Generic properties of dispersion relations for discrete periodic operators

Consider a $Zn$-periodic self-adjoint elliptic operator L in $Rn$⁠. The issue discussed below can be formulated and studied in a more general setting, but the reader can think of the Schrödinger operator L = −Δ + V(x) with a real $Zn$-periodic potential V(x), which we assume to be sufficiently “nice,” e.g., L_∞ (assuming the potential even being C^∞ does not seem to make the problem we discuss any easier).

We recall some notions from the spectral theory of periodic operators and solid state physics (see, e.g., Refs. 3, 33, 34, 43, and 49).

For $k∈Rn$ (called quasimomentum in physics), let us define the twisted Schrödinger operator L(k) to be L applied to functions u(x) on $Rn$ that are k-automorphic (also called Floquet, sometimes Bloch functions, with quasimomentum k), i.e.,

where k ⋅ γ = ∑_jk_jγ_j. In other words, u(x) = e^ik⋅xp(x), where the function p(x) is $Zn$-periodic.

Then, L(k) is an elliptic operator in a line bundle over the torus. [In another Floquet multiplier z incarnation, see Definition 1, L(z) acts in the trivial bundle over the torus, but the operator depends analytically on z.]

The quasimomentum k is well defined up to shifts by vectors from the lattice $2πZn$ (the dual lattice G* to $G≔Zn$⁠, i.e., consisting of all vectors k such that $k⋅γ∈2πZ$ for any γ ∈ G). Thus, it is sufficient to restrict k to the Brillouin zone B = [−π, π)ⁿ.

It is often convenient to factor out the G*-periodicity and consider instead of vectors $k∈Cn$ the complex vectors z with non-zero components,

When the quasimomentum k is real, the corresponding Floquet multiplier belongs to the unit torus,

In terms of Floquet multipliers z, the Floquet functions satisfy

where $zγ=eik⋅γ=ei∑jkjγj$⁠. In other words, u(x) = z^xp(x), where the function p(x) is $Zn$-periodic.

The Floquet theory is a version of Fourier series expansion. One thus is interested in harmonics into which the expansion is done. These are the characters of the group G of periods.

The following statement is well known (and easy to prove):

The above discussion suggests the use of the Fourier series. A standard argument justifies the decomposition of the (unbounded self-adjoint) operator L in $L2(Rn)$ into the direct integral (see Refs. 33, 34, 43, and 49),

As L(k) is an elliptic operator in sections of a line bundle over the torus $Rn/Zn$⁠, it has a discrete spectrum σ(k) ≔ σ(L(k)) that consists of infinitely many eigenvalues, each of finite multiplicity,

See Refs. 33, 34, 43, and 49 for more details.

Thus, the dispersion relation is the graph of the multiple-valued function k ↦ σ(L(k)) (Fig. 1).

The following properties of the dispersion relation (Bloch variety) are well known:

Spectral edges are the endpoints of spectral gaps (see Fig. 2). By Proposition 6, they correspond to some of the extremal values of band functions. We are interested in the generic (with respect to the perturbation of the periodic potential or other parameters of the operator) structure of these extrema. The genericity can be understood in a variety of ways, e.g., holding for a second Baire category set of potentials in an appropriate (Banach) space of potentials (the most likely situation), stronger one—for a dense open subset, or even stronger—in the exterior of an analytic (or even algebraic) subset in the space.

An old conjecture, more or less explicitly formulated in a variety of sources, e.g., in Ref. 34, Conjecture 5.25 or in Refs. 18, 33, 41, and 42, deals with the structure of the dispersion relation of the Schrödinger operator in $Rn$ with periodic potential near the edges of the spectrum, i.e., near (some of) the extrema of the dispersion relation. This well-known and widely believed conjecture says that generically, the band functions are Morse functions. We make this precise below.

Notice that (3) implies (2).

