An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrödinger operator −Δ + *V*(*x*) in $Rn$ with periodic potential near the edges of the spectrum, i.e., near extrema of the dispersion relation. A well-known and widely believed conjecture says that generically (with respect to perturbations of the periodic potential), the extrema are attained by a single branch of the dispersion relation, are isolated, and have nondegenerate Hessian (i.e., dispersion relations are graphs of Morse functions). The important notion of effective masses in solid state physics, as well as the Liouville property, Green’s function asymptotics, and so on hinges upon this property. The progress in proving this conjecture has been slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Moreover, such models are often used for computation in solid state physics (the tight binding model). Alas, counterexamples exist even for Schrödinger operators on simple 2D-periodic two-atomic structures, showing that the genericity fails in some discrete situations. We start with establishing in a very general situation the following natural dichotomy: the non-degeneracy of extrema either fails or holds in the complement of a proper algebraic subset of the parameters. Thus, a random choice of a point in the parameter space gives the correct answer “with probability one.” Noticing that the known counterexample has only two free parameters, one can suspect that this might be too tight for the genericity to hold. We thus consider the maximal $Z2$-periodic two-atomic nearest-cell interaction graph, which has nine edges per unit cell and the discrete “Laplace–Beltrami” operator on it, which has nine free parameters. We then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for this graph. Since the proof is non-trivial and would be much harder for more general structures, we show three different approaches to the genericity, which might be suitable in various situations. It is also proven in this case that adding more parameters indeed cannot destroy the genericity result. This allows us to list all “bad” periodic subgraphs of the one we consider and discover that in all these cases the genericity fails for “trivial” reasons only.

## I. INTRODUCTION

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).

### A. Dispersion relation and spectrum

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 $k\u2208Rn$ (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* = *∑*_{j}*k*_{j}*γ*_{j}. In other words, *u*(*x*) = *e*^{ik⋅x}*p*(*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 $2\pi Zn$ (the **dual lattice** *G** to $G\u2254Zn$, i.e., consisting of all vectors *k* such that $k\u22c5\gamma \u22082\pi Z$ 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 $k\u2208Cn$ the complex vectors *z* with non-zero components,

*Vectors* *z* *in* (2) *are called* **Floquet multipliers** *(the name comes from the Floquet theory for ODEs*^{17,40,50}*, where* *n* = 1*).*

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 $z\gamma =eik\u22c5\gamma =ei\u2211jkj\gamma j$. In other words, *u*(*x*) = *z*^{x}*p*(*x*), where the function *p*(*x*) is $Zn$-periodic.

### B. Characters and quasimomenta

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.

*A*

*character**of a group*

*G*

*is a homomorphism*$\gamma :G\u2192C\{0}$

*, where the set*$C\{0}$

*of non-zero complex numbers is a group with respect to multiplication. In other words,*

*γ*

*satisfies the following conditions:*

*A*

*unitary character**is a character that maps*

*G*

*into the unit circle*$S\u2254{z\u2208C\u2223|z|=1}$

*.*

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

*Every character of*$G=Zn$*can be represented by a vector*$k\u2208Cn$*as follows:*(5)$\gamma (g)=eik\u22c5g,g\u2208Zn,$*where**k*⋅*g*=*∑*_{j}*k*_{j}*g*_{j}*.**The character in*(5)*is unitary if and only if*$k\u2208Rn$*.**Characters (unitary characters) provide all irreducible (unitary irreducible) representations of*$Zn$*.*

### C. Floquet decomposition

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,

*The (real)*

*dispersion relation**(or*

*Bloch variety**)*

*B*

_{L}

*of the periodic operator*

*L*

*is the subset of*$Rkn\xd7R\lambda $

*, where*(

*k*,

*λ*) ∈

*B*

_{L}

*if and only if*

*where*

*p*(

*x*)

*is periodic with the same group of periods as the operator.*

*The* *j*-*th eigenvalue function* *λ*_{j}(*k*) *is the* ** j**-

*th**band*

*function**.*

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

*By allowing both the quasimomentum* *k* *and the spectral parameter* *λ* *to be complex, one defines the* ** complex Bloch variety** $BL,C$

