fix j and drop it from the notation. Set w(y) Since v ∈ S it follows that w ∈ C ∞ is periodic with period h − 1 2 , and The weighted pullback u = h − 1 4 γ * ( w) is 1-periodic and WF h (u) = {(0, ξ 0 )} ⊂ T * S, and we shall show that it also has the other properties stated in the theorem.
Since v ∈ S there is for any N > 0 a constant C > 0 such that where the second estimate follows from Peetre's inequality. It follows that 2 ) by the triangle inequality, and thus u L 2 (S) = 1 + O(h 1 2 ). Since r ∈ S uniformly in 0 < h < 1 we can apply the same arguments to the right-hand side of (3.9) and, in view of (3.3), obtain , and the proof is complete.
3.3. Explicit WKB construction. Before proving Theorem 3.2 in full generality we first discuss a special case for which there exists a rather explicit WKB construction. Recall from (3.8) the harmonic oscillator basis functions where H n is the n:th Hermite polynomial, which is even (odd) when n is even (odd).
Let C d denote the module of homogeneous polynomials of degree d in the ring C = C[y, η]. Let the polynomials of even and odd degree in C[y, η] be denoted by Lemma 3.4. Let v ∈ C ∞ be either even or odd, and let p ∈ C[y, η]. Then p w (y, D)v has the same parity as v when p ∈ P even , and opposite parity when p ∈ P odd .
Proof. By linearity it suffices to consider the case of a homogeneous polynomial p(y, η) = y n η k , where n + k is either even or odd, depending on if p ∈ P even or p ∈ P odd . Since p((y + s)/2, η) = 2 −n n j=0 n j y n−j η k s j we find that p w (y, D)v(y) is a linear combination of terms y n−j D k y (y j v(y)) where 0 ≤ j ≤ n. Since multiplication by y m changes parity if and only if m is odd, and differentiation D k y changes parity if and only if k is odd, it follows that y n−j D k y (y j v(y)) will have the same parity as v when n + k is even and opposite parity to v when n + k is odd.

Introduce the sets
A sym := p 11 p 12 p 12 p 22 : p jk ∈ P even and A asym := p 11 p 12 p 12 p 22 : p jk ∈ P odd .
be the symbol given by Proposition 3.1. By inspecting the proof of the proposition we see that we have T 2j−1 ∈ A asym and T 2j ∈ A sym for j ≥ 1. We then have the following theorem.
Theorem 3.5. Assume that for all ∈ N 0 , the Weyl symbol of T w (y, D) has an Then there exist approximate eigenvalues where u i are polynomials such that and Proof. The eigensystem for T w 0 (y, D) is given by with eigenvectors φ n,ω , 0 t and 0, φ m,ω t .
We therefore make the approximate eigenvalue and quasi-mode ansatz where we present the construction without loss of generality for u 1 0 (y) := (φ n,ω (y), 0) t rather than (0, φ m,ω (y)) t and choose λ 0 = (2n + 1)ω + µ 1 . Recall that the Hermite polynomial H n is an even polynomial if n is even and odd polynomial if n is odd, we may assume without loss of generality that n is odd. Iteratively constructing a WKB solution satisfying (3.10) is equivalent to successively solving In fact, as we will show the u j are polynomials so u j e −ωy 2 /2 ∈ S . Since the remaining terms in (3.10) are h 1+(i+j)/2 T w i u j e −ωy 2 /2 for 2 < i + j ≤ 4 together with h + 3 2 R w 2 ( 2 j=0 h j/2 u j e −ωy 2 /2 ), and since T i , R 2 ∈ S( (y, η) ν ), these terms are all in h + 3 2 S uniformly in 0 < h < 1 by assumption which gives an error of order O S (h + 3 2 ).
Step 1: The first step is to note that holds by assumption.
Step 2: We shall argue by induction as k runs through two consecutive integers. We first notice for k = 1 that To solve this for u 1 , the right-hand side must be orthogonal to ker(T w 0 − λ 0 ). We then observe that according to Lemma 3.4, applying (T w 1 ) j1 to φ n,ω changes the parity of that function, which implies that . From the assumptions on T 1 we see that in either case we can take As mentioned, (T w 1 ) j1 φ n,ω is even since n is odd. Multiplying by e −ωy 2 /2 and applying (T w 0 ) jj − λ 0 to both sides does not change the parity, so (u 1 ) j must also be even. It is easy to check that (u 1 ) j is a polynomial function for j = 1, 2, so u 1 e −ωy 2 /2 ∈ S . This follows as N i=0 a i y 2i e −ωy 2 /2 ; a i ∈ C and N i=0 a i y 2i+1 e −ωy 2 /2 ; a i ∈ C are invariant subspaces of T 0 for any N ∈ N.
where u 0 = (φ n,ω , 0) t and λ 1 = 0 by the previous steps. To solve the equation for (u 2 ) 1 , the right-hand side must be orthogonal to ker((T w 0 ) 11 − λ 0 ) = span{φ n,ω }, which means that λ 2 must satisfy With this choice of λ 2 we then get the solution Note that the expression in brackets is an odd function by Lemma 3.4, and as above we find that (u 2 ) 1 is an odd polynomial.
