Spectra of Discrete Two-Dimensional Periodic Schr\"odinger Operators with Small Potentials

0.8 MB·

by  Mark Embree

SPECTRA OF DISCRETE TWO-DIMENSIONAL PERIODIC
SCHRO¨DINGER OPERATORS WITH SMALL
POTENTIALS
MARK EMBREE AND JAKE FILLMAN
7
1
0
2

n Abstract. We show that the spectrum of a discrete two-dimensional
a
periodicSchro¨dingeroperatoronasquarelatticewithasufficientlysmall
J
potentialisaninterval,providedtheperiodisoddinatleastonedimen-
3
sion. Ingeneral,weshowthatthespectrummayconsistofatmosttwo

intervals and that a gap may only open at energy zero. This sharpens
]
P severalresultsofKru¨gerandmaybethoughtofasadiscreteversionof
S the Bethe–Sommerfeld conjecture. We also describe an application to
. the study of two-dimensional almost-periodic operators.
h
t
a
m 1. Introduction
[ Researchers in mathematics and physics have extensively investigated

1 spectral and quantum dynamical characteristics of one-dimensional Hamil-
v tonians of the form
3
6 (1.1) [Hψ] = ψ +V ψ +ψ , n ∈ Z, ψ ∈ (cid:96)2(Z),
n n−1 n n n+1
8
0 where V : Z → R is a bounded sequence, known as the potential. The
0 most heavily-studied models are those for which V is periodic, almost-
.
1 periodic, or random. Almost-periodic operators can exhibit wild spectral
0 characteristics, such as Cantor spectrum of zero Lebesgue measure and
7
purely singular continuous spectral type. The literature on such operators
1
: is vast; see [9, 11, 25, 26] and references therein. Though such phenom-
v
ena were once thought to be exotic and rare, Cantor spectrum and purely
i
X
singular continuous spectral type turn out to be generic in a rather robust
r sense for many families of one-dimensional operators having the form (1.1)
a
[1, 2, 3, 14, 36]. The more complicated structure of higher-dimensional
analogs of (1.1) makes such models prohibitively difficult to study, even in
simple cases. With some notable exceptions (see, e.g. [21, 22, 28, 29]), spec-
tral properties of aperiodic almost-periodic Schr¨odinger operators in higher
dimensions have proved quite difficult to study.
Recently some success has been achieved by studying operators that are
separable, in the sense that they can be separated into a sum of two com-
muting one-dimensional operators; such separable operators are amenable
M. E. was supported in part by National Science Foundation grant DGE-1545362.
J. F. was supported in part by an AMS-Simons Travel Grant, 2016–2018.
1
2 M.EMBREEANDJ.FILLMAN
to attack, as their spectra can be expressed as the sum of the spectra of
their one-dimensional components, which are well-understood. Even in this
situation, one must deal with delicate challenges, such as arithmetic sums
of Cantor sets and convolutions of singular measures. Initial insight about
these operators and their spectra came from numerical studies, mainly ap-
pearing in the physics literature [10, 15, 16, 17, 24, 39, 40, 41, 42, 47].
Rigorous results have been obtained fairly recently in [13, 19].
The present paper addresses discrete two-dimensional Schr¨odinger oper-
ators on a square lattice, defined by
H = ∆+V, [Vψ] = V ψ
n,m n,m n,m
(1.2) [∆ψ]n,m = ψn−1,m+ψn+1,m+ψn,m−1+ψn,m+1,
n,m ∈ Z, ψ ∈ (cid:96)2(Z2),
with V periodic in the sense that there exist p,q ∈ Z with
+
(1.3) V = V = V for all n,m ∈ Z.
n+p,m n,m+q n,m
When (1.3) holds for some p = (p,q) ∈ Z2, we say V is p-periodic. The
+
study of Schr¨odinger operators on Zd (and more generally on Zd-periodic
lattices) is of interest due to applications in chemistry and physics; see the
survey [5] for instance. Many papers and books have been written about
operators on graphs; see [4, 6, 7, 8, 20, 35] and references therein.
Our main result shows that the spectra of such objects are quite different
from those of operators like (1.1). Concretely, we prove that any periodic
potential in dimension two that is sufficiently small will produce a spectrum
with at most two connected components if p and q are both even, and with
one connected component otherwise. This result contrasts strongly with the
one-dimensional case, in which a generic p-periodic operator has spectrum
with p connected components.
Theorem 1.1. Let p = (p,q) be given. There exists a constant C = C > 0
p
such that the following statements hold true:
(1) If V is p-periodic and (cid:107)V(cid:107) ≤ C, then σ(H ) has at most two
∞ V
connected components.
(2) If at least one of p or q is odd, then σ(H ) is a single interval
V
