Degenerate band edges in periodic quantum graphs
DDEGENERATE BAND EDGES IN PERIODIC QUANTUMGRAPHS
GREGORY BERKOLAIKO AND MINH KHA
Abstract.
Edges of bands of continuous spectrum of periodic structures arise asmaxima and minima of the dispersion relation of their Floquet–Bloch transform. Itis often assumed that the extrema generating the band edges are non-degenerate.This paper constructs a family of examples of Z -periodic quantum graphs wherethe non-degeneracy assumption fails: the maximum of the first band is achievedalong an algebraic curve of co-dimension 2. The example is robust with respect toperturbations of edge lengths, vertex conditions and edge potentials. The simpleidea behind the construction allows generalizations to more complicated graphs andlattice dimensions. The curves along which extrema are achieved have a naturalinterpretation as moduli spaces of planar polygons. Introduction
Periodic media play a prominent role in many fields including mathematical physicsand material sciences. A classical instance is the study of crystals, one of the moststable form of all solids that can be found throughout nature. In a perfectly orderedcrystal, the atoms are placed in a periodic order and this order is responsible formany properties particular to this material. On the mathematical level, the station-ary Schr¨odinger operator − ∆ + V with a periodic potential V is used to describe theone-electron model of solid state physics [1]; here V represents the field created by thelattice of ions in the crystal. The resulting differential operator with periodic coeffi-cients has been studied intensively in mathematics and physics literature for almost acentury. A standard technique in spectral analysis of periodic operators is called theFloquet-Bloch theory (see e.g., [26, 27]). This technique is applicable not only to theabove model example of periodic Schr¨odinger operators on Euclidean space, but alsoto a wide variety of elliptic periodic equations on manifolds and branching structures(graphs). Periodic elliptic operators of mathematical physics as well as their periodicelliptic counterparts on manifolds and quantum graphs do share an important featureof their spectra: the so-called band-gap structure (see e.g., [10, 24, 26, 27]). Namely,the spectrum of a periodic elliptic operator can be represented in a natural way as theunion of finite closed intervals, called spectral bands , and sometimes they may leaveopen intervals between them, called spectral gaps . An endpoint of a spectral gap iscalled a gap edge . For each spectral band, there is also a corresponding band function whose image is exactly that spectral band. The set consisting of all graphs of bandfunctions is called the dispersion relation . The analytical and geometrical propertiesof dispersion relations encode significant information about the spectral features of a r X i v : . [ m a t h . SP ] J un BERKOLAIKO AND KHA the operator. Hence studying structural properties of the dispersion relation mayreveal interesting results for periodic differential operators. A well-known and widelybelieved conjecture in physics literature says that generically (with respect to per-turbations of the coefficients of the operator) the extrema are attained by a singleband of the dispersion relation, are isolated, and have non-degenerate Hessian. Thenon-degeneracy of extrema at the edges of the spectrum is often assumed to establishmany important results such as finding asymptotics of Green’s functions of a peri-odic elliptic operator near and at its gap edge [20, 21, 30], homogenization [12–14], orcounting dimensions of spaces of solutions with polynomial growth [28, 29], just toname a few.In the continuous situation, the generic simplicity of spectral gap edges was ob-tained in [23]. The well-known result in [22] established the validity of the fullconjecture for the bottom of the spectrum of a periodic Schr¨odinger operator in Eu-clidean spaces, however the full conjecture still remains unproven for internal edges.It is worth mentioning that in the two dimensional situation, a “variable period”version of the non-degeneracy conjecture was found in [34] and the isolated nature ofextrema for a wide class of Z -periodic elliptic operators was recently established in[18]. In the discrete graph situation, the statement of the conjecture fails for periodicSchr¨odinger operators on a diatomic lattice (see [18]). However, in the example of[18] there are only 2 free parameters to perturb the operator with and therefore thedegeneracy may be attributed to the paucity of available perturbations. To investi-gate this question further, [15] considered a wider class of Z -periodic discrete graphsand it was found that the set of parameters of vertex and edge weights for whichthe dispersion relation of the discrete Laplace-Beltrami operator has a degenerateextremum is a semi-algebraic subset of co-dimension 1 in the space of all parameters.These examples show that the non-degeneracy of gap edges is a delicate issue evenin the discrete setting.In this paper, we propose two examples of periodic metric (or “quantum”) graphswhose Schr¨odinger operator dispersion relation has a degenerate band edge. Re-markably, this band edge remains degenerate under a continuum of perturbations:one may vary edge lengths, vertex coupling constants and the edge potentials. Ourexamples can be considered quantum-graph versions of the counterexample in [18],and they clearly show that the main reason for the degeneracy is not the small num-ber of perturbation degrees of freedom, but rather the drastic effect a suitably chosenrank-1 perturbation has on the topology of the graph.2. The main result
We now introduce the quantum graph of our main theorem and formulate theresult. The description of principal notions used in the main theorem, such as quan-tum graphs, covers and periodicity, and the Floquet–Bloch transform, are deferredto Sections 3.1, and 3.2 correspondingly. Expanded versions of these descriptions areavailable in several sources, such as [6, 10, 27, 35]. These features are also called “threshold effects” [13] whenever they depend only on the in-finitesimal structure (e.g., a finite number of Taylor coefficients) of the dispersion relation at thespectral edges.
