Perturbation theory for spectral gap edges of 2D periodic Schr\"odinger operators

We consider a two-dimensional periodic Schr\"odinger operator $H=-\Delta+W$ with $\Gamma$ being the lattice of periods. We investigate the structure of the edges of open gaps in the spectrum of $H$. We show that under arbitrary small perturbation $V$ periodic with respect to $N\Gamma$ where $N=N(W)$ is some integer, all edges of the gaps in the spectrum of $H+V$ which are perturbation of the gaps of $H$ become non-degenerate, i.e. are attained at finitely many points by one band function only and have non-degenerate quadratic minimum/maximum. We also discuss this problem in the discrete setting and show that changing the lattice of periods may indeed be unavoidable to achieve the non-degeneracy.


Introduction
Let (1.1) H = −∆ + W be a Schrödinger operator in R d , d ≥ 2, with a smooth periodic potential W = W (x). Let Γ be its lattice of periods and Γ † be the dual lattice. We put T := R d /Γ and T † := R d /Γ † . It is known [10] that the spectrum of H is (1.2) σ(H) = [λ 0 , +∞) \ (⊔ n m=1 (µ m,− , µ m,+ )), where the non-intersecting intervals (µ m,− , µ m,+ ) are called the gaps. There are finitely many of them, and for small V there are no gaps at all, [12,8]. We are interested in the behaviour of the spectrum of H near the spectral edges µ m,± = µ m,± (H). More precisely, consider the Floquet-Bloch decomposition of H into the direct integral: (see Section 2 for more details) and denote by {λ j (k)} ∞ j=0 the collection of eigenvalues of H(k) (in non-decreasing order, taking multiplicities into account). Then each function λ j (called the Bloch function) is smooth, at least outside the values of k where the values of two such functions coincide. Then (1.4) σ(H) = ∪ k ∪ j λ j (k).
Therefore, for any spectral edge µ + = µ m,+ for some m there is a point k 0 and an index j = j(µ + ) such that λ j (k 0 ) = µ + , and k 0 is a point of local (even global) minimum of λ j (similarly, for any edge µ − there is a Bloch function λ j , j = j(µ − ), for which µ − is a maximal value). We are interested in the behaviour of the function λ j in a neighbourhood of k 0 . One may expect that for each spectral edge µ ± the following properties hold, at least generically: A. No other Bloch function takes value µ ± , meaning that if for some l and k ∈ T † we have λ l (k) = µ ± , then l = j.
B. The set of points is finite. C. The quadratic form of λ j around each critical point k 0 ∈ S is non-degenerate, meaning that with positive definite matrix A.
If these properties hold, we say that the spectral edge µ ± is non-degenerate. In the physical papers it is often assumed that generic Schrödinger operators have nondegenerate spectral edges. For example, in solid state physics, the tensor of effective masses is essentially defined as the inverse of the matrix A from (1.6) (see e.g. [1]). This definition makes sense only if all three conditions are satisfied. In one-dimensional situation the spectral edges are always non-degenerate, see e.g. [10]. The bottom of the spectrum is known to be non-degenerate in all dimensions, see [4] (but the same cannot be said about a magnetic Schrödinger operator, see [11], where an example of a magnetic operator the bottom of whose spectrum does not satisfy Condition C is given). It is commonly believed that in multidimensional case (d ≥ 2) the spectral gap edges are non-degenerate for generic potentials, see, for example, [7] and [6], where additional references are given.
Property A has been established to hold generically in [5]. In a recent paper [2] Property B has been proved for all (not just generic) operators if d = 2. It is not known whether Property B holds even generically in higher dimensions. The remarkable simple example discovered recently by N.Filonov [2] shows that for discrete periodic Schrödinger operators Property B does not hold, not even generically: there is a discrete periodic Schrödinger operatorĤ for which the set S corresponding to a spectral edge consists of two intervals, and the same holds for all operators close toĤ. It turns out, however, that this feature ofĤ is generically destroyed if we perturbĤ by a potential with a smaller latticeΓ ⊂ Γ, where Γ is the initial lattice of periods ofĤ. We discuss this and related results in Section 4.
The main result of our paper concerns Property C in the two-dimensional case. We will prove that all spectral edges of H can be made non-degenerate by perturbing it with arbitrarily small periodic potential V , with a smaller lattice of periodsΓ ⊂ Γ. Namely, we will prove the following result: Theorem 1.1. Let W = W (x), x ∈ R 2 , be a smooth function periodic with respect to some lattice Γ. Then for every ǫ > 0 there exist N = N(W, ǫ) ∈ N and a potential V (W, ǫ) periodic with respect toΓ := NΓ and satisfying V ∞ < ǫ such that the following property holds. Suppose, (µ m,− (H 1 ), µ m,+ (H 1 )) is a spectral gap of the operator H 1 := −∆+W +V with µ m,+ (H 1 ) − µ m,− (H 1 ) > ǫ. Then the edges µ m,± are non-degenerate. Remark 1.2. 1. As previous paragraph (and Section 4) show, decreasing the lattice of periods to achieve non-degeneracy may be necessary in the discrete case; we do not know whether it is possible to make spectral edges non-degenerate by perturbing H with a small potential with the same lattice of periods Γ, nor do we know whether this result holds in higher dimension (a substantial part of our proof is based on the fact that Property B holds for all, not just generic operators, and there are no high-dimensional analogues of [2] known so far).