This conjecture asserts that generically, near a spectral edge, the dispersion relation has a parabolic shape and thus resembles the dispersion relation at the bottom of the spectrum of the free operator −Δ. This in turn would trigger the appearance of various properties analogous to those of the Laplace operator. One can mention, for instance, electron’s effective masses in solid state theory,^3,30 Green’s function asymptotics,^6,26,28,37 homogenization,^8,14 Liouville type theorems,^{4,27,35,36,38} Anderson localization,¹ perturbation of discrete spectra in gaps in general,^11–13 and others.

Why would one conjecture this? The existence of a degenerate extremum of the dispersion relation is an “analytic equality type” restriction. It is thus natural to believe that it holds either almost never or almost always with respect to the potential and other parameters of the operator. If such a restriction for (almost) all periodic potentials existed, it would most probably be known. To put it differently, the common idea is that generically, the dispersion relation probably behaves like the spectrum of a “generic” family of self-adjoint matrices.⁵ As we have already mentioned, this has been conjectured in various (explicit or implicit) forms by several authors.

The progress in proving this conjecture has been very slow. We summarize here briefly the successes achieved so far. It is known that the expected parabolic structure always (not only generically) holds at the bottom of the spectrum of any Schrödinger operator with a periodic electric potential²⁹ (which is not necessarily true if a magnetic field is involved⁴⁸). In Ref. 31, the statement (1) of the conjecture was proven. The full conjecture was proven in Ref. 18 in 2D for any fixed number of bands and small smooth potentials. The statement (2) was proven in 2D²¹ in a stronger form, even without the genericity clause.

Besides varying the quasimomentum k, one might vary other parameters of the operator, e.g., of the periodic potential V ∈ L_∞(W), or introduce a periodic diffusion coefficient (metric) in the operator: −∇⋅ D(x)∇u + Vu (assuming that ellipticity is preserved). One can thus consider an extended dispersion relation in an appropriate Banach space of quadruples (D, V, k, λ). As long as ellipticity is preserved, an analog of Proposition 6 still holds.³⁴ 

It is natural to first look at discrete problems. Then, the dispersion relation becomes (in appropriate coordinates) an algebraic variety.^22,32 Such discrete formulations are frequently used, e.g., in solid state physics, when using the tight binding approximation.^3,30 Alas, counterexamples to generic non-degeneracy in the discrete²¹ as well as quantum graph^9,21 case have been known, e.g., the example in Ref. 21 deals with a two-atomic structure on $Z2$⁠, where the potential has just two different possible values: v₀ attained at all vertices $(n,m)∈Z2$ such that n + m is even, and v₁ ≠ v₀ when n + m is odd. Thus, the only free parameters are v₀ and v₁. The impression is that the number of parameters should be large enough in order for the genericity to hold. We conjecture (see Conjecture 18) and prove in a particular case (Theorem 28) that increasing the number of parameters cannot destroy generic non-degeneracy.

Calling a set generic in the strong sense that its complement is contained in a proper algebraic subset, we show that in the discrete periodic case, the following dichotomy holds: either the set of parameters for which there are degenerate critical points is generic or the set of parameters for which there are no such points is generic. Thus, testing a “random” sample of parameters should provide an “almost surely correct” answer.

In Sec. II, we provide a detailed description of the discrete case. The main result (Theorem 16) and its proof are presented in Sec. III. A specific example is considered in Sec. IV, and we present three approaches to establishing Conjecture 7 for this example. These are based on a numerical computation, an “almost surely” verification, and then an actual proof that relies on an exact count of solutions. We also use symbolic computation to find all maximal substructures of this example for which Conjecture 7 does not hold. The final remarks are provided in Sec. VII.

We consider a discrete situation, i.e., when the group $G=Zn$ acts on a graph Γ with the set of vertices V and edges E. We write x ∼ y for two vertices x, y ∈ V when there exists an edge connecting them (it is easy to modify the notions and proofs below for the case when multiple edges are allowed between a pair of vertices).

Let us start with making this notion precise.