whenever V is p-periodic and (cid:107)V(cid:107) ≤ C.
∞
ThisresultcanberegardedasadiscreteversionoftheBethe–Sommerfeld
conjecture (in dimension two), which posits that the spectrum of the oper-
ator −∇2 +V acting in L2(Rd) (d ≥ 2) contains a half-line whenever V is
periodic in the sense that there exists a rank-d lattice Λ ⊂ Rd such that
V(x+γ) = V(x) for all x ∈ Rd,γ ∈ Λ.
In particular, the analysis of the discrete operator H = ∆+λV with λ small
mirrorsthatofthehigh-energyregimeofthe(unbounded)operator−∇2+V
in L2(Rd). The Bethe–Sommerfeld conjecture has inspired intense study,
with substantial contributions from many authors, including (but certainly
SPECTRA OF SMALL PERIODIC POTENTIALS 3
not limited to) [23, 27, 34, 43, 44, 45, 48], and culminating in the paper of
Parnovskii [33]. However, our proof techniques here are a bit different than
those used in the continuum setting. In particular, we employ a pair of soft
arguments: one to count eigenvalues, and one to prove that the eigenvalue
counts forbid small potentials from opening gaps at nonzero energies. These
softargumentsmustberefinedonafiniteexceptionalsetusingperturbation
theory for degenerate eigenvalues, showing that gaps cannot form at such
energies.
On the discrete side, Kru¨ger proved part (2) of Theorem 1.1 under the
morerestrictiveassumptiongcd(p,q) = 1[31]. Healsoconstructedexamples
withp = (2,2)forwhichσ(∆+V)containstwointervalsforarbitrarilysmall
V. In fact, with
Vδ = δ(−1)n+m, δ > 0, n,m ∈ Z,
n,m
he shows that σ(∆ + Vδ) ∩ (−δ,δ) = ∅. Thus, our result improves the
resultof[31]toincorporatetheoptimalrangeofvalidityvis-`a-visarithmetic
conditions on p. Moreover, our proof is substantially simpler than Kru¨ger’s
proof of [31, Theorem 6.1], as he uses some sophisticated algebraic tools (cf.
[31, Section 5]). Finally, in the course of the proof, we answer Questions 6.2
and 6.4 in [31]. Question 6.2 asks for optimal conditions on p and q so that
the conclusion of part 2 of Theorem 1.1 holds; we prove that gcd(p,q) odd
suffices. Question6.4askswhetherthereexistsanothermechanismbywhich
one may open gaps in higher dimensions at small coupling; our arguments
answer this question in the negative.
One immediate consequence of Theorem 1.1 is that it is much more dif-
ficult to produce Cantor spectrum in high dimensions. For example, The-
orem 1.1 immediately implies that if a sequence of periodic potentials con-
verges sufficiently rapidly, then the spectrum of the resulting limit-periodic
operator can have at most two connected components. Again, this draws a
strong contrast with one-dimensional limit-periodic operators, which gener-
ically exhibit zero-measure Cantor spectrum [1, 12, 18].
Corollary 1.2. Supposep isasequenceofperiodssuchthatp |p forallj
j j j+1
(in the sense that each component of p divides the corresponding component
j
of p ). There exist δ > 0 with the following property. If V is a p -periodic
j+1 j j j
potential with (cid:107)V (cid:107) ≤ δ for all j, and
j ∞ j
∞
(cid:88)
V = V ,
j
j=1
then σ(∆+V) consists of at most two intervals. If at least one coordinate
of p is odd for every j, then σ(∆+V) is an interval.
j
Proof. ThisfollowsfromTheorem1.1byrepeatingtheargumentsthatprove
[31, Theorem 7.1] verbatim. (cid:3)
4 M.EMBREEANDJ.FILLMAN
In Section 2, we recall some necessary facts about discrete periodic oper-
ators, which we then use to prove Theorem 1.1 in Section 3.
Acknowledgements
The authors thank George Hagedorn for helpful conversations about this
work. J.F. is grateful to the Simons Center for Geometry and Physics for
their hospitality during the program “Between Dynamics and Spectral The-
ory”, during which portions of this work were completed. J.F. also thanks
Robert Israel for a helpful and insightful answer to a relevant question on
Mathematics Stack Exchange.
2. Discrete Periodic Operators: A Brief Review
Let us briefly review the relevant spectral characteristics of discrete peri-
odic operators. In our arguments, we will need some particular facts about
the discrete one-dimensional Laplacian, so we begin by collecting those.
2.1. The Discrete Laplacian in Dimension One. The discrete Lapla-
cian on (cid:96)2(Z) is defined by
[∆u] = u +u , n ∈ Z, u ∈ (cid:96)2(Z).
n n−1 n+1
Theanalysisthatfollowscomesfromviewing∆asaperiodicJacobimatrix;
for more thorough discussions of periodic Jacobi matrices, see [11, 37, 46].
Given r ∈ Z (r ≥ 3) and θ ∈ R, denote by ∆r the self-adjoint matrix
+ θ
 