*. For complex*

*k*

*, though, numbering the eigenvalues in their order becomes impossible. In many cases, they become branches of the same irreducible analytic function.*

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

*The complex Bloch variety is an analytic subvariety of*$Ckn\xd7C\lambda $*. Namely, it is a set of all zeros of an entire function**f*(*k*,*λ*)*of a finite exponential order on*$Ckn\xd7C\lambda $*. In some instances, the exponential estimate becomes important, see*7*Sec. VII**.**The projection of the real Bloch variety onto the real**λ**-axis is the spectrum**σ*(*L*)*of the operator**L**in*$Rn$*.**In particular, the projection of the graph of the**j*-*th band function (over real quasimomenta) into the real**λ**-axis is a finite closed interval called the*-*j**th spectral band**I*_{j}*. The spectral bands might overlap or leave open spaces in between called**spectral gaps**, see*Fig. 2*.*

**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.

### D. The spectral edge conjecture

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.

*Generically (with respect to the potentials and other free parameters of the operator, e.g., metric in the Laplace–Beltrami operator), the extrema of band functions satisfy the following conditions:*

*Each extremal value is attained by a single band**λ*_{j}(*k*)*.**The loci**k**of extrema are isolated in the quasimomentum space.**The extrema are nondegenerate, i.e., at them the corresponding band functions have nondegenerate Hessians.*

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 potential^{29} (which is not necessarily true if a magnetic field is involved^{48}). In Ref. 31, the statement (1) of the conjecture was proven. The full conjecture was proven in Ref. 18 in 2*D* for any fixed number of bands and small smooth potentials. The statement (2) was proven in 2*D*^{21} in a stronger form, even without the genericity clause.

### E. “Extended” dispersion relations

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}

### F. Discrete version

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 discrete^{21} 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: *v*_{0} attained at all vertices $(n,m)\u2208Z2$ such that *n* + *m* is even, and *v*_{1} ≠ *v*_{0} when *n* + *m* is odd. Thus, the only free parameters are *v*_{0} and *v*_{1}. 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.

### G. The structure of the paper

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.

## II. DESCRIPTION OF THE DISCRETE CASE

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. Periodic graphs

*An infinite graph* Γ *is said to be* *periodic**(or* $Zn$*-periodic) if* Γ *is equipped with an action of the free abelian group* $G=Zn$*, i.e., a mapping* (*g*, *x*) ∈ *G* × Γ ↦ *gx* ∈ Γ*, such that the following properties are satisfied:*

*Group**action:**For any**g*∈*G**, the mapping**x*↦*gx**is a bijection (permutation) of*Γ*onto itself;*0*x*=*x**for any**x*∈ Γ*, where*$0\u2208G=Zn$*is the neutral element;*(*g*_{1}*g*_{2})*x*=*g*_{1}(*g*_{2}*x*)*for any**g*_{1},*g*_{2}∈*G*,*x*∈ Γ*.**Free:**If**gx*=*x**for some**x*∈ Γ*, then**g*= 0*.**Discrete:**For any**x*∈ Γ*, there is a neighborhood**U**of**x**such that**gx*∉*U**for**g*≠ 0*.**Co-compact:**The set of orbits*Γ/*G**is finite. In other words, the whole graph can be obtained by the**G**-shifts of a finite subset.**Structure**preservation:**gu*∼*gv**if and only if**u*∼*v**. In particular,**G**acts bijectively on the set of edges.**If other parameters are present (e.g., weights at vertices or at edges), the action preserves their values.*

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 $g\u2208Zn\u2282Rn$, 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.

*Due to co-compactness* *(4)**, there exists a finite part* *W* *of* Γ *such that:*

*The union of all**G**-shifts of**W**covers*Γ*,*$\u22c3g\u2208GgW=\Gamma .$*Different shifted copies of**W**, i.e.,**g*_{1}*W**and**g*_{2}*W**with**g*_{1}≠*g*_{2}∈*G**, do not share any vertices.*

*Such a compact subset* *W* *is called a* **fundamental domain** *for the action of* *G* *on* Γ*.*

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

*Note that a fundamental domain* *W* *is not uniquely defined.*

### B. Floquet–Bloch theory

#### 1. Floquet transform on periodic graphs

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.