A simple way to visualize this is to think of a graph Γ embedded into $Rn$ in such a way that it is invariant with respect to the shifts by integer vectors $g∈Zn⊂Rn$⁠, which produces an action of $Zn$ on Γ. When n ≥ 3, the graph Γ has such an equivariant embedding in $Rn$ where the action shifts by integer vectors in $Rn$⁠. When n = 2, edges of a graph may cross when embedded into $R2$⁠, and when n = 1, severe overlapping will occur. In the cases when n = 1 or 2, the graph Γ has an embedding in $R3$ that is periodic with respect to a copy of either $Z$ or $Z2$ embedded in a coordinate ray or coordinate plane.

A popular (graphene) example of a periodic graph and its fundamental domain is shown in Fig. 3.

As in the continuous case, the standard idea of harmonic analysis suggests that, as long as we are dealing with a linear problem that commutes with an action of the abelian group $G=Zn$⁠, Fourier series expansion, i.e., expansion into irreducible representations, with respect to this group should simplify the problem. Its implementation leads to what is known as the Floquet transform. Indeed, what one needs to do is to expand functions on the graph Γ into the unitary characters (exponentials) γ_k (we will also use the notation γ_z, where z = exp ik).

Let Γ be a $Zn$-periodic graph and f be a finitely supported (or sufficiently fast decaying) function defined on the set of vertices $V$ of Γ.

The reader can notice that (8) and (9) are just the Fourier transform with respect to the action of $G=Zn$ on the set V of vertices.

Let us formulate some basic properties of the Floquet transform. The following statement follows by a direct inspection of (8) and (9).

The equalities (10) and (11) of the lemma show that, as one would expect, G-shifts after the Floquet transform become multiplication by the corresponding characters. To put it differently, for a fixed z, the function $f̂(v,z)$ on Γ is automorphic with the character z^g = e^ik⋅g. It also shows that the values of $f̂(v,z)$ are determined completely if they are known for vertices v from a fundamental domain W as they can be extended to the whole graph using (10). We thus introduce the following notation.

We also see that the Floquet transformed function is a G*-periodic function of the quasimomentum k according to the identity (12).

Let A be a difference operator on a $Zn$-graph Γ. In other words, A is an infinite matrix with rows and columns indexed by the vertices of Γ. We assume that A has finite order, meaning that in each row it has only finitely many non-zero entries (in other words, only finitely many neighbors of each vertex are involved). We also assume that A is periodic, i.e., commuting with the action of the group $Zn$⁠. After the Floquet transform, this operator becomes the operator of multiplication by a matrix A(z) of size |W| × |W| depending rationally on the Floquet multiplier z (or analytically on the quasimomentum k).

As an example, let us consider the Laplace operator on the regular hexagonal 2D lattice Γ (see Fig. 4). The group $Z2$ acts on Γ by the shifts by vectors p₁e₁ + p₂e₂, where $(p1,p2)∈Z2$ and vectors $e1=(3/2,3/2)$ and $e2=(0,3)$ are shown in Fig. 4. We choose as a fundamental domain (Wigner–Seitz cell) of this action the shaded parallelogram region W. Two black vertices a and b belong to W, while b′, b^″, and b^‴ lie in shifted copies of W. Three edges f, g, h, directed as shown in the picture, belong to W.

We consider the Laplace operator

One can find the “symbol” A(z) (or A(k) in terms of the quasimomenta) by applying A to functions f automorphic with the character z = e^ik. Such a function is determined by its values at the vertices a and b. Indeed, since b′ is obtained from b by a shift of −e₁, one has $f(b′)=z1−1f(b)=e−ik1f(b)$⁠. Similarly, $f(b″)=z2−1f(b)=e−ik2f(b)$⁠. One also finds that the values at two neighbors of the vertex b are $z1f(a)=eik1f(a)$ and $z2f(a)=eik2f(a)$⁠. Hence,

We thus obtain the expression for the “symbol” of A,

The matrix A(z) depends rationally on z-variables. In fact, it is a Laurent polynomial. Since variables z belong to the unit torus $Tn$⁠, no singularities of A(z) appear there. It is thus possible to multiply the matrix A(z) by $z1mz2m…znm$ with a sufficiently high power m, so the resulting matrix $Ã(z)$ has polynomial entries, e.g., in the above example, multiplying by z₁z₂ (i.e., m = 1), the dispersion relation in the (z, λ) space can be given as follows:

where

Thus, the dispersion relation is an algebraic variety of codimension 1 in $C3$⁠.