For (u 2 ) 2 we get the equation − (T w 2 ) 21 φ n,ω so by similar reasoning as above we get where (u 2 ) 2 is an odd polynomial.
Step 3: Now let k ∈ 2N−1 be arbitrary, and assume that λ i and u i have already been determined for 0 ≤ i < k such that for i odd we have λ i = 0 and where all terms on the right are even functions by the induction hypothesis and Lemma 3.4, with the exception of λ k u 0 e −ωy 2 /2 = λ k (φ n,ω , 0) t which is odd. To solve the equation for (u k ) 1 the right-hand side must be orthogonal to ker((T w 0 ) 11 −λ 0 ) = span{φ n,ω } which then gives λ k = 0 and which makes u k an even polynomial by the same arguments as before.
Step 4: Under the same hypothesis as in step 3, but now with k ∈ 2N, we define In analogy with the case k = 2, this allows us to define which makes u k an odd polynomial, and this closes the recurrence scheme.
Note that for the proof of Theorem 3.5 to work, the assumption that µ 1 − µ 2 / ∈ (4Z + 2)ω is crucial. In regards to applications to operators with degenerate wells appearing in the one-dimensional strained moiré lattices, this assumption is violated for both the pseudodifferential Harper model (1.6) and the low-energy model (1.7).
3.4. Low-lying spectral analysis. To prove Theorem 3.2 we must also construct approximate eigenvalues and quasimodes for T w (y, D) when µ 1 − µ 2 ∈ (4Z + 2)ω, i.e., when Theorem 3.5 does not apply. To do so we will adapt a technique of Barry Simon [Sim83]. The idea is to first show that the spectrum is stable in a certain sense, and then use this fact to obtain asymptotic expansions of the eigenvalues and eigenvectors, which by truncation gives approximate eigenvalues and quasimodes. We recall that we in this context regard T w (y, D) as an operator on R with dense domain in L 2 (R). It will be convenient to also be able to express the operator T w (y, D) in the variables x, ξ in order to make use of the semiclassical symbolic calculus. In view of Theorem 3.3 we may without loss of generality assume that ξ 0 = 0 in (3.4), so to avoid additional notation we will simply write and regard P w (x, hD) as an operator on R with dense domain in L 2 (R). In other words, we assume that the well is located at (0, ξ 0 ) = (0, 0) and drop the requirement that P w (x, hD) should act on periodic functions.
Hence, P w mass is microlocally elliptic away from (0, ξ 0 ) ∈ T * R, and it is easy to check that the approximate eigenvalues of order O(h) of P w which correspond to quasimodes microlocalized at (0, ξ 0 ) are precisely the approximate eigenvalues of order O(h) of the massive Weyl operator P w mass . By Definition 1.2 we may choose χ so that in the sense of semi-bounded operators. When P ∈ S 2 (T * S) we can also by (3.1) make sure that where 1 − χ(x, ξ) + V (x) ≥ min(1, Cx 2 ) for some C > 0.
We then let T w mass be the operator so that P w mass (x, hD) = γ * • T w mass (y, D) • (γ −1 ) * . In particular and P w mass (x, hD) and T w mass (y, D) have the same spectrum. Also, since χ ∈ C ∞ 0 (T * R) in (3.14) is independent of h, and G(y, η) = χ(h 1 2 y, h 1 2 η) by (3.17), we see that for some constant δ 0 .