*We define the*

*Floquet transform**of*

*f*

*as*

*where*

*gv*

*denotes the action of*$g\u2208Zn$

*on the vertex*

*v*∈

*V*

*and*$z=(z1,\u2026,zn)\u2208(C\0)n$

*is the Floquet multiplier.*

*Using quasimomenta instead of the Floquet multipliers,*(8)

*becomes*

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 following identities hold:*

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 $f\u0302(v,z)$ on Γ is **automorphic** with the character *z*^{g} = *e*^{ik⋅g}. It also shows that the values of $f\u0302(v,z)$ 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.

*Let* *W* *be a (finite) fundamental domain of the action of the group* $G=Zn$ *on* Γ*. We will denote* $f\u0302(v,z)|v\u2208w$ *by* $f\u0302(z)$*, where the latter expression is considered as a function of* *z* *with values in the space of functions on* *W**. In other words,* $f\u0302(z)$ *takes values in* $C|W|$*.*

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

*One can also interpret the Floquet transform as follows: one takes a function* *f* *on* Γ *and cuts it into pieces by restricting to the shifted copies* *gW* *of a fundamental domain* *W**. Then, these pieces are shifted back to* *W* *and are taken as (vector valued) Fourier coefficients of the Fourier series* (9) *that defines the Floquet transform.*

#### 2. Floquet transform of periodic difference operators

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 2*D* lattice Γ (see Fig. 4). The group $Z2$ acts on Γ by the shifts by vectors *p*_{1}*e*_{1} + *p*_{2}*e*_{2}, where $(p1,p2)\u2208Z2$ 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 −*e*_{1}, one has $f(b\u2032)=z1\u22121f(b)=e\u2212ik1f(b)$. Similarly, $f(b\u2033)=z2\u22121f(b)=e\u2212ik2f(b)$. 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 $z1mz2m\u2026znm$ with a sufficiently high power *m*, so the resulting matrix $A\u0303(z)$ has polynomial entries, e.g., in the above example, multiplying by *z*_{1}*z*_{2} (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 *z*_{1}*z*_{2} being replaced by the product $z1mz2m\u2026znm$.

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 $A\u0303$ 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.

## III. THE CRITICAL POINT DICHOTOMY

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, $A\u0303(\alpha ,z,\lambda )$ 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.

*Let* *F*_{α}: *z* → *λ* *be a (possibly multiple-valued) function depending on the parameter* *α**.*

*The set**DV*(*F*)*(or just**DV**, if no confusion can arise) consists of points**α*,*where**F*_{α}*has a degenerate critical point.**The set**DC*(*F*)*(or just**DC**, if no confusion can arise) consists of points*(*α*,*z*,*λ*)*such that**z**is a degenerate critical point of**F*_{α}*with the critical value**λ**.*

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.

*Let*$U\u2282(C*)n$

*be a neighborhood of the torus*$Tn$

*,*

*P*(

*z*)

*be a polynomial, and*

*A*(

*α*,

*z*)

*be a finite size matrix polynomially dependent on the parameters*$\alpha \u2208Cm$

*and*$z\u2208Cn$

*. Consider for any*

*α*,

*the equation*

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

*that shows the dependence of the [weighted by*

*P*(

*z*)

*] spectrum of*

*A*

*on the parameters*

*z*

*.*

*Then, the set* *DV* ⋂ *U* *either belongs to a proper algebraic subset of* $Cm$ *or contains the complement of such a set.*

As we have mentioned before, projecting *DC* into the *α*-space $Cm$, we obtain the set *DV* of parameters *α* that we are interested in.