An analogous construction holds for general finite order periodic difference operators on graphs with z₁z₂ being replaced by the product $z1mz2m…znm$⁠.

We can consider the equation

as an implicit description of the graph of a multiple-valued function

that shows the dependence of the spectrum of $Ã$ on the parameters z.

Suppose that the matrix A also depends polynomially on extra parameters α (e.g., weights at vertices and/or edges, potentials, and so on),

Correspondingly, we have a family of functions

Thus, the question can be reformulated as follows:

Does non-degeneracy of all critical points of the function F_α on the torus hold generically with respect to the parameters α?

Establishing any weaker type of genericity would be valuable, in particular for the continuous (PDE) situation, but as our results show, in the discrete case, it should be understood in the strongest sense, as being valid outside of a proper algebraic subset. This is clearly not expected to happen.

It is not that clear how to use the information about a value λ being a spectral edge of the operator, so we will address the more general set of all critical values of the dispersion relation.

The matrix A(α, z, λ) introduced above, and thus, $Ã(α,z,λ)$ as well, has a very special structure due to the periodicity. However, an important dichotomy holds in a very general situation without any connection to the periodicity.

It is clear in particular that the image of DC under the natural projection (α, z, λ) ↦ α coincides with DV.

We now formulate and prove the following dichotomy statement.

Our desire is to have the set DV “small,” i.e., the first alternative of Theorem 16 to take place. However, as we have already mentioned, even in the case of a “two-atomic” periodic discrete structure, this is not necessarily the case.²¹ Hence, how can one tell in a particular case which of two options of the dichotomy materializes? While we do not have a complete answer to this, the following “random test” follows from Theorem 16. Let us pick a value of α “randomly” (with respect to an absolute continuous probability distribution) and compute the dispersion relation. If it has no degenerate critical points, then we know “with probability one” that DV is contained in a proper algebraic subset, and thus, non-degeneracy is generic. If instead we determine that α ∈ DV, then we know that “with probability 1,” degeneracy is generic. Indeed, the chances for a randomly selected point to belong to a given proper algebraic subset are zilch.

We also formulate the following conjecture:

One can ask whether one can avoid a random choice of parameters. The answer is “yes,” but it is not that easy to implement.

Namely, there are n + 2 polynomial equations (20) determining the set DC. If the codimension of DC were exactly n + 2, then projecting onto the space $Cαm$ along the (n + 1)-dimensional $U×C$ would produce a set DV of at least codimension 1, and thus, for generic parameters α, all critical points are nondegenerate. This dimension-counting was part of our intuition behind Conjecture 7.

Unfortunately, the codimension of an algebraic set (or at least of some of its irreducible components) could be less than the number of defining equations (in our case, n + 2). Hence, one can try to figure out the dimensions (and thus codimensions) of the irreducible components, which is, as the example below shows, sometimes possible but far from being easy.

We consider the discrete periodic graph Γ shown in Fig. 5. The square fundamental domain W contains two vertices (“atoms”) a and b and nine edges shown with solid lines. We allow connections inside W and to its four adjacent copies, introducing thus more free parameters, which hopefully would make the conjecture more likely to hold. No loops or multiple edges are allowed. Shifts of W by integer linear combinations of basis vectors e₁ and e₂ tile the plane. We write V and E for the sets of vertices and edges of Γ, respectively.

The graph Γ is equipped with a periodic weight function α (an analog of a metric or an anisotropic diffusion coefficient) that assigns to each edge a non-negative number.