Recall that ϕ n = ϕ n,ω , n ≥ 0, are the harmonic oscillator basis functions given by (3.8), where we omit ω to shorten notation. From (3.5), we notice that where e 1 n = (2n + 1)ω + µ 1 , e 2 n = (2n + 1)ω + µ 2 , n ∈ N 0 . Let (e n ) n∈N denote a monotonically increasing ordering of the two sets of eigenvalues. The spectrum of T w mass is covered in the following sense: Theorem 3.6. Let λ n (h) be the n:th eigenvalue, counting multiplicity, of T w mass and let e n be the n:th eigenvalue, counting multiplicity, of T w 0 , viewed as densely defined operators on L 2 (R). Then for n fixed and h small, T w mass has at least n eigenvalues and In view of Theorem 3.5 we may without loss of generality assume that (µ 1 , µ 2 ) = (−ω, ω) in the sequel. The eigenvalues (e n ) n∈N of T w 0 are then given by e 2n+k = e k n = (2n + 1)ω + (−1) k ω, n ∈ N 0 , k = 1, 2. (3.20) In particular, e 1 = 0 is a simple eigenvalue while e 2m = e 2m+1 for m ≥ 1.
Theorem 3.6 has an analog for scalar self-adjoint Schrödinger operators on the line, and as mentioned we will adapt a proof by Simon [Sim83, Theorem 1.1] to our situation. One difference is that we shall use a microlocal cutoff function instead of a local one which allows applications to operators P w with bounded symbols (such as the pseudodifferential Harper model) when the domain of P w is all of L 2 , while the domains of the operators T w 0 and R w 0 in the expansion T w = hT w 0 + h 3/2 R w 0 are strictly smaller. To this end, fix J ∈ C ∞ 0 (R) with 0 ≤ J ≤ 1 and J(y) = 1 (resp. 0) if |y| ≤ 1 (resp. |y| ≥ 2), and let J 1 (y, η; h) = J(h 1/10 y)J(h 1/10 η). (3.21) Lemma 3.7. If R 0 ∈ S( (y, η) 3 ) uniformly for 0 < h < 1 then Proof. Since |y|, |η| ≤ 2h −1/10 on the support of J 1 we have is O(h 6/5 ) by the calculus.
To shorten notation below we will always understand G w and J w 1 to mean nonsemiclassical Weyl quantizations in the variable y (as in, e.g., G w (y, D y )), while χ w is understood as the semiclassical Weyl quantization χ w (x, hD x ) in the variable x.
We begin by establishing an upper bound.
Proposition 3.8. With notation and assumptions as in Theorem 3.6, for n fixed and h small, T w mass has at least n eigenvalues and lim h→0 + λ n (h)/h ≤ e n . (1, 0) t , n ∈ N 0 , k = 1, 2. (3.23) We claim that where δ nm is the Kronecker delta. Clearly (ψ 2n , ψ 2m+1 ) = 0 for all n and m, and for pairings where both indices are either even or odd the claim follows from the definitions of ϕ n and J 1 . In fact, for 2n + k, 2m + k both even or both odd we get where the first term on the right equals δ nm . We have where |h 1/10 η| ≥ 1 if 1 − J 1 ((y + s)/2, η) ≡ 0 due to (3.21) and the definition of J. A standard integration by parts using ih 1/10 (h 1/10 η) −1 ∂ s e i(y−s)η = e i(y−s)η then shows that We also claim that h −1 (T w mass ψ n , ψ m ) = e n (ψ n , ψ m ) + O nm (h 1/5 ). (3.25) We prove this when n, m are both odd (the case when they are both even is similar and when one is even and one is odd it is trivial). We have where the second term on the right is O(h 1/5 ) by Lemma 3.7. To analyze the first term on the right we note that . Since J w 1 T w 0 (ϕ n , 0) t = e 1 n ψ 2n+1 = e 2n+1 ψ 2n+1 by (3.20) we find, by using the Weyl calculus to compute the commutators, that where it is easy to see that the last four terms on the right are O nm (h ∞ ) since ∂ k y J 1 (y, η) = 0 when |y| ≤ h −1/10 and ∂ k η J 1 (y, η) = 0 when |η| ≤ h −1/10 for k ≥ 1. Hence, By arguments similar to those used to obtain (3.24) we see that we may replace T w by T w mass = T w + (1 − G w ) id C 2×2 without changing the right-hand side, which gives (3.25). In fact, the symbol 1 − G is bounded on R and on supp(1 − G) we have |h 1/2 η| ≥ const. by (3.18) so an integration by parts using ih for k ≥ 0. This proves the claim.