_{α}(

*z*,

*λ*), by using the condition that this function as well as implicitly computed gradient and Hessian of

*λ*with respect to

*z*all vanish. This, indeed, can be done by implicit differentiation using the equation Φ

_{α}(

*z*,

*λ*) = 0,

*λ*with respect to

*z*can be obtained by implicitly differentiating the equation Φ

_{α}(

*z*,

*λ*) = 0. This produces rational expressions whose vanishing is equivalent to the vanishing of their polynomial numerators. Thus, one obtains the system of

*n*+ 2 polynomial equations,

*DC*. We ask the following question: how large can the projection

*DV*≔

*πDC*of

*DC*into the space $C\alpha m$ of parameters

*α*be? As a projection of an algebraic set, its closure is algebraic of dimension not exceeding the dimension of

*DV*(see Theorem 1.25 of Sec. 6.3 in Ref. 45). If the dimension of the projection is less than

*m*, then it is a proper subset of $Cm$, and we have the genericity of the parameters for which all critical points are nondegenerate. The alternative is that the closure of

*DV*is $Cm$, so that for generic

*α*there are degenerate critical points. In this last case, there may yet be a proper algebraic (i.e., “small”) set of parameters

*α*for which all critical points are nondegenerate.□

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.

*If a random (with respect to an absolute continuous probability distribution) choice of parameters provides non-degeneracy, then “almost surely” non-degeneracy holds outside of a proper algebraic subset (i.e., “almost always,”). Analogously, if a random choice of parameters produces a degenerate example, then “almost surely” degeneracy holds generically.*

*Due to the Zariski density of integers, in our (algebraic) situation, a random integer point would also suffice.*

We also formulate the following conjecture:

*Generic non-degeneracy survives under extending the set of parameters, e.g., if in addition to varying the potential, we start varying the metric as well. In other words, changing more parameters cannot make the situation worse.*

## IV. AN EXAMPLE AND THREE ALTERNATIVE APPROACHES

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 $C\alpha m$ along the (*n* + 1)-dimensional $U\xd7C$ 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.

### A. An example of a discrete structure

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 *e*_{1} and *e*_{2} 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 $\alpha =(\alpha 1,\u2026,\alpha 9)\u2208R+9$, 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 *e*_{1}, *e*_{2}). 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* = (*k*_{1}, *k*_{2}) 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 $(p1,p2)\u2208Z2$ 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*)) = {*λ*_{1}(*k*), *λ*_{2}(*k*)}, where *λ*_{1}(*k*) ≤ *λ*_{2}(*k*).

Our second main result is the following.

*The dispersion relation of the operator* *L*_{α} *generically (i.e., outside of an algebraic subset of the parameters* *α**) satisfies all three conditions of Conjecture* 7*.*

## V. PROOF OF THEOREM 20

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\u2282C\alpha 9\xd7(C\{0})2\xd7C\lambda $ 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.

### A. Analytic reduction

We express the condition that the dispersion relation of *L*_{α} has a degenerate extremum in terms of the Floquet multipliers $z=(z1,z2)\u2254(eik1,eik2)$ instead of quasimomenta *k* ∈ *B*. To start, notice that

and

For each $z\u2208T2\u2254{(z1,z2)\u2208C2:|z1|=|z2|=1}$, 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 $z=(z1,z2)\u2208T2$ and $\lambda \u2208R$ 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:

$DCR$ is a subset of an algebraic variety $DC\u2282C\alpha 9\xd7(C\{0})2\xd7C\lambda $.

*DC*has dimension eight.The projection

*DV*of*DC*to the $C\alpha 9$ of complex weights*α*has dimension at most eight. This implies that the semi-algebraic (see the remark below) set $DvR\u2282R+9\u2229Dv$ has a positive codimension in $R+9$, which will complete the Proof of Theorem 20.

*Projections of algebraic sets are not necessarily algebraic, but semi-algebraic, i.e., defined not only by equations, but also by inequalities.*

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

We derive equations that vanish on $DCR$. For $z\u2208T2$, in the ordered basis (*f*(*a*), *f*(*b*)), the operator Λ_{α}(*z*) is represented by the 2 × 2 matrix $A\alpha (z)=(ajl)j,l=12$ 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 $\u2202\lambda \u2202zj$ as a rational function. The vanishing of its numerator is equivalent to $\u2202\lambda \u2202zj=0$, giving the consequence of (24),