Let us denote the set of non-negative real numbers by $R+$⁠. Given $α=(α1,…,α9)∈R+9$⁠, we can assign the weights α_j, j = 1, …, 9 to the edges from the fundamental domain W, as shown in Fig. 5. The entire structure Γ and all edge weights can be obtained from W by $Z2$-shifts (with the basis e₁, e₂). Define a divergence (or Laplace–Beltrami) type operator L_α acting on the graph Γ as follows:

where u, v ∈ V and α(e) is the weight of edge e. When this does not lead to confusion, we will use the notation L instead of L_α. This is a discrete analog of a second-order divergence type elliptic partial differential operator.

For each k = (k₁, k₂) from the Brillouin zone B = [ − π, π)², let L(k) be the Bloch Laplacian that acts as (21) on the set of functions defined on Γ that satisfy the Floquet condition

for all $(p1,p2)∈Z2$ and all u ∈ V. Such a function f is determined by its restriction to the fundamental domain W. Due to the direct integral decomposition (6), one obtains

Since there are two vertices inside the fundamental domain, each operator L(k) acts on a two-dimensional space of functions defined on the two atoms and thus has a spectrum σ(L(k)) = {λ₁(k), λ₂(k)}, where λ₁(k) ≤ λ₂(k).

Our second main result is the following.

We provide three arguments for Theorem 20. The first uses the paradigm of numerical algebraic geometry.⁴⁷ While it yields a detailed understanding of the set $DC⊂Cα9×(C\{0})2×Cλ$ of degenerate critical points of the dispersion relation, it is not a traditional proof as the results of the numerical computation are not certified in the sense of Ref. 24. The second argument uses Theorem 16, and it is “probabilistic” in the sense of Corollary 17. We give a third argument that is a proof in the traditional sense. The value of these arguments is that they illustrate the possibilities of ideas and methods from computational algebraic geometry for studying such questions.

We express the condition that the dispersion relation of L_α has a degenerate extremum in terms of the Floquet multipliers $z=(z1,z2)≔(eik1,eik2)$ instead of quasimomenta k ∈ B. To start, notice that

and

For each $z∈T2≔{(z1,z2)∈C2:|z1|=|z2|=1}$⁠, write Λ_α(z) for the operator that acts as (21) on the set of functions defined on Γ that satisfy the condition

The spectrum σ(Λ_α(z)) of Λ_α(z) coincides with σ(L_α(k)) for k ∈ B such that $(z1,z2)=(eik1,eik2)$⁠.

From (22) and (23), if the dispersion relation of L_α has a degenerate extremum, then there exist $z=(z1,z2)∈T2$ and $λ∈R$ such that λ ∈ σ(Λ_α(z)) with λ being a critical point,

and at which the Hessian vanishes,

Note that $DCR$ contains all points of the dispersion relation of Λ_α where λ is a degenerate extremum. Consequently, its projection $DvR$ into the space $R+9$ of weights α includes the set of weights for which there is a degenerate extremal value of a band function for Λ_α.

Our Proof of Theorem 20 proceeds in three steps:

1. $DCR$ is a subset of an algebraic variety $DC⊂Cα9×(C\{0})2×Cλ$⁠.

2. DC has dimension eight.

3. The projection DV of DC to the $Cα9$ of complex weights α has dimension at most eight. This implies that the semi-algebraic (see the remark below) set $DvR⊂R+9∩Dv$ has a positive codimension in $R+9$⁠, which will complete the Proof of Theorem 20.

Steps (1), (2), and (3) are consequences of Lemmas 23, 24, and 25, proven below.