We now turn to a lower bound, which combined with Proposition 3.8 gives Theorem 3.6. Proposition 3.9. With notation and assumptions as in Theorem 3.6, we have For the proof it will be more convenient to work with the unitarily equivalent P w mass (x, hD) rather than T w mass (y, D), where we will use a pseudodifferential version of the IMS localization formula (so dubbed by Barry Simon [Sim83] after Ismagilov, Morgan, Simon and I. M. Sigal). To state it we let With respect to the standard rescaling we have on operator level that where χ 1 is supported for (x, ξ) such that |x|, |ξ| ≤ 2h 2/5 . 1 We observe that χ 1 ∈ S 0,−∞ 2/5 (T * R), so χ w 1 (x, hD) ∈ Ψ 0,−∞ 2/5 (R). Next, define χ 0 ∈ C ∞ (T * R) by the condition that (χ 0 (x, ξ)) 2 + (χ 1 (x, ξ)) 2 = 1. (3.28) 1 The precise value 2/5 of the exponent is not important -what is needed is that χ 1 is supported in a ball or radius ∼ h ν for some 1 3 < ν < 1 2 .
Lemma 3.10 (IMS). Let χ 0 and χ 1 be as above. Then there are X w 0 (x, hD) ∈ Ψ 0,0 2/5 (R) and X w 1 (x, hD) ∈ Ψ 0,−∞ 2/5 (R) such that X j = χ j modulo S −∞,−∞ (T * R) and As with χ w we shall, to shorten notation, always understand X w 0 and X w 1 to mean semiclassical Weyl quantizations in the variable x (as in, e.g., X w 0 (x, hD)). Recall from (3.5) and (3.13) that P w = γ * • (hT w 0 + h 3/2 R w 0 ) • (γ −1 ) * and let us write for n ∈ N 0 and k = 1, 2. By Lemma 3.10 we then have The middle term is O(h 6/5 ), too: Lemma 3.11. For X w 1 as in Lemma 3.10 and H w 0 as above, we have X w We postpone the proofs of Lemmas 3.10 and 3.11 to Appendix A. With these preparations at hand, we are now able to give the proof of the lower bound on the eigenvalue asymptotics.
Proof of Proposition 3.9. We first assume that n ≥ 2. It then suffices to prove the proposition when n is even. Indeed, suppose it holds for even n, and recall from (3.20) that e 2m = e 2m+1 for m ≥ 1. Then which proves the claim.
We will prove the statement in the proposition with n replaced by n + 1, so suppose therefore that n is odd, and fix a number e n < r < e n+1 and let P n be the projection onto the eigenvalues below rh for H w 0 so that P n has rank n. It is then easy to see that (3.31) Also, by (3.15) the symbol of P w mass is semi-bounded from below by Ch 4/5 + O(h) on supp χ 0 , and since X w 0 = χ w 0 mod Ψ −∞,−∞ an application of the sharp Gårding inequality for systems (see Hörmander [Hör79, Theorem 6.8]) gives for h small if P ∈ S(1). If P ∈ S 2 (T * S), we use (3.16) and the sharp Gårding inequality in the scalar case instead to obtain (3.32). Summing up using (3.30) and (X w 0 ) 2 + (X w 1 ) 2 = 1 mod Ψ −∞,−∞ we get P w mass ≥ hr + R + o(h) where R = X w 1 (H w 0 − hr)P n X w 1 has rank at most n. We can then pick ψ in the span of the first n + 1 eigenvectors of P w mass with ψ = 1 and ψ ∈ ker R. Then λ n+1 ≥ (P w mass ψ, ψ) ≥ hr + o(h) and, since r ∈ (e n , e n+1 ) was arbitrary, this shows that lim h→0 + λ n+1 /h ≥ e n+1 .
It remains to consider n = 1. However, inspecting the arguments above we see that if we fix r < e 1 = 0 then (3.31)-(3.32) hold trivially, and R = 0, so λ 1 ≥ hr + o(h) from which the result easily follows.
By combining the ideas used in the proof of Theorem 3.3 with the method used to prove Theorem 3.6 it is possible to obtain a stability result for the eigenvalues of P w mass also in the periodic setting when P w is viewed as a densely defined operator on L 2 (S). Since it might be of independent interest we state such a result here but leave the proof to the interested reader.
Theorem 3.12. Let Λ n (h) be the n:th eigenvalue, counting multiplicity, of P w mass viewed as a densely defined operator on L 2 (S). Let e n be the n:th eigenvalue, counting multiplicity, of T w 0 viewed as a densely defined operator on L 2 (R). Then for n fixed and h small, P w mass has at least n eigenvalues and lim h→0 + Λ n (h)/h = e n .