If we now compute $\u22022\lambda \u2202zi\u2202zj$ implicitly using (26) and that $\u2202\lambda \u2202z1=\u2202\lambda \u2202z2=0$ using (24), we obtain

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

Let *g*_{1}, *…*, *g*_{4} 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 *α*_{1}, …, *α*_{9}, *z*_{1}, *z*_{2}, *λ* with an interesting structure. They are polynomials of degrees 2, 2, 2, and 4 in *α*_{1}, …, *α*_{9}, *λ* (homogeneous in *α*) and Laurent polynomials in *z*_{1}, *z*_{2}—their denominators all have the form $z1n1z2n2$ for some $n1,n2\u2208N$. Thus, *g*_{1}, *…*, *g*_{4} are the Laurent polynomials that are defined on $C\alpha 9\xd7(C\{0})2\xd7C\lambda $. Let *DC* be the algebraic variety defined by *g*_{1} = *g*_{2} = *g*_{3} = *g*_{4} = 0. This implies our first lemma.

*The set* $DCR$ *of points on the dispersion relation for the family* *L*_{α} *having degenerate critical points is the set of real points of the algebraic variety* *DC**.*

To study the variety $DC\u2282C9\xd7(C\{0})2\xd7C$, for each *j* = 1, *…*, 4, let *f*_{j} be the numerator of *g*_{j}. Then, *f*_{1}, *…*, *f*_{4} are the ordinary polynomials in *α*, *z*, *λ*. Let $P\u2282C12$ be the algebraic variety defined by the vanishing of *f*_{1}, *…*, *f*_{4}. Then, $DC=P\u2229(C9\xd7(C\{0})2\xd7C)$, but we may have *DC* ≠ *P*, as *P* may have components where *z*_{1}*z*_{2} = 0. Indeed, that is the case.

*The dimension of* *P* *is nine and the dimension of* *DC* *is eight.*

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.

*The image* *DV* *of* *DC* *under the projection to* $C\alpha 9$ *has dimension at most eight.*

### B. Numerical algebraic geometry verification

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 *f*_{1}, *…*, *f*_{4}, which are the numerators of (26), (27), and (28) and define the algebraic variety $P\u2282C12$. Bertini used the algorithms of regeneration,^{23} numerical irreducible decomposition,^{46} and deflation^{39} 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 dim*Y* + dim*X* = 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 *z*_{1} or *z*_{2} vanishes. One component has degree 4 and multiplicity 2, and on it, *z*_{1} = *z*_{2} = 0. The other two components have multiplicity 1 and each has degree 8. On one, *z*_{1} = 0, and on the other, *z*_{2} = 0. These components do not lie in *DC* as *z*_{1}*z*_{2} ≠ 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, (*z*_{1}, *z*_{2}) = (1, −1), and on the other, (*z*_{1}, *z*_{2}) = (−1, 1). The remaining four components are singular. One has degree 8 and multiplicity 2, and on it, (*z*_{1}, *z*_{2}) = (−1, −1). Another has degree 3 and multiplicity 2, and on it, (*z*_{1}, *z*_{2}) = (1, 1) and *λ* is not constant. On the remaining two components, *λ* = 0 and (*z*_{1}, *z*_{2}) = (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.

### C. Symbolic computation verification

We provide an “almost surely” verification of Theorem 20, using symbolic computation, which is exact and certified. For this, we select a “random” point $\alpha \u2208R+9$ and to check that *α* ∉ *DV* and hence that $\alpha \u2209DvR$ by showing that $DC\u2229({\alpha}\xd7(C\{0})2\xd7C)$ 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.

*Let* *α* = (1, 2, 3, 4, 5, 6, 7, 8, 1)*. Then, there are no points* $(z,\lambda )\u2208(C\{0})2\xd7C$ *such that* *g*_{1}, *…*, *g*_{4} *all vanish at* (*α*, *z*, *λ*)*.*

A complex number *z* is non-zero $(z\u2208C\{0})$ if and only if there exists $u\u2208C$ with *zu* = 1. For *α* = (1, 2, 3, 4, 5, 6, 7, 8, 1), a Gröbner basis computation in both Maple and Singular^{19} shows that the ideal *I* in $Q[\lambda ,z1,z2,u1,u2]$ generated by *f*_{1}, *…*, *f*_{4}, *z*_{1}*u*_{1} − 1, *z*_{2}*u*_{2} − 1 contains 1. However, then *I* defines the empty set by Hilbert’s Nullstellensatz.□

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 $k\u2208[\u2212\pi 2,32\pi ]$.