2. There are two problems we have to deal with when increasing the lattice of periods. The first one is that the perturbed operator H 1 can have more spectral gaps than H. If the lattice of periods of V is Γ, the number of new gaps has an upper bound depending only on H and ||V || ∞ . If the lattice of periods of V is NΓ, then the number of new gaps may also depend on N, and our construction of the perturbation W has no control on the size of N. Therefore, what can happen in principle is the following. We introduce a perturbation V 1 periodic with respect to N 1 Γ so that H + V 1 has edges of 'old' gaps non-degenerate, but some 'new' gaps (of very small length) may appear. Then we may deal with these gaps by adding another, even smaller perturbation V 2 with lattice of periods N 2 Γ, but a further set of 'new new' gaps may be opened, etc. We do not have control over whether this process can last indefinitely long. Therefore, in our theorem we can guarantee only that all the edges of 'old' gaps (i.e. gaps the length of which is not small) become non-degenerate.
3. Another problem of dealing with the perturbations with increasing lattice of periods is the stability issue. It follows from the standard perturbation theory that the nondegeneracy of the spectral edges is stable under further perturbations with the same lattice of periods. More precisely, let ǫ, N, V be as in Theorem 1.1 then the conclusion of the theorem holds for operator H 2 = −∆ + W + V + Q with any smooth potential Q periodic with respect to NΓ, provided Q ∞ ≤ δ with δ = δ(W, V, N, ǫ) being sufficiently small. We, however, cannot guarantee the same result if the lattice of periods of Q can increase further and become NMΓ with M ∈ N (we can probably achieve this stability only at the edges of the 'old' gaps by introducing an extremely weird-looking norm in the class of all periodic operators with lattices being a sub-lattice of Γ, but the proof of corresponding statement is rather long and unhelpful, so we do not include it here). 4. As can be seen from the construction, the potential V is a finite trigonometric polynomial. As a result, we can choose our perturbation satisfying V s < ǫ, where s is a fixed real number and || · || s is a Sobolev norm.
5. The intuition behind the behaviour of the Bloch functions by the perturbations with increasing lattices of periods comes from studying the almost-periodic Schrödinger operators.
6. The same conclusions will hold if we consider a more general class of unperturbed operators, say the periodic magnetic Schrödinger operators (or even periodic second order coefficients); effectively, the only property we need from a class of operators we consider is the finiteness of the set S, see [2].
The rest of the paper is constructed in the following way. In section 2, we introduce the necessary notation and discuss how the decomposition (1.3) changes when we increase the lattice of periods Γ. In Section 3 we prove Theorem 1.1 and also give a simple proof of Property A (proved originally in [5]). Finally, in Section 4, we discuss the discrete situation when Property B is violated.

General facts
Suppose, is a Schrödinger operator with periodic potential W acting in R d . For simplicity we assume W to be smooth, but in fact we do not require this assumption for our results. Let Γ be its lattice of periods and Γ † be the dual lattice. We put T := R d /Γ and T † := R d /Γ † . For each quasi-momentum k ∈ T † we denote by H(k) the fibre operator of H corresponding to k so that The domain of H(k) consists of functions from H 2 (T) satisfying k-quasi-periodic boundary conditions; let us denote this space by H 2 (T; k). The action of H(k) (considered as an unbounded operator acting in L 2 (T)) is given by the formula The other, more convenient way of defining these operators is the following. First, we put We then denote by H s (R d ; Γ; k) the space of all infinite Fourier series of the form where a θ = f, e θ+k L 2 (T) |T| −1 ∈ C satisfy Finally, we say that H s (T; k) is the restriction of H s (R d ; Γ; k) to the torus T, and the LHS of (2.6) multiplied by |T| defines the square of the norm of f in H s (T; k). It will be convenient to define L 2 (R d ; Γ; k) as the collection of functions of the form (2.5) with (2.7) θ∈Γ † |a θ | 2 < +∞ and L 2 (T; k) as the restriction of L 2 (R d ; Γ; k) to the torus T. Note that when we change k, the space L 2 (R d ; Γ; k) does not change as the collection of elements (but the form in which we write these elements does change), whereas H 2 (R d ; Γ; k) changes with k.
Given another function from L 2 (T; k), we obviously have Suppose, the Fourier decomposition of W has the following form: Then the action of H(k) on the function f is given by: We denote by {λ j (k)} (j = 0, 1, . . . ) the collection of eigenvalues of H(k) (counted with multiplicities; it will be convenient from now on to stop assuming that λ j (k) are listed in the increasing order) and by ψ j = ψ j (k) = ψ j (k; x) corresponding orthonormal eigenfunctions. We will also assume (as we can without loss of generality) that λ j are piecewise continuous. We also denote (2.12) ψ j (k), e θ+k L 2 (T) /|T| =:ψ j (θ; k), so that These formulas also show that Now we discuss how this decomposition changes when we increase the lattice (in the sense that we increase the size of a cell of the lattice). Let N be a natural number and putΓ := NΓ. Then (Γ) † = Γ † /N. We also putT := R d /Γ andT † := R d /Γ † . The quotient group (Γ) † /Γ † consists of M := N d elements; let us denote by {p 1 = 0, . . . , p M } representatives of the elements of this group inΓ † . Then each element ofΓ † has a unique representation in the form p l + θ, θ ∈ Γ † . Moreover, every element k ∈ T † can be uniquely written as p l + κ, l = 1, . . . M and κ ∈T † . In this case we say that κ = κ(k) and l = L(k). This defines a mapping L : T † → {1, . . . , M} and a mapping κ : T † →T † ; each point κ ∈T † will have exactly M pre-images under mapping κ. Sometimes we will call coordinate k the old quasimomentum and κ the new quasimomentum.