3.5. Asymptotic series. We now use Theorem 3.6 to prove asymptotic expansions of eigenvalues and eigenvectors of T w (y, D) on R. We first consider the massive operator T w mass = T w (y, D) + 1 − G w (y, D). We assume that the Weyl symbol T satisfies the conditions in Proposition 3.1. In particular, writing z = (y, η) we then have Recall that T 0 (z) = (η 2 + ω 2 y 2 ) id 2 +ω diag(−1, 1) and let e n be the n:th eigenvalue of T w 0 on R. Choose ε (depending only on ω) so that for each m, either e m = e n or |e m − e n | ≥ ε. Let (1, 0) t in accordance with (3.23). We will use a version of [Sim83, Theorem 2.3] proved in the same way which we state here using our notation.
Note that under the assumption (3.20) we have n = 1 in the theorem if e n is simple. For other values of µ 1 , µ 2 in (3.6) this is of course not necessarily true so we have written the statement in this way to emphasize the assumption that e n is simple.
Proof. Let ϕ ∈ L 2 (R; C 2 ) be the eigenfunction corresponding to the simple eigenvalue e n of T w 0 on R, i.e., ϕ :=φ n in the notation above. Then B ε (e n ), with ε independent of h, contains precisely one eigenvalue and is the projection onto an eigenfunction v(h) = Π(h)ϕ/ ϕ, Π(h)ϕ corresponding to a single eigenvalue of T w mass /h for h small enough, see Theorem 3.12. The denominator is non-vanishing by (3.35). Due to our assumptions on T we clearly have an asymptotic expansion of T w ϕ, so in view of the expressions for Π(h), v(h) and λ(h) we see that if we obtain an asymptotic expansion for (T w mass /h − ζ) −1 ϕ which is uniform in ζ then we also get asymptotic expansions for v(h) and hλ(h). (The contribution of h −1 (1 − G w )ϕ to λ(h) is negligible by (3.26).) We then use the standard geometric series As above we find by (3.26) that the terms involving 1 − G w in ψ 1 , . . . , ψ m and r m all belong to S by the calculus and are O(h ∞ ) there uniformly for h small, i.e., we can redefine ψ 1 , . . . , ψ m and r m so that (3.36) holds with Let b ≥ m + 1, then from (3.33) follows that Z w ( z −b ) is bounded on L 2 (R) with norm of size √ h. We then rewrite r m using Z w k := Z w ( z −kb ) w as is a bounded (even compact) operator on L 2 (R) by the calculus, with norm of size √ h. Hence, r m = O S (h (m+1)/2 ). If we then define ψ i just like ψ i but with Z w replaced by Q w m , we get Then, by comparing the difference of ψ i and ψ i we find by using that Z − Q m = R m that ψ i − ψ i is a finite sum of terms of the form where A w k = R w m for at least one 1 ≤ k ≤ i. By (3.31), each term above defines an element in S with L 2 norm of size h (m+1)/2 since R m is order h (m+1)/2 . Hence, we obtain the desired expansions of v n (h).
We now turn to degenerate eigenvalues.
Theorem 3.15. Let e n be an eigenvalue of T w 0 on R of multiplicity 2 with e n = e n+1 and let λ n (h), λ n+1 (h) be the eigenvalues of T w mass on R which, when divided by h, tend to e n . Then for j ∈ {n, n + 1} there exists an asymptotic expansion with coefficients a (i) Proof. Letφ j be as above so they span the eigenspaces of T w 0 on R, and let Π n (h) be the projection (3.34) onto the span of all eigenvectors of T w mass /h associated to eigenvalues approaching e n as h → 0. Since e n = e n+1 we thus have Π n (h) = Π n+1 (h). By (3.35) together with the proof of Theorem 3.14 we see that 2 ) for i, j ∈ {n, n + 1}. In particular, Π n (h)φ n and Π n (h)φ n+1 are linearly independent for h small. We then let ∆ − 1 2 be the square root of the inverse of ∆ = (∆ ij ) 2 i,j=1 which exists for h small. Since ∆ ij has an asymptotic expansions by the proof of Theorem 3.14, and since the eigenvalues of a Hermitian matrix have asymptotic expansions in h provided that the elements of the matrix do (see [Sim83,Lemma 5.2]), it follows that ∆ − 1 2 also has an asymptotic expansion. (We diagonalize ∆ −1 = U EU * with E diagonal consisting of the eigenvalues of ∆ −1 which are positive for h small. has an asymptotic expansion by the proof of Theorem 3.14. Hence, C(h) has an asymptotic expansion, and thus the eigenvalues of C(h) do as well, which we claim are precisely λ n and λ n+1 .
We consider two cases for h small but fixed: 1) λ n (h) = λ n+1 (h). Then H = λ n ∆ so C(h) = λ n (h) id 2 which proves the claim in this case.