EGENERATE BAND EDGES IN PERIODIC QUANTUM GRAPHS 3
We will in fact describe two variants of our graph, X and X ; the main theoremwill apply equally to both. We start by describing one layer of the graph, which lookslike planar hexagonal lattice shown in Figure 1. It has vertices of two types, type A and type B denoted by red filled and blue empty circles correspondingly. The graph X will have δ -type conditions at vertices A and B , with real coupling constants γ A and γ B , γ A (cid:54) = γ B . The graph X will have only Neumann–Kirchhoff (NK) conditionsbut the vertices of type A are decorated by attaching a “tail”, i.e. an edge leadingto a vertex of degree one, shown as a smaller black circle in Figure 1(right). Eitherversion is a Z -periodic graph in R and its period lattice is generated by the twobrown dashed vectors. The edges of the same color (parallel edges) are related by Z -shifts. They are assumed to have the same length and to have the same potential(if any) placed on them. Figure 1.
Two layers of graphs X (left) and X (right) respectively.These layers are Z -periodic with respect to the Bravais lattice gener-ated by the two brown dashed. The only difference between these twolayers is the extra black tails added in the right layer.The Z -periodic graphs X and X are obtained by stacking the correspondinglayers infinitely many times in both directions of the height axis, see Figure 2. Thelayers are connected in a periodic fashion by edges (shown in green) between verticesof type B in a lower level and vertices of type A in the upper level. Roughly speaking,one may think of the result as an infinite sheeted cover of the layers in Figure 1. Inparticular, X is a 3-dimensional topological diamond lattice, see [35]. In Figure 3we sketch a choice of the fundamental domain of the graph X with respect to the Z -periodic lattice.The graphs X and X we defined above are actually the maximal abelian covers offinite graphs (see e.g., [2, 35] for more details on maximal abelian covers of graphs).Taking the quotient of X with respect to the periodic lattice we obtain the respectivegraphs in Figure 4. The graph Γ = X / Z has two vertices, A and B , which arethe images of the vertices of type A and B in X under the canonical covering mapfrom X to Γ . The four edges of Γ are the images of the sets of parallel edges in X . The graph Γ = X / Z has three vertices and five edges. For either graph Γ, thefirst integral homology group is H (Γ , Z ) ∼ = Z . We will be using notation X whena statement applies equally to both X and X ; similarly we use Γ to refer to bothgraphs Γ and Γ .The graphs X are metric graphs: each edge e in X is identified with the interval[0 , (cid:96) ( e )], where (cid:96) ( e ) is the length of the edge e . The lengths of edges related by a BERKOLAIKO AND KHA
Figure 2.
The graph X is generated by stacking together infinitelymany copies of the layer graph along the height axis. A layer is con-nected to the next layer by certain green edges. To get the graph X ,one just simply adds black tails at the red filled vertices of X . Figure 3.
A fundamental domain for the graph X . Here the threegray vertices are not included in the fundamental domain. The graph X can be obtained by shifting this fundamental domain along thethree dashed directions, which are its periods.periodic shift (i.e. belonging to the same Z -equivalence class or having the samecolor) are the same. We denote by (cid:96) j , j ∈ { , . . . , } the distinct lengths of edges EGENERATE BAND EDGES IN PERIODIC QUANTUM GRAPHS 5
A Be e e e e AC Be e e e Figure 4.