0 1 e−iθ
 1 0 1 
 
∆r =  ... ... ...  ∈ Cr×r.
θ  
 
 1 0 1 
eiθ 1 0
For r = 1,2, one has to be a little careful, defining
(cid:20) 0 1+e−iθ(cid:21)
∆1 = 2cosθ, ∆2 = .
θ θ 1+eiθ 0
Proposition 2.1. Let r ∈ Z be given. Then,
+
(cid:26) (cid:18) (cid:19) (cid:27)
πj
σ(∆r) = 2cos : 0 ≤ j ≤ r and j is even
0 r
(cid:26) (cid:18) (cid:19) (cid:27)
πj
σ(∆r ) = 2cos : 0 ≤ j ≤ 2r and j is odd
π/2 2r
(cid:26) (cid:18) (cid:19) (cid:27)
πj
σ(∆r) = 2cos : 0 ≤ j ≤ r and j is odd .
π r
For ∆r and ∆r, the eigenvalues ±2 are simple; the other eigenvalues all
0 π
have multiplicity two. All eigenvalues of ∆r are simple.
π/2
SPECTRA OF SMALL PERIODIC POTENTIALS 5
Proof. For each j with 0 ≤ j ≤ r, define the vectors (cid:126)v±(j) by
(cid:126)v±(j) = e±ijrkπ, 1 ≤ k ≤ r.
k
One can readily verify that (cid:126)v±(j) is an eigenvector of ∆r for even j and of
0
∆r for odd j, corresponding to the eigenvalue 2cos(πj/r). Moreover, for
π
0 < j < r,(cid:126)v+(j)and(cid:126)v−(j)arelinearlyindependent, whichgivesthedesired
statements on multiplicities of the eigenvalues of ∆r and ∆r.
0 π
For ∆r and 1 ≤ j ≤ r, define θ = (−1)jπ(2j −1)/(2r), and put
π/2 j
w(cid:126) (j) = e−ikθj 1 ≤ k ≤ r.
k
One can check that w(cid:126)(j) is an eigenvector of ∆r corresponding to the
π/2
eigenvalue 2cos(θ ) for each 1 ≤ j ≤ r. These r distinct points are precisely
j
the eigenvalues given in the proposition. (cid:3)
To handle exceptional energies in arguments that follow, we will use a
perturbative analysis that involves the derivatives of the eigenvalues of ∆r
θ
with respect to θ.
Lemma 2.2. Fix r ∈ Z , and denote the eigenvalues of ∆r by
+ θ
λ (θ) ≤ ··· ≤ λ (θ).
1 r
For every 1 ≤ j ≤ r:
(a) λ is right-differentiable at 0 and left-differentiable at π;
j
(b) for all θ ∈ (0,π), λ is differentiable and (−1)r−jλ(cid:48)(θ) < 0;
j j
(c) for φ ∈ {0,π},
1(cid:113)
(2.1) |λ(cid:48)(φ)| = 4−λ (φ)2.
j r j
Proof. That λ is differentiable (even real-analytic) on (0,π) with
j
(−1)r−jλ(cid:48) < 0 thereupon is well-known [37, Theorem 5.3.4]. Moreover,
j
by general eigenvalue perturbation theory, it is known that λ enjoys a con-
j
tinuously differentiable extension through the points 0 and π; see, e.g. [30,
Theorem II.6.8]. Thus, we need only concentrate on proving (2.1). We will
provethisinthecasewhenr iseven. Theproofforoddr isidentical, except
−2 = λ (π) instead of λ (0).
1 1
Let D denote the associated discriminant, defined by
(cid:20) (cid:21)
z −1
D(z) = tr(Tr), T = , z ∈ C.
z z 1 0
(Tr denotestherthpowerofthematrixT .) Givenanormalizedeigenvector
z z
w(cid:126) of ∆r corresponding to the eigenvalue λ (θ) of ∆r, it is straightforward
θ j θ
to verify that
(cid:20) (cid:21) (cid:20) (cid:21)
w(cid:126) w(cid:126)
Tr 2 = eiθ 2 ,
λj(θ) w(cid:126)1 w(cid:126)1