2) λ n (h) = λ n+1 (h). Since Π n (h) has rank 2 we can find orthonormal v n (h), v n+1 (h) such that Ran Π n = span{v n , v n+1 } and T w for some invertible matrix D. It is straightforward to check that id 2 = D∆D * so (3.37) Now observe that So U is unitary and therefore U * = U −1 , which in view of (3.37) means that C(h) has eigenvalues λ n , λ n+1 .
Theorem 3.16. Let e n be an eigenvalue of T w 0 on R of multiplicity 2 with e n = e n+1 and let λ n (h), λ n+1 (h) be the eigenvalues of T w mass on R which, when divided by h, tend to e n . If v n (h) and v n+1 (h) are the corresponding eigenvectors of T w mass then v n , v n+1 ∈ S and have asymptotic series in √ h to any order, and each term in the expansions is a polynomial in y times e −ωy 2 /2 . Proof. We assume that µ 1 − µ 2 = −2ω, which, since e n = e n+1 , means that n is even so n = 2n + 2 for some integer n . Withφ n =φ 2n +2 = ϕ n (0, 1) t andφ n+1 = ϕ 2(n +1)+1 = ϕ n +1 (1, 0) t we then have T w 0φ n+j = e n+jφn+j , j = 0, 1, so {φ n ,φ n+1 } is an orthonormal basis for the eigenspace of T w 0 associated to the double eigenvalue e n .
If Π n is given by (3.34) then Π n = Π n+1 and as above we have that Π nφn and Π nφn+1 are linearly independent for h small.
We consider two cases: 1) The asymptotic series for λ n (h) and λ n+1 (h) provided by Theorem 3.15 are not identical. We can then define spectral projections P n and P n+1 of rank 1 onto the span of v n (h) and v n+1 (h), respectively. Since Π n is the projection onto the span of {v n (h), v n+1 (h)} and Π nφn and Π nφn+1 are linearly independent for h small, we must have P nφi = 0 and P n+1φj = 0 for some i, j ∈ {n, n+1}. Indeed, if for example P nφn = P nφn+1 = 0 then Π nφn = P n+1φn and Π nφn+1 = P n+1φn+1 are linearly dependent, a contradiction. For the same reason we cannot have P n+1φn = P n+1φn+1 = 0. Hence, for j = n, n + 1 we have v j ∈ span{P j ϕ j } for some eigenvector ϕ j ∈ {φ n ,φ n+1 } of T w 0 . We now obtain an asymptotic expansion of P j ϕ j by arguments similar to those in the proof of Theorem 3.14, which gives the desired expansion of v j .
2) The asymptotic series for λ n (h) and λ n+1 (h) are identical. In this case any vector in Ran Π n is an approximate eigenvector of both λ n (h) and λ n+1 (h) to any order. Since Π nφn and Π nφn+1 is a basis for Ran Π n for h small, and both Π nφn and Π nφn+1 have asymptotic expansions by the proof of Theorem 3.14 we obtain the expansion in this case as well.
Finally we note by the above that the eigenvectors are spectral projections ofφ j , so as in the proof of Theorem 3.14 we find that v j ∈ S and each term in the expansion is a polynomial times e −ωy 2 /2 .
We can now give Proof of Theorem 3.2. We assume that µ j = (−1) j ω as before. For n ∈ N 0 and k ∈ {1, 2} let e 2n+k = e k n be the eigenvalues of T w 0 arranged as in (3.20), with e 1 simple and e n double for n ≥ 2. Let λ 1 (h) be the eigenvalue of T w mass tending, after division by h, to e 1 as h → 0, and let v 1 ∈ S be the corresponding eigenvector. Also let λ 2n , λ 2n+1 be the eigenvalues of T w mass tending, after division by h, to e 2n = e 2n+1 as h → 0, and let v 2n , v 2n+1 ∈ S be the corresponding eigenvectors. Now, as in the proof of (3.26) we find that T w contains a factor h we then find by Theorems 3.14-3.16 that where v (k) j (n) is a polynomials times e −ωy 2 /2 , and 0 (n) and v (k) 0 (n) having leading asymptotics as in the statement. Hence, i (n), and the result follows.

Wells in chiral strained moiré lattices
Here we shall apply the results of Section 3 to the low-energy model H w c (k x ) in (1.7) and the pseudodifferential operator H ΨDO in (1.6). To do so we must first show that each model can be written in the appropriate normal form, which we will do by verifying the assumptions in Proposition 3.1.

4.1.