Suppose, κ ∈T † . Then the space L 2 (T; κ) consists of all the expansions of the form Obviously, we can treat the RHS of (2.17) as a sum of functions from L 2 (R d ; Γ; k l ), where k l runs over all the pre-images of κ under the mapping κ. Suppose, f ∈ L 2 (T; k). Then expansion (2.5) can be looked upon as the element from the space L 2 (T; κ) with κ = κ(k). This defines a mapping F : L 2 (T; k) → L 2 (T; κ(k)); obviously, this mapping maps also H s (T; k) to H s (T; κ(k)).

Description of the approach: main tools
Suppose that d = 2 and our operator H has a gap (µ − , µ + ) in its spectrum. Definition. We say that µ + (resp. µ − ) is a non-degenerate end of the spectral gap, if there are finitely many points k 0 , k 1 , . . . , k n ∈ T such that λ j (k l ) = µ + (resp. λ j (k l ) = µ − ) for some j (not depending on l), for any m = j the equation λ m (k) = µ + (resp. λ m (k) = µ − ) has no solutions k ∈ T, and in the neighbourhood of each k l the function λ j behaves quadratically: as k → k l for some positive definite matrix A = A l .
We want to prove that, generically, each end of the gap is non-degenerate. For the sake of definiteness, we will be working with the top end of the gap, but the proof will easily extend to the bottom end of the gap and similar results will hold for µ − . Denote A recent result of Filonov and Kachkovskiy [2] shows that the set S is finite. Suppose, ν is arbitrary non-zero vector from R d . We denote δ := |ν|, n := νδ −1 , and v := e ν + e −ν . The perturbations we consider will be of type H ε := H + εV, where V is the operator of multiplication by v and ε > 0 a small parameter. We will always assume that that δ is smaller than the distances between different points in S and that the perturbed operator H ε is still periodic with, possibly, a new lattice of periodsΓ ⊂ Γ, i.e. that ν is a rational multiple of a vector from Γ † : ν ∈ QΓ † . After adding εV to our initial operator H, the new operator becomes periodic with respect to a new lattice of periodsΓ; we define this lattice as the lattice dual toΓ † -the lattice generated by Γ † and ν. Note that the new latticeΓ † contains more elements than the old one and therefore it may happen that some points from S (different modulo old lattice Γ † ) become 'glued together' after introducing the shift by ν, i.e. there may be two points k 1 , k 2 ∈ S so that k 1 + nν = k 2 ( mod Γ † ) for some integer n. We impose an additional condition that such 'gluing' does not occur. Later, we will explain why such a choice of ν is always possible (see Subsection 3.4).
For sufficiently small ε the operator H ε will have a gap (µ ε− , µ ε+ ), where |µ ε± − µ ± | = o(1) as ε → 0. The set S ε := S(H ε ) of quasimomenta where one of eigenvalues of H ε equals µ ε+ lies in a small neighbourhood of S. From now on we will always assume for simplicity that µ + = 0, but we will nevertheless often write µ + to emphasise that we are at the upper edge of the spectral gap.
Remark 3.1. In this section, as well as in the rest of the paper, we will work only with the dual space -the space where the quasimomentum k is located. We, thus, no longer need letters x or y to denote the original spacial variables. Therefore, we sometimes will be using this fact and introduce new coordinates denoted by x or y, etc. in the dual space (the space where the quasimomenta live).
We will also need the following simple result which easily follows from the analytical perturbation theory, see e.g. [3]. Suppose, λ n (k 0 ) is a simple eigenvalue of H(k 0 ). Then for k near k 0 there exists a unique eigenvalue of H(k) close to λ n (k 0 ); denote it λ n (k). We also can find a neighbourhood of k 0 (denoted by O(k 0 )) such that λ n (k) is a simple eigenvalue of H(k) whenever k is inside the closure of O(k 0 ). Obviously, the same will hold for H ′ (k) := H(k) + V , where ||V || -the L ∞ -norm of V is sufficiently small; we denote the corresponding eigenvalue by λ ′ n (k). In these notations we have: Suppose that x, y are some orthogonal coordinates around k 0 , l, m ≥ 0 are integers and ε > 0 is given. Then there exist a real number η = η(ε, O(k 0 )) > 0 such that if ||V || < η and k ∈ O(k 0 ), then we have Proof. Indeed, (3.3) with l = m = 0 is an immediate consequence of analytic perturbation theory and holds in a slightly bigger neighbourhood than O(k 0 ). This together with analyticity of (λ ′ n − λ n ) implies (3.3) for arbitrary l, m. This lemma shows that in order to prove our main result, it is enough to prove the following statement: and s are two fixed real numbers. Then there exists a periodic potential V with H s -norm smaller than ε such that H + V is periodic with a spectral gap [µ ′ − , µ ′ + ] with |µ ′ ± − µ ± | < ε and both spectral ends µ ′ ± being non-degenerate. Indeed, if the spectrum of H has several gaps in it, we first make the edges of the first gap non-degenerate with the help of sufficiently small perturbation V 1 . Then, we deal with the edges of the second gap by introducing a (even smaller) second perturbation V 2 . Note that Lemma 3.2 implies that if V 2 is small enough, then the edges of the first gap remain non-degenerate. Of course, while we are doing this, we may open 'new' gaps, but their length will be smaller than the sum of the norms of our perturbations V j . It remains to notice that the number of gaps of H is finite, see [9] or [8].