The graph Γ (left) and the graph Γ (right). In bothgraphs, the vertices A , B correspond to red-filled and blue-empty typevertices in X , while the vertex C corresponds to the decorated verticesin the black tails in X . Here e is the line CA and e j , 1 ≤ j ≤ A and B .in the graph X ; the graph X has an additional length — the length of the tail— which we denote by (cid:96) . This metric information on X can be viewed as a pull-back of the metric on Γ via the covering map π : X → Γ. Notice that unlike theperiodic realization of graphene and its multi-layer variants, we do not assume thateach hexagon in the layer graph is regular, i.e. the lengths of edges with distinctcolors may be different.On the edges of the graph X we consider the Laplacian − ∆ X = − d dx or, moregenerally, the Schr¨odinger operator − ∆ X + q e ( x ) with piecewise continuous potential q e ( x ). The potential is assumed to be the same on the edges of the same equivalenceclass (color), taking into account the edge’s orientation. This ensures the potentialis Z -periodic like the rest of the graph; we do not impose any other symmetryconditions on q e . Regularity of the potential also plays no role in our examples, thesame results can be extended to L potentials with minor modifications.At every vertex of the graph X , we impose the standard Neumann-Kirchhoffboundary condition; we impose δ -type conditions with distinct coupling constants γ A and γ B (one of them may be zero) on the corresponding vertices of the graph X . Forthe precise definition of vertex conditions, the reader is referred to Section 3.1. Thegraphs X are non-compact, Z -periodic quantum graphs. According to the Floquet-Bloch theory, the spectrum of the operator − ∆ X is the union of the ranges of theband functions λ j = λ j ( k ), j ≥
1, where the quasimomentum k ranges over the torus T := ( R / π Z ) = ( − π, π ] and(1) λ ( k ) ≤ λ ( k ) ≤ · · · for any k ∈ T . Now we state our main result.
Theorem 2.1. (a) The spectrum of the operator − ∆ X has an open gap betweenthe first and the second band functions, i.e. max k λ ( k ) < min k λ ( k ) . (b) If the lengths (cid:96) j ( ≤ j ≤ ) are approximately equal, then there exists anon-trivial one-dimensional algebraic curve µ in T such that λ attains itsmaximum value on µ . Consequently, there exists a degenerate band edge in the spectrum of − ∆ X . BERKOLAIKO AND KHA (c) The degenerate band edge in the spectrum is persistent under a small pertur-bation of edge lengths, vertex coupling constants or edge potentials.
Theorem 2.1 will be proved in Section 4 after reviewing relevant definitions andtools in Section 3. It will become clear during the proof that the phenomenon de-scribed in the Theorem is very robust. Informally speaking, the extremum responsiblefor a band edge is frequently degenerate for any graph where removing a single vertex(but not the edges incident to it) reduces the rank of the fundamental group by 3 ormore. In particular, the condition on the edge lengths in part (b) of the Theoremserves only to insure a degenerate band edge particularly for the first band. Foralmost all choices of edge lengths one can show that a finite proportion of bands willhave degenerate edges. The decorations introduced at vertices A and B ( δ -type conditions in X and thetail edge in X ) serve to break symmetry in the periodic graph and thus create aband gap. If the symmetry is not broken, one would expect the bands to touchalong the curve µ ; see [7] for related results. Finally, the topology of the degeneracysubmanifold µ may be non-trivial in the higher-dimensional analogues of our example.We touch upon it in in Section 5.3. Some preliminaries and notations
Quantum graphs and vertex conditions.