and hence, since det(T ) = 1,
z
(2.2) D(λ (θ)) = 2cosθ for all θ ∈ [0,π].
j
6 M.EMBREEANDJ.FILLMAN
By a straightforward induction, one can check that
(2.3) D(2cosη) = 2cos(rη) for every η ∈ [0,π].
Concretely, it is easy to verify that (2.3) holds when r = 1,2. Inductively,
if (2.3) holds for r and r−1, then, by the Cayley–Hamilton theorem,
tr(Tr+1 ) = 2cos(η)tr(Tr )−tr(Tr−1 )
2cos(η) 2cos(η) 2cos(η)
= 4cos(η)cos(rη)−2cos((r−1)η)
= 2cos((r+1)η).
In view of (2.3), every point of the form 2cos(πm/r) with 0 < m < r an
integer is a critical point of D. Hence, every eigenvalue of ∆ or ∆ except
0 π
±2 is a critical point of D. Differentiate both sides of (2.2) twice (with
respect to θ) to obtain
D(cid:48)(cid:48)(λ (θ))λ(cid:48)(θ)2+D(cid:48)(λ (θ))λ(cid:48)(cid:48)(θ) = −2cosθ.
j j j j
Since, when j (cid:54)= 1,r, λ (0) is a critical point of D, we deduce
j
D(cid:48)(cid:48)(λ (0))λ(cid:48)(0)2 = −2.
j j
Consequently,
(cid:115)
2
|λ(cid:48)(0)| = − ,
j D(cid:48)(cid:48)(λ (0))
j
for 1 < j < r. Similarly,
(cid:115)
2
|λ(cid:48)(π)| =
j D(cid:48)(cid:48)(λ (π))
j
for all 1 ≤ j ≤ r. Thus, we need to compute D(cid:48)(cid:48) at the critical points of
D. Differentiate (2.3) twice with respect to η and plug in η = πm/r with
1 ≤ m ≤ r−1 an integer to get
(cid:16) (cid:16)πm(cid:17)(cid:17) r2
D(cid:48)(cid:48) 2cos = (−1)m+1 .
r 2sin2(πm/r)
Thus, we obtain (2.1) for φ = 0 and 1 < j < r, as well as for φ = π and
1 ≤ j ≤ r.
It remains to check the derivative at the eigenvalues ±2: for even r, this
amounts to showing that λ(cid:48)(0) = λ(cid:48)(0) = 0. We can explicitly compute the
1 r
derivative at those points using first-order perturbation theory for simple
eigenvalues. Concretely, w(cid:126) ≡ 1 supplies an eigenvector of ∆r corresponding
j 0
to the eigenvalue 2. An explicit calculation gives
∂ (cid:16) (cid:17)
∆r = i eiθ(cid:126)e (cid:126)e(cid:62)−e−iθ(cid:126)e (cid:126)e(cid:62) ,
∂θ θ r 1 1 r
so, by the Feynman–Hellmann theorem (see [38, Theorem 1.4.7] or [30,
Chapter II]), we get
(cid:16) (cid:17)
λ(cid:48)(0) = iw(cid:126)(cid:62) (cid:126)e (cid:126)e(cid:62)−(cid:126)e (cid:126)e(cid:62) w(cid:126) = 0.
r r 1 1 r
SPECTRA OF SMALL PERIODIC POTENTIALS 7
Similar considerations work for the eigenvalue −2 using (cid:126)u = (−1)j, which
j
is an eigenvector of ∆r since r is even. When r is odd, the proof is identical
0
except that (cid:126)u is an eigenvector of ∆r instead of ∆r. (cid:3)
π 0
Remark 2.3. The identity (2.3) shows that D is a (rescaled) Chebyshev
polynomial. This is a special case of a more general fact for periodic Jacobi
matrices; see, e.g. [37, Example 5.7.3].
2.2. Periodic Operators in Dimension Two. Webrieflyrecallthemain
tools that we will need; for a more complete review and enjoyable reading,