3.1. Description of the approach: The main idea. Let us for now assume for simplicity that there is a unique point k 0 such that (3.4) λ j (k 0 ) = µ + for some j, but (3.1) does not hold. Results of [5] show that (3.4) generically can hold only for one value of j (in the next subsection we will give a short proof of this); we will assume WLOG that j = 0, so that λ 0 (k 0 ) = µ + (remember that the labelling of eigenvalues is not necessarily done in increasing order, but λ 0 (k) continuously depends on k in some neighbourhood of k 0 ). We will never use this specific labelling and have chosen it only for convenience. We also assume that there is a vector n of unit length such that ∂ 2 λ 0 ∂n 2 (k 0 ) = 0 (otherwise there is nothing left to do). This means that as δ → 0 with even α ≥ 4 and C 0 > 0. We denote n ⊥ to be any of the two unit vectors orthogonal to n. Let ν be a vector that belongs toΓ † with some choice of sufficiently large N, but does not belong to Γ † /2 (so that κ 0 + ν and κ 0 − ν are different points modulo Γ † ). We denote l 0 := L(k 0 ) so that k 0 = κ 0 + p l 0 . Denote v := e ν + e −ν and H ε := H + εV, where V is the operator of multiplication by v. Whenever κ lies outside of a small neighbourhood of κ 0 := κ(k 0 ), all eigenvalues of H ε (κ) are located far away from µ + , therefore we are interested only in quasimomenta κ located in an o(1)-neighbourhood of κ 0 as ε → 0. Therefore, we have to study the perturbation of the eigenvalues of the fibre operators H ε (κ) when κ is close to κ 0 . We will write down the action of this operator in the orthonormal basis The matrix of H in this basis is diagonal, as we have established above. Let us compute the matrix of V (κ) (the fibre operator corresponding to V at the point κ). We denote by l + 0 the unique index satisfying p l + 0 = p l 0 + ν(modΓ † ); similarly, we define l − 0 by requiring . We will assume, as we can without loss of generality, that in fact we have p l ± 0 = p l 0 ± ν (in order to achieve this we need to choose proper representatives p l from corresponding classes of equivalence). Then we obviously have (3.8) V : e k → e k+ν + e k−ν .
Using (2.13) and (2.14), this implies This means that if we denote by the matrix element of operator V in the basis (3.6) then we have: Suppose, κ is close to κ 0 so that the eigenvalue λ 0 (κ + p l 0 ) is a simple eigenvalue of H(κ). Then for small ε the operator H ε (κ) will have a single eigenvalue τ = τ ε (κ) which is inside 2ε-neighbourhood of λ 0 (κ). Let us write a perturbation theory expansion of τ . Let P 0 be the projection onto φ 0,l 0 and P ′ := I − P 0 . Using P 0 V P 0 = 0 (see Corollary 2.2 or (3.11)) we obtain: as ε → 0. Here, is analytic in κ in some neighbourhood of κ 0 . Next, let P 1 be the orthogonal projection onto the two-dimensional subspace generated by φ 0,l ± 0 ; we also put P ′ 1 := P ′ − P 1 . Then we can rewrite Z as follows. (3.14) Note that expressions (3.15) and (3.16) depend only on k and ν and do not depend on N andΓ † .
Lemma 3.4. There exists a vector ν of arbitrarily small positive length such that Proof. We will consider vectors ν := δn with small δ and write the expansion of ∂ 2 Z ∂n 2 (k 0 , ν) in terms of δ when δ → 0. We will transform the formula for Z in the following way: The second equality in (3.18) is due to (2.15). Note that λ 0 (k ± ν),ψ 0 (θ; k ± ν), and R 0 are analytic in δ near 0 and in k near k 0 . It is a straightforward calculation that R = R(δ) = R(δ; k, ν) satisfies R(δ) = O(δ −α+1 ), R ′ (δ) = O(δ −α ), and R ′′ (δ) = O(δ −α−1 ) as δ → 0 (compare with calculations (3.19) and (3.20) and the proof of Lemma 3.5 below; recall that α is defined in (3.5)). Let us calculate now the second derivative of the principle term in (3.18). We have (here, by f ′ we denote ∂f ∂n ): (3.19) Therefore, (3.20) It is easy to check that the contribution from the derivative of the second term in (3.18) is the same. Therefore, as δ → 0, we have which shows that for some ν this second derivative is non-zero. Now, we are going to make the formula for Z more specific by introducing the proper coordinates.
Lemma 3.5. Let us denote the coordinates of k around k 0 by (x, y) so that x := k − k 0 , n is the coordinate along n, and y := (k−k 0 ), n ⊥ is the coordinate along n ⊥ . Then we have: where f is a real analytic function in a neighbourhood of the origin. Function f (x, y) also depends on δ but actual dependence is not important.