In this section we recall somenotations and basic notions of quantum graphs; for more details the reader is encour-aged to consult [10, 33]. Consider a graph G = ( V , E ) where V and E are the setsof vertices and edges of G , respectively. For each vertex v ∈ V , let E v be the set ofedges e incident to the vertex v . The degree d v of the vertex v is the cardinality ofthe set E v . The graph G is a metric graph if each edge e of the graph is give a length, (cid:96) e and can thus be identified with the interval [0 , (cid:96) e ]. A function f on the graph G is henceforth a collection of functions { f e } e ∈E , each defined on the correspondinginterval.Let us denote by L ( G ) (correspondingly H ( G )) the space of functions on thegraph G such that on each edge e in E , f e belongs to L ( e ) (corresp. H ( e )) and,moreover, (cid:88) e ∈E (cid:107) f (cid:107) L ( e ) < ∞ (cid:32) corresp. (cid:88) e ∈E (cid:107) f (cid:107) H ( e ) < ∞ (cid:33) . G is called a quantum graph if it is a metric graph equipped with a self-adjointdifferential operator H of the Schr¨odinger type acting in L ( G ). We will take H toact as − ∆ G + q e ( x ) on the edge e , where q e are assumed to be piecewise continuous.The domain of the operator will be the Sobolev space H ( G ) further restricted by aset of vertex conditions which involve the values of f e ( v ) and the derivatives df e dx ( v )calculated at the vertices. We list some commonly used vertex conditions below. This is a consequence of Barra–Gaspard ergodicity of quantum graphs: informally, what happensonce for one choice of lengths will happen with finite frequency for almost all choices of lengths. Formore precise statements, see [3, 5, 11, 16]
EGENERATE BAND EDGES IN PERIODIC QUANTUM GRAPHS 7 • Dirichlet condition at a vertex v ∈ V requires that the function f vanishes atthe vertex, f ( v ) = 0 . This is an example of a decoupling condition. Namely, if the Dirichlet condi-tion is imposed at a vertex of degree d >
1, it is equivalent to disconnecting theedges incident to the vertex and imposing Dirichlet conditions at the resulting d vertices of degree 1. • δ -type condition at a vertex v ∈ V requires the function to be continuous at v in addition to the condition(2) (cid:88) e ∈E v df e dx ( v ) = γ v f ( v ) , γ v ∈ R , where df e dx ( v ) is the derivative of the function f e taken in the direction intothe edge. We note that the value f ( v ) is well-defined because of the assumedcontinuity. The real parameter γ v is called the vertex coupling constant . Thespecial case of the δ -type condition with γ v = 0 is the Neumann-Kirchhoff (NK) or “standard” condition. The Dirichlet condition defined above can benaturally interpreted as γ v = + ∞ . • quasi-NK or magnetic condition at a vertex v ∈ V : Assume that the degree ofthe vertex v is d v , E v = { , . . . , d v } and we are given d v unit complex scalar s z , . . . , z d v ∈ S . We impose the following two conditions:(3) (cid:40) z f ( v ) = z f ( v ) = . . . = z d v f d v ( v ) (cid:80) d v j =1 z j df j dx ( v ) = 0 , Of course, the NK condition is a special case of (3) when all z j are equal.If every vertex of the graph G is equipped with one of the above conditions, theoperator H is self-adjoint (see [10, Theorem 1.4.4] and references therein). The lastset of conditions allow one to introduce magnetic field on the graph without modifyingthe operator (see [25] and also [32, 33] for more recent appearances). They also ariseas a result of Floquet–Bloch reduction reviewed in the next section.3.2. Floquet-Bloch reduction.
Let us now return to our periodic graph X . Recallthat the δ -type conditions are imposed at all vertices of X and hence the operator − ∆ X is self-adjoint. A standard Floquet-Bloch reduction (see e.g., [10, 26, 27]) al-lows us to reduce the consideration of the spectrum of − ∆ X to a family of spec-tral problems on a compact quantum graph (a fundamental domain). More pre-cisely, denote by g , g , g some choice of generators of the shift lattice Z . For each k = ( k , k , k ) ∈ ( − π, π ] =: T , let − ∆ ( k ) X be the Laplacian that acts on the domainconsisting of functions u ∈ H loc ( X ) that satisfy the δ -type conditions at vertices alongwith the following Floquet conditions,(4) u g e ( x ) = e ik u e ( x ) , u g e ( x ) = e ik u e ( x ) , u g e ( x ) = e ik u e ( x ) , BERKOLAIKO AND KHA for all x ∈ X and n = ( n , n , n ) ∈ Z . Then − ∆ X is the direct integral of − ∆ ( k ) X and therefore,(5) σ ( − ∆ X ) = (cid:91) k ∈ T σ ( − ∆ ( k ) X ) . The operator − ∆ ( k ) X has discrete spectrum σ ( − ∆ ( k ) X ) = { λ j ( k ) } ∞ j =1 where we assumethat λ j is increasing in j , see (1). The dispersion relation of the operator − ∆ X isthe multivalued function k (cid:55)→ { λ j ( k ) } and the spectrum of − ∆ X is the range of thedispersion relation for quasimomentum k in T . Hence, it suffices to focus on solvingthe eigenvalue problems − ∆ ( k ) X u = λu where λ ∈ R for u in the domain of − ∆ ( k ) X .This problem is unitarily equivalent to the eigenvalue problem on the compact graphΓ,(6) − d dx u = λu, λ ∈ R , where u satisfies the respective vertex conditions at the vertices A and C and thequasi-NK conditions at the vertex B :(7) (cid:40) e ik u ( B ) = e ik u ( B ) = e ik u ( B ) = u ( B ) e ik u (cid:48) ( B ) + e ik u (cid:48) ( B ) + e ik u (cid:48) ( B ) + u (cid:48) ( B ) = γ B u ( B ) , where γ B is taken to be 0 for the graph Γ and u j are the restrictions of the function u to the edges e j . We will use the notation Γ k and Γ k (or Γ k if the distinction betweenthe two graphs is irrelevant) to denote the eigenvalue problem with condition (7) atthe vertex B .From now on, we shall emphasize the vertex conditions pictorially by replacing thenames of the vertices by their corresponding boundary conditions, see Fig. 5. We willuse γ A , NK , D and Q k ,γ B to indicate the δ -type, Neumann–Kirchhoff, Dirichlet andquasi-NK vertex conditions respectively. We will also occasionally use this conventionin the text, e.g., the vertex B in the above graph Γ k will be mentioned as the Q k -vertex. Finally, we will use the symbol λ j (Γ k ) for the j th -eigenvalue of the quantumgraph Γ k . In particular, we have(8) σ ( − ∆ X ) = (cid:91) j ≥ ,k ∈ T (cid:8) λ j (Γ k ) (cid:9) .γ A A Be e Q k ,γ B e e NK e NK AC Be e Q k , e e Figure 5.
The quantum graphs Γ k (left) and Γ k (right) and theirvertex conditions. In the figures, the types of the boundary conditionsare bold letters while the labels of the vertices are regular letters. EGENERATE BAND EDGES IN PERIODIC QUANTUM GRAPHS 9
Eigenvalue comparison under some surgery transformations.
In thissection we list some eigenvalue comparison results that will be useful to prove theexistence of a gap in the dispersion relation in Theorem 2.1(a).The following interlacing inequality is often useful when variation of a coupling con-stant is used to interpolate between different δ -type conditions and also the Dirichletcondition (which is interpreted as the δ -type condition with coupling + ∞ ). Theorem 3.1 (A special case of [8, Theorem 3.4]) . If the graph (cid:98) G is obtained from G by changing the coefficient of the δ -type condition at a single vertex v from γ v to (cid:98) γ v ∈ ( γ v , ∞ ] . Then their eigenvalues satisfy the interlacing inequalities (9) λ k ( G ) ≤ λ k ( (cid:98) G ) ≤ λ k +1 ( G ) ≤ λ k +1 ( (cid:98) G ) , k ≥ . If a given value Λ has multiplicities m and (cid:101) m in the spectra of G and (cid:98) G respec-tively, then the Λ -eigenspaces of G and (cid:98) G intersect along a subspace of dimension min( m, (cid:101) m ) . Note that by (9) , (cid:101) m must be equal to m − , m or m + 1 . For simplicity, from now on, if the graph G is obtained from G by changing the δ -type conditions to Dirichlet conditions at a single vertex, we will say that G is arank one Dirichlet perturbation of the graph G .We now consider the effect on the eigenvalue of the enlargement of a graph, whichis realized by attaching a subgraph at a designated vertex. The following theorem isquoted in the narrowest form that is sufficient for our needs. Theorem 3.2 (A special case of [8, Theorem 3.10]) . Suppose that (cid:98) G is formed fromgraphs G and H by identifying or “gluing” two Neumann–Kirchhoff vertices v ∈ G and w ∈ H . If λ ( H ) < λ ( G ) and the eigenvalue λ ( G ) has an eigenfunction whichdoes not vanish at v then λ ( (cid:98) G ) < λ ( G ) . Topology of moduli spaces of polygons.
Given n positive real numbers { a j } one can ask what is the topology of the space of all planar polygons whoseside lengths are { a j } . Two polygons are identified if they can be mapped into eachother by a composition of rotation and translation. The resulting spaces may not besmooth and their full classification is surprisingly rich, see [17] and references therein.These spaces make an appearance in our question as the degenerate curves on whichthe dispersion relation has an extremum.For our example we will only require the following simple lemma (which followsfrom the results of [17]) addressing the topology of the set of quadrangles with givenfour edge lengths, see Figure 6. Lemma 3.3.