see [31, 32]. We consider operators H of the form (1.2) where the potential
is p = (p,q)-periodic in the sense of (1.3). In this situation, we can compute
the spectrum of H using a direct integral decomposition as follows. Let Γ
denote the fundamental domain
Γ = Γ d=ef (cid:0)[0,p)×[0,q)(cid:1)∩Z2.
p
The fibers of the direct integral are given by
H = CΓ d=ef {(ψ ) : ψ ∈ C for each (n,m) ∈ Γ}.
Γ n,m n,m
For each θ,ϕ ∈ [0,π], let HΓ be the operator given by restricting H to H
θ,ϕ Γ
with boundary conditions of phase θ at the vertical boundaries and phase ϕ
on the horizontal boundaries. Concretely,
(cid:104) (cid:105)
(cid:2)HΓ ψ(cid:3) = Hψθ,ϕ , (n,m) ∈ Γ,
θ,ϕ n,m n,m
where ψθ,ϕ : Z2 → C is defined by the conditions ψθ,ϕ = ψ if (n,m) ∈ Γ,
n,m n,m
and
(2.4) ψθ,ϕ = eiθψθ,ϕ , ψθ,ϕ = eiϕψθ,ϕ for all n,m ∈ Z.
n+p,m n,m n,m+q n,m
Equivalently, one can define the space
(cid:96)∞ (Z2) = (cid:8)ψ ∈ (cid:96)∞(Z2) : (2.4) holds(cid:9),
Γ,θ,ϕ
which is isomorphic to H in a canonical fashion. Under this isomorphism,
Γ
HΓ coincides with the restriction of H to (cid:96)∞ (Z2).
θ,ϕ Γ,θ,ϕ
We denote the eigenvalues of HΓ (counted with multiplicity) by
θ,ϕ
λ (θ,ϕ) ≤ λ (θ,ϕ) ≤ ··· ≤ λ (θ,ϕ).
1 2 pq
Together, the spectra of the HΓ operators give a nice characterization of
θ,ϕ
the spectrum of H.
Theorem 2.4. If V is p-periodic, then
pq
σ(H) = (cid:91) σ(cid:0)HΓ (cid:1) = (cid:91) B ,
θ,ϕ j
θ,ϕ∈[0,π] j=1
where Γ = Γ and
p
B = {λ (θ,ϕ) : θ,ϕ ∈ [0,π]}
j j
is called the jth band of σ(H).
8 M.EMBREEANDJ.FILLMAN
Proof. This result is standard; for a proof in the discrete setting, see [31,
Theorem 3.3]. (cid:3)
As a consequence of Proposition 2.1, we can compute the multiplicities
of the eigenvalues of ∆Γ and ∆Γ explicitly on square lattices. Here and
0,0 π,π
throughout, we use ∆Γ to denote HΓ with V ≡ 0.
θ,ϕ θ,ϕ
Lemma 2.5. Let Γ = Γ with r even. Then
(r,r)
(a) ±4 are simple eigenvalues of ∆Γ ;
0,0
(b) 0 is an eigenvalue of ∆Γ with multiplicity congruent to two modulo
0,0
four;
(c) every other eigenvalue of ∆Γ has multiplicity divisible by four;
0,0
(d) every eigenvalue of ∆Γ has multiplicity divisible by four.
π,π
Proof. These observations follow immediately from Proposition 2.1. (cid:3)
We will frequently have recourse to a specific implication of this Lemma:
all eigenvalues of ∆Γ have even multiplicity, while all eigenvalues of ∆Γ
π,π 0,0
have even multiplicity except the extreme eigenvalues ±4, which are simple.
One may identify H with Cpq via the “vectorization” map vec : H →
Γ Γ
Cpq defined by
ψ (cid:55)→(cid:126)v = vec(ψ), (cid:126)v = ψ for (k,(cid:96)) ∈ Γ.
kq+(cid:96)+1 k,(cid:96)
In view of this identification, HΓ enjoys the matrix representation
θ,ϕ
H(0) I e−iθI 
ϕ
 I H(1) I 
 ϕ 
vec◦HΓ ◦vec−1 =  ... ... ...  ∈ Cpq×pq,
θ,ϕ
 