Wells for the chiral low-energy Hamiltonian. Let us start with H w c (k x ). In the chiral limit w 0 = 0 we get with H(k x ) understood to be a densely defined operator on L 2 (S).
Lemma 4.1. Consider the chiral limit w = (0, w 1 ). Then H w c (k x ) in (4.1) is unitarily equivalent to the system Proof. Let Then U U * = U * U = id 2 and U U * = U * U = id 4 . By first conjugating by diag (1, σ 1 , 1) and then conjugating by U we see that H is equivalent to L c in (4.2).
as claimed.
If λ is an eigenvalue of L c then clearly λ 2 is an eigenvalue of (L 2 c ) 11 . In view of the converse correspondence between eigenvalues of (L 2 c ) 11 and L c given by Lemma 4.1 we can therefore study the spectrum of (L 2 c ) 11 in place of L c . We then use the following description of the Weyl symbol of L 2 c . Lemma 4.2. Let L c be given by (4.2). Then the square L 2 c is the Weyl quantization of the symbol σ(L 2 c ) = σ 0 (L 2 c ) + hσ 1 (L 2 c ) + h 2 σ 2 (L 2 c ) where the principal symbol σ 0 (L 2 c ) has block-diagonal form where f (x) = w 1 (1 − cos(2πx)) and g(x) = w 1 √ 3 sin(2πx), and with lower order terms By using the Weyl calculus and noting that k x = O(h) by (1.8) it is now straightforward to check that L 2 c is an operator having Weyl symbol as described in the statement.
Proof. That the symbol of (L 2 c ) 11 has the stated expansion follows from Lemma 4.2. It is also clear that at (0, 0) the subprincipal symbol has the stated diagonal from, so we only need to show that the principal symbol P 11 has a degenerate well at (0, 0), where P 11 is as in Lemma 4.2. It is straightforward to check that P 11 (x, ξ) is positive semi-definite for all (x, ξ) ∈ T * S, and by Taylor's formula we have, with f (x) = w 1 (1 − cos(2πx)) and g(x) = w 1 √ 3 sin(2πx), that . This gives the result.
We also record that the chiral Hamiltonian satisfies (3.1).
As a consequence of Theorem 3.3 we obtain quasimodes for the chiral low energy model.
Fix j ∈ {1, 2} and n ∈ N 0 and omit them from the notation. In the notation of Lemma 4.1 with ψ = (ψ 1 , ψ 2 ) t = (u, ±λ − 1 2 D * c u) t we then have D c D * c = (L 2 c ) 11 and the correspondence where the last identity follows from (L 2 Remark 4.6. Let H w c (k x ) be given by (4.1) with k ⊥ = 0. Since k x ∈ [0, 2πh) we may write k x = hξ 0 with ξ 0 ∈ [0, 2π). If we make the symplectic change of variables (x, ζ) = (x, ξ + k x ) = (x, ξ + hξ 0 ) then the proof of Proposition 3.1 shows that . This shows that the approximate eigenvalues of H w c (k x ) in Corollary 4.5 are independent of k x . Note also that multiplying by e ixξ 0 does not affect the wavefront set of the associated quasimodes, since WF h (e ixξ 0 ) = R n × {0}, see [Zwo12,Section 8.4].

4.2.
Wells for the chiral Harper model. We now establish the existence of degenerate wells for the operator H ΨDO in (1.6). We shall then use the following result, analogous to Lemma 4.1 and with identical proof.
Lemma 4.7. Let b(x, ξ) be given by (2.4). In the chiral limit, b w (x, hD) is unitarily equivalent to a Hamiltonian on off-diagonal block form, where the symbol of D c is given by Let H c be as in Lemma 4.7. Using the lemma it is easy to see that Thus in case that k ⊥ / ∈ 1 2 Z, then for the imaginary part to vanish we require ξ ∈ 1 2 Z+ 1 4 . For then the real part to vanish as well, we require 1 = w 2 1 U + U − (x), which, since the range of U + U − is [−2, 4], admits a solution once w 1 ≥ 1/2.
Conversely, for k ⊥ ∈ Z there is the special solution x = 0, ξ = ± 1 3 + Z which exists independent of w 1 , together with the level set (4.6) For k ⊥ = 1 2 + Z there is the special solution x = 0, ξ = ± 1 6 + Z together with the level set (4.7) As we shall see, there are no degenerate wells unless k ⊥ ≡ 0 or k ⊥ ≡ 1 2 mod Z in which case there are degenerate wells precisely at these special solutions.