The curve µ of solutions k = ( k , k , k ) ∈ T of (10) (cid:88) ≤ j ≤ e ik j a j + a = 0 is an algebraic curve of co-dimension 2 if and only if (11) a m < (cid:88) j (cid:54) = m a j for every m = 1 , . . . , . P a Qa Ra Sa k k k Figure 6.
Quadrangle corresponding to equation (10). k2k1 -3-2-1 k k2 k1 -3-2-10123 0-3-2-1 k Figure 7.
The set of roots of (10) for two choices of { a j } ; two viewsof the same plot are shown. The ranges are adjusted to k ∈ (0 , π ] and k , k ∈ ( − π, π ] for a smoother plot. Straight red lines correspond to a j = 1 for all j ; Black stars (appear as a thick fuzzy line) are producedusing a = 1 . a = 0 . a = 0 . a = 1. If there is an m with the inequality reversed, the set of solutions µ is empty. Ifthere is an m with inequality turning into equality, the set of solutions is a singlepoint. The topology of µ in this particular case has been described, for example, in [19, Sec12]. The curve is smooth unless there is a linear combination of { a j } with coefficients ± µ is smooth it is either a circle or a disjointunion of two circles. The non-generic cases when µ is not a smooth manifold are ofthe following types: two circles intersecting at a point, two circles intersecting at twopoints and three circles with one intersection among each pair.The latter case arises when all a j are equal. It is shown in red solid line in Figure 7.Note that the plot is on a torus, therefore each pair of parallel lines is actually a singleline forming a circle. A smooth curve µ for a generic choice of a j ≈ EGENERATE BAND EDGES IN PERIODIC QUANTUM GRAPHS 11 Proof of the main result
In this section, we present the details of the proof of Theorem 2.1. Without lossof generality, for the graph Γ we will make the assumption(12) γ A < γ B . Starting with the graph Γ , we introduce two of its modifications. The graph Γ A is obtained by changing the condition at the vertex A to Dirichlet; the graph Γ B isobtained similarly by placing a Dirichlet condition at the vertex B . Rememberingthat a Dirichlet condition is decoupling, we can picture the result as shown in Fig. 8. γ B DDDD γ A DDDD
Figure 8.
The two “star” graphs Γ A (left) and Γ B (right) after dis-connecting the corresponding Dirichlet vertices ( D ).By placing Dirichlet conditions at vertices A or B of the graph Γ , we analogouslyconstruct the two graphs Γ A and Γ B . We remark that the graph Γ A has two connectedcomponents, see Fig. 9 and Fig. 10. Using the tools introduced in Section 3.3 weestablish the following comparison result, which compares the first eigenvalue Γ Aj with the first eigenvalue of Γ Bj , where j is either 1 or 2. DNK NKDDDD
Figure 9.
The quantum graphs Γ A, (left) and Γ A, after disconnect-ing from the Dirichlet vertex of their union Γ A . NK NK DDDD
Figure 10.
The quantum graph Γ B . Lemma 4.1.
The first eigenvalue of Γ B is always strictly less than than the firsteigenvalue of Γ A , (13) λ (Γ B ) < λ (Γ A ) Proof.
The graphs Γ B and Γ A differ only in the coefficient of the δ -type condition atthe vertex of degree 4 (we are in the situation of pure Laplacian, with no potential).Since the coefficient of Γ B (which is γ A ) is smaller than the coefficient of Γ A , seeequation (12), we immediately get from Theorem 3.1 that λ (Γ B ) ≤ λ (Γ A ). Thecase of equality is excluded because the ground state must be non-zero on the vertex ofdegree 4 which means it cannot satisfy δ -type conditions with two different constants(hence it cannot be a common eigenfunction).For the graph Γ we establish two inequalities, λ (Γ B ) < λ (Γ A, ) and λ (Γ B ) <λ (Γ A, ). The first follows by changing the condition at vertex A of the graph Γ B from NK to Dirichlet: the eigenvalue strictly increases (since the eigenfunction of Γ B is non-zero at A ) and the graph decouples into several disjoint parts one of whichcoincides with Γ A, .To prove the second inequality, we start with λ (Γ A, ) > B , and attach to B a Neumann interval of length (cid:96) whosefirst eigenvalue is 0 < λ (Γ A, ). The strict inequality follows from Theorem 3.2. (cid:3) In our terminology, the graphs Γ A and Γ B are the rank one Dirichlet perturbationsof the corresponding graph Γ. The next important observation is that they are also,in fact, the rank one Dirichlet perturbations of the corresponding graph Γ k for any k . Lemma 4.2.