Proof. In our new coordinates formula (3.18) looks as follows where R 0 is analytic in (x, y, δ) near zero and (3.23) We recall thatψ 0 and λ 0 are real-analytic in (x, y) near zero. Equation (3.5) has the form λ 0 (δ, 0) = C 0 δ α + O(δ α+1 ) or, in other words, . Now, the direct calculations similar to the ones from the proof of Lemma 3.4 complete the proof of (3.22) which is just the Taylor series for Z in x and y near zero with δ being a parameter. We emphasize that while δ is going to be small (for example, to ensure that the coefficient in front of x 2 is negative and the eigenvalue λ 0 is still simple after the shift of the argument by δ) it will be fixed; then x and y are considered to be in a small neighborhood (depending on δ) of the origin which ensures the convergence of the Taylor series (3.22).
Remark 3.6. Since formula (3.22) is an identity of analytic in x, y functions (for every sufficiently small δ > 0), we can differentiate this identity with respect to x and y arbitrary many times. Now we discuss the broad strategy of our approach. Suppose, H =: H 0 is our initial operator with a degenerate minimum of the spectral edge (meaning that S consists of several isolated points, but the quadratic form of the Bloch function λ(k) at one of them is degenerate). We then start perturbing H by adding potentials of the form ε j (e ν j + e −ν j ), so that H j = H j−1 + ε j (e ν j + e −ν j ). Each H j is a periodic operator with the lattice of periods Γ j , where each Γ j is a sub-lattice of Γ j−1 . At each step we will achieve that a certain partial derivative (or a certain combination of partial derivatives) of a perturbed Bloch function at the new extremal point (or points) becomes non-zero. Lemma 3.2 shows that, once some combination of the partial derivatives on the Bloch function is non-zero for the operator H j , we can choose ε n , n > j so small that the same combination is non-zero for all operators H n , n ≥ j (notice that the choice of how small we require each ε n to be depends also on the lattice Γ n ). At the end our objective is to achieve that at all local minima of the Bloch functions λ(κ) of H n located near µ + we had ∂ xx λ = 0, ∂ yy λ = 0, and ∂ xx λ∂ yy λ − (∂ xy λ) 2 = 0 in some coordinate system (x, y); then, all minima will be non-degenerate. Of course, we will also make sure that ε j are so small that j ε j < µ + − µ − , so we have not closed the spectral gap. We will also assume that the perturbed operator is still periodic with, possibly, a new lattice of periodsΓ ⊂ Γ. In order to achieve this, it is enough to require that each vector ν j belongs to the set QΓ † . This set is dense and the objective of our perturbation will always be making certain quantities (like combinations of partial derivatives) non-zero. Since these quantities will always depend continuously on ν, once we have found any vector ν for which these quantities are non-zero, we can always find a vector inside QΓ † with these quantities still being non-zero. For example, in Lemma 3.4 we can always find ν satisfying the requirements of that Lemma such that, additionally, we have ν ∈ QΓ † . Therefore, we will always assume that our choice of ν j will be rational multiples of a vector from Γ, without specifying it explicitly.

3.2.
Several band functions have the same minimum at the same point. In this section, we will get rid of a situation when a point k 0 is a minimum of two or more different band functions simultaneously. It was proved in [5] that (generically) this cannot happen; here, we will give (an outline of) a different proof, which seems to us to be rather shorter. We will need the following Lemma, sometimes known as the Shur complement Lemma.
Lemma 3.7. Suppose, P 1 and P 2 are two orthogonal projections in a Hilbert space H with P 1 + P 2 = I, and H is the self-adjoint operator which has the following block form with respect to P 1 and P 2 . This means that U jl = P j HP l . We put H j := P j (H) and assume that λ ∈ σ(U 22 ). Proof. This is a straightforward computation.
To begin with, let us assume that µ + is the minimum of two band functions reached at the same point, say λ 1 (k 0 ) = λ 2 (k 0 ) = µ + . In this case we will not be taking vector ν from a finer latticeΓ † , instead we will assume that ν ∈ Γ † \ {0}. We also take slightly more complicated potential than before, namely we put v = ae ν +āe −ν and denote by V the operator of multiplication by v. Let us check what will happen with the eigenvalues at k 0 after this perturbation. We apply the Shur complement Lemma to study eigenvalues of H ε (k) with k close to k 0 . We denote by P 1 the orthogonal projection onto H(k)the two-dimensional subspace generated by ψ 1 (k; ·) and ψ 2 (k; ·) and P 2 = I − P 1 . Then the Shur complement Lemma shows that the perturbed eigenvalues coincide with the eigenvalues of the 2 × 2 matrix A = A(ε, k) = (a mn ) 2 m,n=1 with the coefficients given by if m = n. We notice that the choice of the basis in H(k 0 ) is not uniquely determined; we just fix some orthonormal basis φ := (ψ 1 (k 0 ; ·), ψ 2 (k 0 ; ·)) of H(k 0 ).
This lemma shows that off-diagonal elements of A(ε, k 0 ) are non-zero for a certain choice of a and ν = 0. Therefore, its eigenvalues are different and we have achieved the required splitting. This simple argument is already sufficient to prove that Condition A is generic if eigenfunctions ψ j are continuous in k, since then off-diagonal elements of A will be non-zero for all k in a neighbourhood of k 0 .