We derive equations that vanish on $DCR$⁠. For $z∈T2$⁠, in the ordered basis (f(a), f(b)), the operator Λ_α(z) is represented by the 2 × 2 matrix $Aα(z)=(ajl)j,l=12$ with entries

The eigenvalues of the matrix A_α(z) form the spectrum of Λ_α(z). Dropping the subscript α, λ belongs to the spectrum of Λ(z) if and only if it satisfies the characteristic equation,

For j = 1, 2, the implicit differentiation of (26) leads to an expression of $∂λ∂zj$ as a rational function. The vanishing of its numerator is equivalent to $∂λ∂zj=0$⁠, giving the consequence of (24),

If we now compute $∂2λ∂zi∂zj$ implicitly using (26) and that $∂λ∂z1=∂λ∂z2=0$ using (24), we obtain

Thus, the vanishing of the Hessian determinant (25) implies that

Let g₁, …, g₄ be, respectively, the left-hand sides of the characteristic equation (26), the two equations (27) for critical points of the dispersion relation, and the Hessian equation (28). These are rational functions in the variables α₁, …, α₉, z₁, z₂, λ with an interesting structure. They are polynomials of degrees 2, 2, 2, and 4 in α₁, …, α₉, λ (homogeneous in α) and Laurent polynomials in z₁, z₂—their denominators all have the form $z1n1z2n2$ for some $n1,n2∈N$⁠. Thus, g₁, …, g₄ are the Laurent polynomials that are defined on $Cα9×(C\{0})2×Cλ$⁠. Let DC be the algebraic variety defined by g₁ = g₂ = g₃ = g₄ = 0. This implies our first lemma.

To study the variety $DC⊂C9×(C\{0})2×C$⁠, for each j = 1, …, 4, let f_j be the numerator of g_j. Then, f₁, …, f₄ are the ordinary polynomials in α, z, λ. Let $P⊂C12$ be the algebraic variety defined by the vanishing of f₁, …, f₄. Then, $DC=P∩(C9×(C\{0})2×C)$⁠, but we may have DC ≠ P, as P may have components where z₁z₂ = 0. Indeed, that is the case.

In Subsection V B, we describe the decomposition of P into irreducible components, which proves Lemma 24.

The dimension of the image of an algebraic variety X under an algebraic map is contained in an algebraic variety whose dimension is at most that of X (see Sec. I.6.3 in Ref. 45). Combined with Lemma 24, this implies our third lemma, which together with the result of Ref. 31 completes the Proof of Theorem 20.

We describe computations that establish Lemma 24 and therefore Theorem 20. They, as well as the derivations of Subsection V A, are archived on the website www.math.tamu.edu/sottile/research/stories/dispersion/ that accompanies this article.

We used the software Bertini,⁷ which is freely available and implements many algorithms in numerical algebraic geometry.⁴⁷ We started with the polynomials f₁, …, f₄, which are the numerators of (26), (27), and (28) and define the algebraic variety $P⊂C12$⁠. Bertini used the algorithms of regeneration,²³ numerical irreducible decomposition,⁴⁶ and deflation³⁹ to study P, determining its decomposition into irreducible components, as well as the dimension and degree of each component, and the multiplicity of P along that component. A component is singular if the multiplicity is at least 2. For each component X of P, it computes the points of Y ∩ X, where Y is a general affine linear subspace of $C12$ in general position with dimY + dimX = 12. The number of points is the degree of X, and examining the coordinates of the points reveals information about the component X. We sketch the consequence of that computation.

First, P is defined by four equations in $C12$ and has ten irreducible components of dimensions eight and nine. Specifically, it has seven components of dimension eight and three of dimension nine. On all three components of dimension nine, one of z₁ or z₂ vanishes. One component has degree 4 and multiplicity 2, and on it, z₁ = z₂ = 0. The other two components have multiplicity 1 and each has degree 8. On one, z₁ = 0, and on the other, z₂ = 0. These components do not lie in DC as z₁z₂ ≠ 0 on DC. This already implies that DC has dimension eight and therefore implies Theorem 20.