 I H(p−2) I 
 ϕ 
eiθI I H(p−1)
ϕ
where
 
V 1 e−iϕ
j,0
 1 Vj,1 1 
 
H(j) =  ... ... ...  ∈ Cq×q.
ϕ  
 
 1 Vj,q−2 1 
eiϕ 1 V ,
j,q−1
3. Proof of Theorem
3.1. Proof Strategy. Given p = (p,q), let Γ = Γ and view ∆ as a p-
p
periodic operator. For each (θ,ϕ) ∈ B d=ef [0,π]2, we denote the eigenvalues
of ∆Γ by
θ,ϕ
λΓ(θ,ϕ) ≤ ··· ≤ λΓ (θ,ϕ).
1 pq
SPECTRA OF SMALL PERIODIC POTENTIALS 9
The jth band of σ(∆) is then
BΓ = (cid:8)λΓ(θ,ϕ) : (θ,ϕ) ∈ B(cid:9).
j j
Ofcourse,wecaneschewthecalculationofbandsandcomputethespectrum
of ∆ directly, as it is diagonalized by the Fourier transform on Z2. One can
check that
σ(∆) = [−4,4].
Then, in view of Theorem 2.4, we deduce that
(3.1) maxBΓ ≥ minBΓ for every 1 ≤ j < pq.
j j+1
One key fact that we will use frequently is that ∆Γ has the structure
θ,ϕ
∼
of a separable operator. Conceretely, under the natural identification H =
Γ
Cp⊗Cq,
∆Γ ∼= ∆p⊗I +I ⊗∆q.
θ,ϕ θ q×q p×p ϕ
Thus, for every 1 ≤ j ≤ pq, λΓ(θ,ϕ) is of the form λp (θ)+λq (ϕ) for some
j k1 k2
choice of 1 ≤ k ≤ p and 1 ≤ k ≤ q.
1 2
By standard eigenvalue perturbation theory for Hermitian matrices, the
edges of the bands are 1-Lipschitz functions of the underlying potential
(viewed as an element of RΓ equipped with the uniform norm topology;
compare [31, Lemma 3.9]). Heuristically, this means that if λ is small and
V isp-periodic,thenH = ∆+λV canbeviewedasaninfinitesimalperturba-
tiontotheoperator∆thatpreservesthenumberofbandsbutinfinitesimally
shifts their locations. The only place such a perturbation can open a gap
is at the interface between bands, not in the interior of any band. Thus, to
prove Theorem 1.1, it suffices to show that every energy in (−4,4) is in the
interior of at least one spectral band of ∆ (by compactness); see Figure 1.
Away from energy zero, it will be most convenient to treat periods p
that are equal and even, so one may introduce r = lcm(p,q,2), and denote
r = (r,r)throughoutthissection. Obviously, anyp-periodicpotentialisalso
20 20
19 19
18 18
17 17
16 16
15 15
14 14
13 13
12 12
11 11
10 10
9 9
8 8
7 7
6 6
5 5
4 4
3 p=(5,4) 3 p=(10,2)
2 2
1 1
-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
Figure 1. ThetwentyspectralbandsB whoseuniongives
j
σ(∆) = [−4,4] for p = (5,4) (left) and p = (10,2) (right). If
p or q is odd, each E ∈ (−4,4) is in the interior of some B .
j
If p and q are even, E = 0 is only only point in (−4,4) that
is not in the interior of any B .
j
10 M.EMBREEANDJ.FILLMAN