In general situation we proceed as follows. We denote by {v mn (k)} 2 m,n=1 the matrix of V in the basis ψ j (k), j = 1, 2, and for any choice φ = (φ 1 , φ 2 ) of an orthonormal basis of H(k 0 ) we denote by {v mn (k 0 ; φ)} 2 m,n=1 the matrix of operator V in this basis. Let us fix a and ν so that the left-hand side of (3.27) is equal to one, i.e. v 12 (k 0 ; ψ) = 1. This means that for any other choice of the basis φ we either have |v 12 (k 0 ; φ)| ≥ 1/4 or |v 11 (k 0 ; φ) − v 22 (k 0 ; φ)| ≥ 1. Since the projection onto H(k) is analytic in k, for every η > 0 there exists δ > 0 such that whenever |k − k 0 | ≤ δ the matrix {v mn (k)} 2 m,n=1 is ηclose to the matrix {v mn (k 0 ; φ)} 2 m,n=1 with some choice of the basis φ. As a consequence, taking η = 1/8 we obtain the following statement. For every k in some neighbourhood M of k 0 the elements of the matrix {v mn (k)} 2 m,n=1 satisfy either |v 12 (k)| ≥ 1/8 or |v 11 (k) − v 22 (k)| ≥ 3/4. Now, we consider the matrix of A(ε, k) in the basis ψ j (k), j = 1, 2. We have Here, O(ε 2 ) is uniform in k ∈ M. If k is such that |v 12 (k)| ≥ 1/8 then we obviously don't have a multiple eigenvalue for ε small enough to dominate O(ε 2 ). Assume the second alternative, i.e. |v 12 (k)| ≤ 1/8 but |v 11 (k) − v 22 (k)| ≥ 3/4. Assume for definiteness v 11 (k) > v 22 (k) + 3/4 and λ 1 (k 0 ) = λ 2 (k 0 ) = 0, and 0 is the minimal value of these functions. We also have |v jj (k) − v jj (k 0 ; φ)| ≤ 1/8 for some choice of the basis φ. The matrix of A(ε, k 0 ) in this basis has the form Now, assume that we have a multiple eigenvalue at k ∈ M. Then a 22 (k) = a 11 (k), which, together with the observation that λ j (k) ≥ 0, implies λ 2 (k) ≥ 3ε/4 + O(ε 2 ) and thus this means that k is not a point of minimum. Thus we proved that in some neighbourhood of k 0 we cannot have the edge of the spectrum attained by more than one band function.
Suppose now that µ + is the minimum of t band functions reached at the same point, λ 1 (k 0 ) = λ 2 (k 0 ) = · · · = λ t (k 0 ) = µ + . Then we proceed as above and our perturbation will be described by a t × t matrix the off-diagonal elements of which are non-zero for some choice of parameters. Arguments similar to those above imply that after this perturbation, the resulting operator will have an eigenvalue of multiplicity at most t − 1. Repeating this procedure t − 1 times if necessary, we will achieve that no two different bands can have a minimum at the same point. The arguments which justify the subsequent elimination of the multiple minima at different points or/and at different edges are standard. This proves the following Theorem (originally due to Klopp-Ralston, [5]): Theorem 3.9. Condition A is generic for two-dimensional periodic potentials.

3.3.
Minimum is a minimum of only one band function. First, we choose coordinates around k 0 so that x goes along ν. In this section we will change these coordinates many times; in order to avoid cumbersome notation, we will call both old and new set of coordinates by the same letters (x, y) (sometimes writing x old and x new to avoid confusion). Each time we perform a change of coordinates, we will have to check that the perturbation Z in the new coordinates still satisfies (3.22) (or, at least, (3.17)).
Step 1. Obtaining a quadratic term in one direction. Let k 0 be a point of local minimum of λ 0 ; we will introduce the orthogonal coordinates (x, y) around k 0 so that the Taylor expansion of λ 0 at k 0 in these coordinates has a form (3.29) λ 0 (x, y) = x 2n + α,β d αβ x α y β and the sum is over all (α, β) with α + β ≥ 2n with the exception of (α, β) = (2n, 0). If n = 1, we move to the next step, so now we assume that n ≥ 2. We apply the Weierstrass Preparation Theorem and obtain that λ 0 has the following form: where a j are analytic functions such that a j (y) = O(y 2n−j ) and c(0, 0) = 1. Making a change of variables x new = x old − a 2n−1 (y)/(2n), we can assume that a 2n−1 = 0. We notice that this change of variables does not affect the representation (3.22). So, for simplicity we will use the same notation (x, y) for the new variables. Then the Bloch function after the perturbation has a form Here, b is analytic function in all variables. Each a j is analytic function of one variable and therefore has a simple form a j (y) = c j y k j (1 + O(y)), c j = 0 (either this, or a j ≡ 0, in which case we put k j := ∞). Obviously, k j ≥ 2n − j. Let (x * , y * ) be a point where the minimum of τ ε is attained in a small neighbourhood of the origin, i.e. x * , y * = o(1) as ε → 0. We prove that there is an improvement after the perturbation, namely: Lemma 3.10. There is a partial derivative of τ ε of order smaller than 2n that does not vanish at (x * , y * ).
Thus, all partial derivatives ofτ := τ ε /c(x, y) of order smaller than 2n are O(ε 2 ) at (x * , y * ). We get and this implies y * = O(ε 2/(k j −2n+1+j) ) for j = 0, . . . , 2n − 2. Now we notice that formula (3.22) implies that where b j are functions of δ and b 2 > 0 for small δ. Then our assumption that all derivatives of τ ε of order smaller than 2n disappear implies assuming k j , j = 0, 1, 2, are finite. But since k j > k j − 2n + 1 + j, j = 0, 1, 2, for 2n ≥ 4, this is the contradiction. The case when one or more k j = ∞ is even simpler and can be considered in the same way.