By Lemma 4.8 we have ) 2 ) so the eigenvalues of σ 0 (H 2 c ) jj for j = 1, 2 are given by has vanishing imaginary part only when ξ = 1 4 + 1 2 Z in which case Υ k ⊥ (ξ) ≡ 1 for all ξ. Thus, λ + (x, ξ) never vanishes when k ⊥ / ∈ 1 2 Z. If k ⊥ ∈ Z then λ + (x, ξ) = 0 precisely when x = 0 and ξ = ± 1 3 mod Z. By the analysis preceding the lemma, the characteristic set of λ − is the level set (4.6). Since w 2 1 U + U − = f 2 − g 2 we find in view of (4.5) that this level set has several connected components in R 2 /Z 2 : the points (0, ± 1 3 ) and one or two closed curves depending on w 1 , see Figure 5. If k ⊥ = 1 2 + Z then λ + (x, ξ) = 0 precisely when x = 0 and ξ = ± 1 6 mod Z. In this case the characteristic set of λ − is the level set (4.7) which again has three connected components in R 2 /Z 2 . In particular, when k ⊥ ∈ 1 2 Z the eigenvalues are distinct near all points in the characteristic set of λ − except at a set of discrete points.
We now restrict to the case k ⊥ ≡ 0 mod 1 2 Z. Since we then have Υ k ⊥ = Υ −k ⊥ the result of Lemma 4.8 takes on a simpler form. We first compute the full symbol of H 2 c where we include a restatement of Lemma 4.8 for convenience. Lemma 4.9. Let H c be as in Lemma 4.7 and k ⊥ = 0 or k ⊥ = 1 2 . Then H 2 c = σ(H 2 c ) w (x, hD) where σ 0 (H 2 c ) = diag(P 11 , P 22 ) with where f (x) = w 1 (1−cos(2πx)), g(x) = w 1 √ 3 sin(2πx), and Υ (ξ) = 2 cos(2πξ)(−1) 2k ⊥ + 1, and with lower order terms Proof. The result follows by inspecting the proof of Lemma 4.8 and using Υ k ⊥ = Υ −k ⊥ together with properties of the Weyl calculus. 3 ) and one or two closed curves in T * S mod Z 2 . Here, k ⊥ = 0 and the top panels show the zero set for w 1 = 2/5 (left), w 1 = 1 (middle) and w 1 = 2 (right), while the bottom panels show the special values w 1 = 1/2 (left) and w 1 = 3/2 (right) where the number of closed curves in the zero set switches between one and two.
Since a = Υ #f + f #Υ − 2Υ f , similar computations also show that so after computing the derivatives we obtain the result.
Proof. The asymptotic expansion of σ(H 2 c ) follows from Lemmas 4.9 and 4.10, and by the discussion preceding Lemma 4.10 we see that P 0 has the stated form. Next, with which in view of Lemma 4.9 shows that P 1 also has the stated form.
Here,V w is defined as V w but with matrix-valued self-adjoint potentialŝ Proof. Consider the unitary map U q : L 2 (T * 1 ; C 4 ) → L 2 (T * 1 ; C 4 ) ⊗ C q , defined by (U q u) = diag(u, T 1 u, ..., T q−1 u) For u(x) = e −2πix , this map satisfies U q uU * q = uJ p q and U q e −2πihDx U * q = e −2πih Dx K * q . (4.8) The result then follows immediately, as the operator consists of such primitive Fourier modes.
A more detailed analysis of this model close to commensurable moiré lengths is an open problem and should be compared to the magnetic case [HS90]. Using the results of Proposition 4.13, it is possible to show that for example for p = 1, q = 2, the chiral Hamiltonian also exhibits a potential well at zero energy.
Proof of 3.11. As in the proof of Lemma 3.10 we may replace P w mass by P w without changing the estimate. Since X w 1 (P w − H w 0 )X w 1 = γ * • (h 3/2 J w 1 R w 0 J w 1 ) • (γ −1 ) * is unitarily equivalent to h 3/2 J w 1 R w 0 J w 1 the result then follows from Lemma 3.7.
Remark B.3. In contrast to the pseudodifferential Harper model discussed in Theorem B.2, there are no wells in the anti-chiral low energy model. Indeed, let H w ac (k x ) be given by (1.7) with w = (w 0 , 0) and k ⊥ = 0. Conjugating by U in (B.1) shows that H w ac (k x ) is unitarily equivalent to the semiclassical operator L w ac (x, hD) with symbol −1, 1, 1) where k x = O(h). Each component of the principal symbol is a scalar symbol of real principal type so there are no point-localized states near zero energy, see [Zwo12,Theorem 12.4].