The rank one Dirichlet perturbation of the graph Γ k at the vertex A (corresp. B ) is unitarily equivalent to Γ A (corresp. Γ B ) for any k ∈ T .Proof. Since the Dirichlet perturbation is decoupling, the resulting graphs have nocycles and therefore any quasi-momenta can be removed by a gauge transform, see[10, Thm 2.6.1]. To put it another way, replacing the vertex condition (7) at B withDirichlet removes all dependence on the quasi-momenta k . Similarly, the quasi-NKconditions could be equivalently imposed at the vertex A , where replacing them withDirichlet also removes all dependence on k . (cid:3) Lemma 4.3.
The first eigenvalue λ of − ∆ on Γ B is simple. If (cid:96) = (cid:96) = (cid:96) = (cid:96) , theeigenfunction corresponding to λ is identical on these four edges, φ ≡ φ ≡ φ ≡ φ ,and non-zero except at B .Proof. The proof is identical for Γ B and Γ B . Simplicity of the eigenvalue follows fromgeneral variational principles [32] (or can be deduced from the secular equation forthe corresponding graphs, see also the proof of Proposition 4.4 below). The firsteigenfunction is known to be positive, except where a Dirichlet condition is enforced,for a large family of vertex conditions [32]. Symmetry can be deduced by, for example,restricting − ∆ to the symmetric subspace of the operator’s domain [4], observingthat the first eigenfunction of the restricted operator is positive and concluding thatit corresponds to a positive eigenfunction of the full operator and therefore must bethe ground state. (cid:3) Proof of Theorem 2.1.
Since by Lemma 4.2 Γ A and Γ B are obtained by a rank-1Dirichlet perturbation from the quantum graph Γ k for any k , Theorem 3.1 yields the EGENERATE BAND EDGES IN PERIODIC QUANTUM GRAPHS 13 inequalities(14) λ (Γ k ) ≤ λ (Γ B ) ≤ λ (Γ k ) , and(15) λ (Γ k ) ≤ λ (Γ A ) ≤ λ (Γ k ) , which hold of all k ∈ T . Adding the result of Lemma 4.1, we get(16) λ (Γ k ) ≤ λ (Γ B ) < λ (Γ A ) ≤ λ (Γ k ) , obtaining part (a) of Theorem 2.1.We will now show that the first inequality in (16) turns into equality(17) λ (Γ k ) = λ (Γ B )for k in a one-dimensional curve γ in T .Let ϕ be the λ (Γ B )-eigenfunction of Γ B . By Theorem 3.1, equality (17) holds ifand only if ϕ is also an eigenfunction of Γ k . We denote by ϕ j the restriction of ϕ on e j for 0 ≤ j ≤
4. Obviously, ϕ satisfies the first condition in (7) at the vertex B .Therefore, equality (17) holds if and only if k = ( k , k , k ) ∈ [ − π, π ) is such that(18) (cid:88) ≤ j ≤ e ik j ϕ (cid:48) j ( B ) + ϕ (cid:48) ( B ) = 0 . By Lemma 3.3, the set of solutions of (18) is a non-trivial algebraic curve of co-dimension 2 if(19) 2 max j (cid:12)(cid:12) ϕ (cid:48) j ( B ) (cid:12)(cid:12) < (cid:88) ≤ j ≤ (cid:12)(cid:12) ϕ (cid:48) j ( B ) (cid:12)(cid:12) . If the lengths (cid:96) j (1 ≤ j ≤
4) are approximately equal then (by eigenfunction con-tinuity and Lemma 4.3) all (cid:12)(cid:12) ϕ (cid:48) j ( B ) (cid:12)(cid:12) are approximately equal and condition (19) issatisfied. This completes the proof of part (b).Finally, the robustness of the degenerate gap edge under a small perturbation ofedge lengths or edge potentials follows directly from continuity of eigenvalue andeigenfunction data [9, 31] and the fact that conditions for the degenerate gap edgeare inequalities (13) and (19). (cid:3) With a little extra effort we can provide a quantitative condition on the lengths (cid:96) j to ensure the validity of the quadrangle inequalities (19) whenever all of the deriva-tives ϕ (cid:48) ( B ) , . . . , ϕ (cid:48) ( B ) are not zero. Proposition 4.4.