This lemma shows that after the perturbation we get a non-zero derivative of order smaller than 2n in the new (and therefore in the old) variables. Repeating this procedure, we obtain a new Bloch function for which the second derivative in one direction does not vanish. Now we move to the next step.
Step 2. Obtaining a nondegenerate quadratic form. Suppose now that the second derivative at our minimum in one direction is nondegenerate, i.e.
Since we have an isolated minimum, f (x) = bx 2n (1 + O(x)) with some b > 0 and n ≥ 2. Now, rescaling we obtain that λ 0 has the form λ 0 = (x 2n (1 + O(x)) + y 2 )c(x, y) with c(0, 0) > 0. Finally, we make another change of variables x 2n new = x 2n old (1+O(x old ))c(x old , y old ) and y 2 new = y 2 old c(x old , y old ) so that in the new coordinates we have (3.34) where we have denotedτ (x new , y new ) = τ (x old , y old ). Below (see Lemma 3.11) we will show that Z from (3.22) still admits similar representation in new variables: As before, we assume that (x * , y * ) is a point of a local minimum forτ ε near point (0, 0) (in particular, x * , y * = o(1)). We consider three cases. Case 1. Suppose, b 1 = 0. Then from ∇τ (x * , y * ) = 0 we get y * = O(ε 2 ), (1)). This implies Thus, we have a nondegenerate minimum at (x * , y * ). Case 2. Suppose, b 1 = 0 and n ≥ 3. We notice that if x * = O(ε) then (τ ) ′′ xx (x * , y * ) < 0 which leads to a contradiction. So, |x * /ε| → ∞ as ε → 0. Then, similar to the previous case one gets Thus, we again have nondegenerate minimum at (x * , y * ). Case 3. Finally, we consider the case b 1 = 0 and n = 2. It is convenient to rescale x old = εx new , y old = ε 2 y new and divide by ε 4 . Then we have to consider Here we have used the fact that Y (ε) from (3.12) is analytic in (x, y). We are using the following convention: O(1) is a bounded function of ε only, and O x,y (1) is a bounded analytic function of ε, x, y. Calculating the derivatives at point (x * , y * ) we get (3.37) Since we have a minimum, the worst scenario is when our quadratic form is degenerate. This means that 6x 2 * − b 2 = O(ε) and corresponding form becomes zero in the direction (y − y * ) = O(ε)(x − x * ). In this direction the cubic term becomes (4x * + O(ε))(x − x * ) 3 which contradicts to minimum condition since x 2 * ∼ b 2 /6. Now, let us show that the change of variables we use above does not destroy our achievements, i.e. that (3.35) holds in the new variables. Lemma 3.11. All changes of the variables described above do not change the representation (3.22) for sufficiently small δ. In particular, (3.35) holds.
Proof. First, we discuss the change of variables (3.38) x new = x old , y new = y old + f 1 (x old ).
If f 1 (x) = O(x 3 ) then the statement immediately follows from (3.22). So, we assume that f 1 (x) = sx 2 (1 + O(x)), s = 0. Then, since we have a minimum at point (0, 0) we have f 2 (x) = as 2 x 4 (1 + O(x)), a ≥ 1. Using this explicit form for λ 0 = (y 2 + 2f 1 (x)y + f 2 (x))p(x, y) and repeating the calculations similar to (3.19), (3.20), it is not difficult to obtain more detailed version of (3.22). Namely, 3.4. Several minima. All the results of this Section obtained so far prove our main Theorem under assumptions that on each step of the procedure we have |S| = 1 (i.e. the minimum of the band function is attained at one point). Let us discuss the changes we need to make if S consists of several points. Then we have to be slightly more careful with the choice of ν. The properties we need are summarised in the following statement: Lemma 3.12. Suppose S is finite and arbitrarily sufficiently small δ > 0 is fixed. Then we can find a vector ν ∈ QΓ † arbitrarily close to a given direction with the length |ν| ∈ (δ/2, δ) and such that there are no two different points k 1 , k 2 ∈ S satisfying k 1 + nν = k 2 + θ, where n ∈ Z and θ ∈ Γ † .
Let us first discuss why these properties are sufficient for our purposes (and where exactly in our procedure these properties are required). We need to be able to choose ν close to any direction to be able to perform Step 2. Here it is important to have uniform control of the length of ν so that all the estimates from Step 2 still hold for sufficiently close direction. Since δ is arbitrarily small we can also ensure that, e.g., C 1 > 0 in (3.22). Let us denote byΓ † the lattice generated by Γ † and ν (this lattice is discrete due to the assumption ν ∈ QΓ † ). Our last assumption means that all points k j ∈ S are different modulo this new latticeΓ † (i.e. κ j are different). This guarantees that (3.12) holds. Indeed, without this assumption Y is infinite and with this assumption Y , although depending on ν in an uncontrolled way, is still O(ε 3 ) and analytic in κ in a neighbourhood of κ 0 .