The seven components of P of dimension eight are all components of DC. One component has degree 744 and is non-singular. On all other components, we have $z12=z22=1$⁠. There are two further non-singular components of degree 8. On one, (z₁, z₂) = (1, −1), and on the other, (z₁, z₂) = (−1, 1). The remaining four components are singular. One has degree 8 and multiplicity 2, and on it, (z₁, z₂) = (−1, −1). Another has degree 3 and multiplicity 2, and on it, (z₁, z₂) = (1, 1) and λ is not constant. On the remaining two components, λ = 0 and (z₁, z₂) = (1, 1). One has degree 3 and multiplicity 2, and the other has degree 1 and multiplicity 4.

This computation does not constitute a traditional proof, as Bertini does not certify its output. We give an alternative verification using the dichotomy of Theorem 16 that relies on a symbolic computation and then a rigorous proof that uses geometric combinatorics.

We provide an “almost surely” verification of Theorem 20, using symbolic computation, which is exact and certified. For this, we select a “random” point $α∈R+9$ and to check that α ∉ DV and hence that $α∉DvR$ by showing that $DC∩({α}×(C\{0})2×C)$ is empty. An application of Theorem 16 then shows that Theorem 20 is almost surely valid in the sense of Corollary 17.

Below we choose an integer point, which simplifies computations significantly (in fact, many such points have been tested). One wonders why this is a “random” choice. Integer points are Zariski dense in the whole space, and thus, in our problem, a “random” choice of an integer point is as good as choosing any “random” point.

Figure 6 shows the dispersion relation for L_{(1,2,3,4,5,6,7,8,1)}. The horizontal plane is at λ = 0, and the domain is $k∈[−π2,32π]$⁠.

We now present a valid proof of Theorem 20. Since A_α(z) is a 2 × 2 matrix, its characteristic polynomial (26) is quadratic in λ with leading coefficient 1, and thus, the variety it defines in $C9×(C\{0})2×C$ (the dispersion relation) has the property that its projection to the (α, z) parameters $C9×(C\{0})2$ is a proper map with each fiber consisting of either two points or a single point of multiplicity 2.

Consider now the set CP that is defined by the characteristic equation (26) and the two critical point equations (27). Then, CP consists of points (α, z, λ) such that λ is a critical point of the dispersion relation. The critical points $(z,λ)∈(C\{0})×C$ for any given $α∈C9$ are the set of solutions to three equations in the three variables z₁, z₂, λ. A celebrated result of Bernstein¹⁵ gives a strict upper bound for the number of isolated solutions counted with multiplicity.

Let us explain this. The exponent of a monomial $z1a1z2a2λa3$ is an integer vector $(a1,a2,a3)∈Z3$⁠. The exponents that occur in the nonzero terms in a Laurent polynomial f form its support. Their convex hull N(f) is the Newton polytope of f. The bound in Bernstein’s theorem is given by Minkowski’s mixed volume of the Newton polytopes of the equations.

The polynomials f₁, f₂, f₃ that define CP have the Newton polytopes shown in Fig. 7. For the characteristic equation f₁, this is the pyramid with vertex (2, 2, 2) and the base square with vertices (0, 2, 0), (2, 0, 0), (4, 2, 0), and (2, 4, 0). The side length of each edge in the base is $22$ and its height is 2, so that its volume is 32/6. The other two are reflections of each other. The first has as the base the hexagon with vertices

and its other vertices are (1, 1, 1) and (3, 1, 1). If all polytopes are translated so that the centers of their bases are at the origin (0, 0, 0), then the second two lie inside the pyramid.

An interesting outcome from this version of the proof is the following result:

Indeed, the crucial mixed-volume computation of Lemma 27 does not react to increasing the number of parameters α.

Lemma 26 and Corollary 17 provide an efficient method to study Conjecture 7 on sufficiently simple discrete periodic graphs. We illustrate this on a case study involving all 2⁹ subgraphs of the graph of Fig. 5, corresponding to choosing a subset S of the nine edges.