r-periodic, and hence it suffices to show that any E (cid:54)= 0 is in the interior
of at least one BΓr. When p is not even, we must show a similar result for
j
E = 0. In this case it will be technically convenient for at least one period
to be divisible by four, so if p is not even, introduce
(cid:40)
(p,lcm(q,4)), if p is odd;
p(cid:48) = (p(cid:48),q(cid:48)) =
(lcm(p,4),q), otherwise.
Of course, any p-periodic potential is also p(cid:48) periodic. Finally, we note that
interchangingtherolesofpandq leavesthebandsinvariant,andhencethere
is no generality lost in assuming that p is odd.
In view of the foregoing discussion, our goal in this section is to prove the
following theorem.
Theorem 3.1. LetΓ = Γ withr ∈ Z even. For everyE ∈ (−4,4)\{0},
(r,r) +
E ∈ intBΓ for some 1 ≤ j ≤ r2.
j
If p = (p,q) with p odd and q divisible by four, then
0 ∈ intBΓp for some 1 ≤ j ≤ pq.
j
We will prove this result in a sequence of steps. Away from a suitable
exceptional set, a soft eigenvalue counting argument will show that E must
be in the interior of at least one band. The exceptional set corresponds to
theenergiesthatoccuratthecornersoftheBrillouinzone,whichthemselves
correspond to sums of eigenvalues of truncations of the corresponding one-
dimensional Laplacian. Let Γ and r be as in the statement of the theorem,
put
(cid:26) (cid:18) (cid:19) (cid:27)
πj
C = 2cos : j ∈ Z, and 0 ≤ j ≤ r = σ(∆r)∪σ(∆r),
r r 0 π
and define the set of exceptional energies to be
C = C +C = {a+b : a,b ∈ C }.
Γ r r r
The key idea is to construct pairs of points in the Brillouin zone with
different “eigenvalue counts.” More precisely, for each E, we want to find
(θ ,ϕ ),(θ ,ϕ ) ∈ B with E ∈/ σ(∆Γ )∪σ(∆Γ ) and
1 1 2 2 θ1,ϕ1 θ2,ϕ2
(3.2) #(cid:0)σ(∆Γ )∩(−∞,E)(cid:1) (cid:54)= #(cid:0)σ(∆Γ )∩(−∞,E)(cid:1).
θ1,ϕ1 θ2,ϕ2
Figures 2–3 suggest the overall strategy. First, the defintion of the excep-
tional energies is such that σ(∆Γ ),σ(∆Γ ) ⊆ C . The right plot in Fig-
0,0 π,π Γ
ure 2 illustrates that (3.2) holds for (θ ,ϕ ) = (0,0) and (θ ,ϕ ) = (π,π) for
1 1 2 2
E = −0.01 ∈ (−4,4)\C . Indeed, it is simple to show that the eigenvalue
Γ
counts differ for (0,0) and (π,π) whenever E ∈ (−4,4) \ C : Lemma 2.5
Γ
implies that the left-hand side of (3.2) is odd while the right-hand side is
even! (See Proposition 3.2.)
For nonzero exceptional energies, E ∈ C \ {0}, the question is more
Γ
delicate, as E may be an eigenvalue of ∆Γ or ∆Γ having high multiplicity.
0,0 π,π