Let ρ be the unique solution in (2 , to the equation ρ − ρ π , and assume further that min (cid:26)(cid:18) ρ · min ≤ j ≤ (cid:96) j (cid:19) , (cid:96) (cid:27) ≥ max ≤ j ≤ (cid:96) j Then · | ϕ (cid:48) j ( B ) | < | ϕ (cid:48) ( B ) | + | ϕ (cid:48) ( B ) | + | ϕ (cid:48) ( B ) | + | ϕ (cid:48) ( B ) | for each ≤ j ≤ . As aconsequence, the same conclusion in part (b) of Theorem 2.1 holds. Proof.
Without loss of generality, assume that (cid:96) ≥ (cid:96) ≥ (cid:96) ≥ (cid:96) . On the edge e j where 1 ≤ j ≤
4, we write ϕ j ( x ) = α j sin( βx ), where 0 ≤ x ≤ (cid:96) j , α j ∈ R and β = (cid:0) λ (Γ B ) (cid:1) / . Here we identify the vertex B as x = 0 on each edge e j . Observethat(20) 0 < β = λ (Γ B ) / ≤ min ≤ j ≤ (cid:26) π (cid:96) , π(cid:96) j (cid:27) = π (cid:96) This implies that β(cid:96) j ∈ (0 , π/
2] for each j . So min ≤ j ≤ | sin( β(cid:96) j ) | = sin( β(cid:96) ). More-over, from the fact that ϕ j ( (cid:96) j ) (cid:54) = 0 and the continuity of ϕ at the vertex A , wehave β − ϕ (cid:48) j ( B ) = α j = α · sin( β(cid:96) )sin( β(cid:96) j )Therefore, it is enough to show(21) sin( β(cid:96) ) · (cid:88) i =2 β(cid:96) i ) > ρ := (cid:96) (cid:96) ∈ [1 , ρ ] then we get 1 − π ρ > ρ β(cid:96) < π(cid:96) (cid:96) ≤ π ρ and so it implies(22) 1 − ( β(cid:96) ) > (cid:96) (cid:96) Since sin( β(cid:96) ) ≥ β(cid:96) − ( β(cid:96) ) / β(cid:96) j ) ≤ ( β(cid:96) ), (21) follows from (22). (cid:3) Discussion
Our Theorem 2.1 provides a quantum graph counterexample to the mentionedconjecture at the beginning of the paper, about the genericity of non-degeneratespectral edges in spectra of Z d -periodic quantum graphs, where d >
2. Note that thisconstruction can also be modified to provide an example of a Z d -discrete graph whosedispersion relation of the discrete Laplacian operator contains a degenerate band edge.Indeed, let Γ d be the graph with two vertices such that there are exactly d + 1 - edgesbetween them and therefore, its maximal abelian covering X d is a d -dimensionaltopological diamond. One can write down explicitly the dispersion relation of X d andthen proceed a similar calculation as in [18] to derive the degeneracy of the extremaof the band functions.Our construction of the graphs Γ required that the dimension of the dual torus ofquasimomenta k be of dimension at least three. The same method and proof willstill work if we increase the number of edges connecting the two vertices A and B (atleast four edges). In fact, the entire mechanism of the proof is extremely robust: tworank-one perturbations that reduce the number of cycles by 3 or more help createthe gaps between conductivity bands, while a continuum of solutions to an equationsimilar to (18) will make the band edge degenerate. The degeneracy curve thus still EGENERATE BAND EDGES IN PERIODIC QUANTUM GRAPHS 15 has a natural interpretation as the set of possible ( n − n -gon with the given edge lengths; here n is the number of cycles broken by therank one perturbation. For n > Acknowledgments
The work of the first author was partially supported by NSF DMS–1815075 grantand the work of the second author was partially supported by the AMS–Simons Travelgrant. Both authors express their gratitude to Peter Kuchment for introducing themto this exciting topic and to Lior Alon and Ram Band for many deep discussions.We thank an anonymous referee for several improving suggestions.
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EGENERATE BAND EDGES IN PERIODIC QUANTUM GRAPHS 17
G.B., Department of Mathematics, Texas A&M University, College Station, TX77843-3368, USA
E-mail address : [email protected]
M.K., Department of Mathematics, The University of Arizona, Tucson, AZ 85721-0089, USA
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