### D. Combinatorial algebraic geometry proof

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 $C9\xd7(C\{0})2\xd7C$ (the dispersion relation) has the property that its projection to the (*α*, *z*) parameters $C9\xd7(C\{0})2$ 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 $(z,\lambda )\u2208(C\{0})\xd7C$ for any given $\alpha \u2208C9$ are the set of solutions to three equations in the three variables *z*_{1}, *z*_{2}, *λ*. A celebrated result of Bernstein^{15} gives a strict upper bound for the number of isolated solutions counted with multiplicity.

Let us explain this. The exponent of a monomial $z1a1z2a2\lambda a3$ is an integer vector $(a1,a2,a3)\u2208Z3$. 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 *f*_{1}, *f*_{2}, *f*_{3} that define *CP* have the Newton polytopes shown in Fig. 7. For the characteristic equation *f*_{1}, 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.

*The mixed volume of the three polytopes (*Fig. 7) *is 32.*

This is a consequence of a result of Rojas (Ref. 44, Corollary 9), which is explained in Ref. 16, Corollary 3.7. Let the three translated polytopes be *P*, *Q*, and *R* with *P* being the pyramid. Then, *P* = *P* ∪ *Q* ∪ *R*.

Observe that at least one of the three polytopes (*P*) meets every vertex of *P*. Also, for every edge *e* of *P*, at least two have an edge lying along *e*. Finally, for every facet *F* of *P*, all three polytopes have a facet lying along *F*. A consequence of Ref. 44, Corollary 9 is that the mixed volume of *P*, *Q*, *R* is 3! = 6 times the volume of *P*, which is 32.□

For the point *α* = (1, 2, 3, 4, 5, 6, 7, 8, 1), we use Maple or Singular to show that there are 32 nondegenerate critical points of the dispersion relation. By Lemma 27, this is the maximal number of critical points, and so we conclude that *α* is a regular value of the projection map $\pi :CP\u2192C9$. Furthermore, there is a neighborhood *U* of *α* such that over it the map *π* is a 32-sheeted cover and therefore is proper near *α*.

The set *DC* of degenerate critical points is a closed subset of *CP*, and thus, its projection to $C9$ is proper near *α*. Thus, its image *DV* is closed in a neighborhood of *α*. Since *α* ∉ *DV*, this implies that the complement of *DV* in $C9$ contains a neighborhood of *α*. However, any nonempty classical open subset of $C9$ is Zariski dense, and therefore, we conclude that the complement of *DV* in $C9$ contains a nonempty Zariski open set, which completes the proof.□

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

*Under the conditions of Theorem* 20*, the statement of Conjecture* 18 *holds true. In other words, increasing the number of parameters in (e.g., adding more edges to) the two-atomic structure shown in* Fig. 5 *does not change the conclusion on the genericity of Theorem* 20*.*

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

## VI. DEGENERATE SUBGRAPHS

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 2^{9} subgraphs of the graph of Fig. 5, corresponding to choosing a subset *S* of the nine edges.

Let $f1,\u2026,f4\u2208Q[\alpha ,\lambda ,z1,z2]$ be the polynomials that define the variety *P* following Lemma 24. Given a subset *S* of the nine edges, let $IS\u2282Q[\alpha ,\lambda ,z1,z2,u1,u2]$ be the ideal generated by *z*_{1}*u*_{1} − 1, *z*_{2}*u*_{2} − 1, and the polynomials obtained from *f*_{1}, *…*, *f*_{4} 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.

## VII. CONCLUSIONS AND FINAL REMARKS

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.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.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.

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.

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

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.

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.

## ACKNOWLEDGMENTS

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.

## VI DATA AVAILABILITY

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

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