Proof. We assume that δ is smaller than 1 100 min{|γ|, 0 = γ ∈ Γ † }. Let us start by choosing any vectorμ from QΓ † with direction close to a given one. Let µ be the smallest vector in Γ † having the same direction asμ. We put ν :=p µ p , wherep and p are natural numbers defined as follows. Suppose, k j , k s ∈ S are two points such that k j − k s = m js n js µ + θ, where n js > 1 and m js are co-prime integers and θ ∈ Γ † ; note that n js is uniquely determined by k j and k s and does not depend on θ ∈ Γ † . If there are no such points k j , k s , we just definep := 1 and choose p to be any natural number such that |ν| ∈ (δ/2, δ). Otherwise, we first choose p to be any large prime number (namely, p > 100|µ|/δ) co-prime with all n js . Then we choosep such that |ν| ∈ (δ/2, δ). Obviously,p is smaller than p and thus it is co-prime with p. We claim that this choice of ν satisfies all the required conditions. Indeed, assume that k 1 + nν = k 2 + θ, where n ∈ Z and θ ∈ Γ † . This means that m 12 +qn 12 n 12 µ + np p µ = 0 for integer n, q with |n| < p, n = 0. However, this implies that npn 12 = −(qn 12 + m 12 )p, which is a contradiction since n 12 andp are co-prime with p, and 0 < |n| < p.

Counter-examples
First of all, in Subsection 4.1, we will give several examples of discrete periodic Schrödinger operators for which property B is violated on an open set of potentials. This obviously shows that property B cannot possibly be generic in the discrete setting. Then, in Subsection 4.2, we will discuss how property B can be forced to hold by a small perturbation of our example if this perturbation is periodic with a sublattice of our original lattice of periods (of index two).
Then H can be represented as the direct integral where H(k) acts in C 2 and is represented by the following matrix Then it is easy to see that the spectrum of H consists of two bands and these intervals are disjoint unless V 0 = V 1 . Moreover, the upper edge of the first band and the lower edge of the second band both are attained on the boundary ofΩ. Thus, we have degenerate edges of the gaps with the corresponding degeneracy undestroyable with any small perturbation of the potential. Of course, for the continuous Schrödinger operator the degeneracy on the lines is impossible by Thomas construction, moreover, as recently was proved in [2], even degeneracy on the curves is impossible for 2-dimensional continuous Schrödinger operators. However, this example is important as it shows that the question is not as obvious as it may look. Here we would also like to mention [11] where the magnetic Schrödinger operator was constructed with degenerate lower edge of the spectrum (still attained at one point) while the proof from [2] excludes degeneracy on the curves for 2D magnetic operators too.
Remark 4.1. The example described above can be adjusted to obtain degeneracy even if the number of parameters is very large. For example, assume that n ≥ 3 and consider the periodic operator with the lattice of periods generated by (n, 0) and (1, 1) and the potential (V 0 , V 1 , . . . , V n−1 ) satisfying V 0 < V j − 2. Then, if we put s := e ik 1 and t := e ik 2 , the matrix of the fibre operator has the following form: Note that the quadratic form of H(k) equals V 0 on the vector (1, 0, . . . , 0), which shows that the point V 0 is the right edge of the first spectral zone, attained at {s + t = 0}. Thus the degeneracy of the spectral edge is an interval (not the union of two intervals as in the case n = 2).

4.2.
How to destroy the degeneracy by changing the lattice of periods. Now we discuss how the degeneracy in the example from the previous subsection will be destroyed using our approach. We consider the initial operator with doubled period (in vertical direction); we also assume (as we can without loss of generality) that V 0 = V > 0, We will show that now the small perturbation of the potential destroys the degeneracy of the edges. We consider in details the lower edge of the second band, the construction for the upper edge of the first band is similar. Our perturbation has the form B := diag{2ε, 0, 0, 0}, ε > 0. Equation det(H(k) + B − λ) = 0 reads as follows We put t := [(λ 2 − V 2 ) − (a 2 + b 2 )] and solve the quadratic equation for t. We get First, let us show that the minimum of the second band is situated near the center of the square a = b = 0. Indeed, obviously it must be near the diagonals a = ±b. In a neighborhood of any point on the diagonals which is not the center of the square we have |ab| ≫ 1. Then (4.3) gives and thus λ 2 − ελ − (a ± b) 2 − V 2 − εV + O(ε 2 ) = 0.
At the same time, for a = b = 0 there is the solution t = 0 for (4.3) which corresponds to λ = V , and thus, the lower edge of the second band occurs near the point a = b = 0, i.e. k 1 = k 2 = π/2. For the unperturbed operator we have λ 2 = V 2 + (a ± b) 2 . This leads to We consider the two-dimensional subspace of the eigenvectors corresponding to the perturbed eigenvalues when ε ≥ 0. Let P be the orthogonal projection onto this subspace. We notice that while the eigenvectors, generally, are not analytic in a, b and ε, the projection P is analytic and can be represented by the convergent series P = P 0 + ∞ n=1 ε n P n with P n = P n (a, b) being analytic in the small neighborhood of (0, 0). Moreover, it is not hard to see that one can choose the analytic orthonormal basis f 1 (a, b, ε), f 2 (a, b, ε) in the range of P such that f j (0, 0, 0) = (1, 0, ±1, 0)/ √ 2. Indeed, first we notice that the unperturbed matrix has analytic eigenvectors f j (a, b, 0) with f j (0, 0, 0) = (1, 0, ±1, 0)/ √ 2 and then one should just apply Gram-Schmidt orthogonalization to P f j (a, b, 0). Since the range of P is, obviously, an invariant subspace of H(k) + B, it is enough to consider the and therefore (4.7) x Thus, we proved that x * = O(ε), y * = O(ε). Now direct calculation of the derivatives of τ gives Thus, we have obtained the non-degenerate minimum.