Let $f1,…,f4∈Q[α,λ,z1,z2]$ be the polynomials that define the variety P following Lemma 24. Given a subset S of the nine edges, let $IS⊂Q[α,λ,z1,z2,u1,u2]$ be the ideal generated by z₁u₁ − 1, z₂u₂ − 1, and the polynomials obtained from f₁, …, f₄ by setting all parameters α_j equal to zero for j ∉ S. Then, for parameters α_S = (α_i∣i ∈ S), I_S vanishes on the set DC of degenerate critical points on the dispersion relation for the graph Γ_S corresponding to S with parameters α_S.

We have a Maple script that, for each subset S, evaluates the ideal I_S at ten random instances of the parameters α_S. If, for each of these instances of the parameters α_S, it finds that 1 ∉ I_S (so that the corresponding dispersion relation has a degenerate critical point), then it adds S to a set DSG of degenerate subgraphs. This set DSG contains 87 subsets S. This set according to Theorem 28 has the structure of a simplicial complex. That is, if S ∈ DSG and T ⊂ S, then T ∈ DSG. There are 11 maximal subsets (simplices) S in DSG, corresponding to 11 maximal subgraphs of the graph in Fig. 5, which always have degenerate dispersion relations. We display them in Figs. 8–10.

Observe that each of these graphs consists either of one or more $Z$-periodic graphs together with their disjoint copies under translation by $Z$ (thus providing jointly a $Z2$-periodicity) or two disjoint isomorphic copies of a $Z2$-periodic graph. It is a simple exercise that both situations lead to degeneration (even in the continuous case). Thus, all degenerations occur for “obvious” reasons only.

1. In the continuous case, the dispersion relations are not algebraic, and the operators L(z) are unbounded, and thus, the projection of the set DC is not a proper map. One thus needs to deal with projections of sets of zeros of entire functions into subspaces, such as for instance in the classical theorem by Julia.²⁵ The situation there is complicated, and to get any reasonable results about projections of such analytic sets, one needs more information, e.g., assumptions on the growth of the defining function.^2,20 Although such growth estimates do exist (see, e.g., Refs. 33 and 34), the authors have not succeeded in establishing similar results for, say, Schrödinger operators with periodic potentials. We, however, conjecture that an analog of the dichotomy Theorem 16 should hold with the genericity understood in the (unavoidably weaker) Baire category sense.

2. It may be possible to extend our analysis in Sec. V D to other periodic graphs. A starting point could be to determine the Newton polytopes and mixed volumes of the polytopes corresponding to the equations defining the set CP of critical points of the dispersion relation.

3. The initial impression is (see Conjecture 18) that one needs to have sufficiently many free parameters in the operator to expect generic non-degeneracy. It would be, however, interesting to have better understanding of what makes some discrete periodic problems degenerate. The examples of Sec. VI may be instructive.

4. There are 98 disconnected subgraphs of the graph of Fig. 5, which also form a simplicial complex. Of those that do not appear in Figs. 8–10, we show the three that are maximal in Fig. 11.

5. Theorem 28 confirms Conjecture 18 in our specific example. It would be very interesting to know to what extent the conjecture holds in general.

6. It is easy to create (see Ref. 10), using the non-trivial graph topology, compactly supported eigenfunctions. This is known to lead to the appearance of flat components in the Bloch variety and thus degenerate extrema. However, this situation is non-generic: it is destroyed by generic small variations of the lengths (weights) of edges.

7. The reader should notice that we quickly abandon the discussion of the spectral edges only and target a loftier goal—all critical points. It might be easier to understand the generic structures of (much fewer) spectral edges, but the authors have not figured out how to use this distinction.

The authors acknowledge support provided by the NSF. N.D. and P.K. were supported by Grant No. DMS-1517938, P.K. by Grant No. DMS-2007408, and F.S. by Grant No. DMS-1501370. The authors are also grateful to G. Berkolaiko, N. Filonov, J. Hauenstein, I. Kachkovskiy, Minh Kha, L. Parnovsky, and R. Shterenberg for discussions and information.

The data that support the findings of this study are available at https://www.math.tamu.edu/sottile/research/stories/dispersion/index.html.