Localized perturbations of periodic structures and the bifurcation of defect modes

0.92 MB·English

by  Vincent Duchêne

Localized perturbations of periodic structures and the
bifurcation of defect modes
Vincent Duchêne, Iva Vukićević and Michael I. Weinstein
December 11, 2013
3
1
0
2 Abstract

n
Waves in extended periodic structures are well-known to spatially disperse and decay in am-
a
plitude as time advances. This dispersion is associated with the continuous spectrum of the
J
underlying differential operator and the absence of discrete eigenvalues. The introduction of lo-
4
calizedperturbations,leadstodefect modes,statesinwhichenergyremainstrappedandspatially

localized. Inthispaperwepresentresultsonthebifurcationofsuchdefectmodes,associatedwith
]
h the emergence of discrete eigenvalues from the continuous spectrum.
p
-
h 1 Introduction
t
a
m Wavesinextendedperiodicstructuresarewell-knowntospatiallydisperseanddecayinamplitude
as time advances. This dispersion (Floquet-Bloch dispersion) is associated with the continuous
[
spectrum (extended states) of the underlying differential operator and the absence of discrete

1 eigenvalues (localized bound states). The introduction of localized perturbations leads to defect
v modes,statesinwhichenergyremainstrappedandspatiallylocalized. Thisphenomenonisofgreat
7 importance in fundamental and applied science - from the existence of stable states of matter in
3
atomicsystemstotheengineeringofmaterialswithdesirableenergytransportpropertiesthrough
8
localized doping of ordered materials.
0
Theprocessbywhichthesystemundergoesatransitionfromonewithonlypropagatingdelo-
.
1 calized states to one which supports both localized and propagating states is associated with the
0 emergenceorbifurcationofdiscreteeigenvaluesfromthecontinuousspectrumassociatedwiththe
3 unperturbed periodic structure. In this paper, we discuss this bifurcation phenomenon in detail
1
for the Schrödinger operator
:
v HQ = −∂x2+Q(x) (1.1)
i
X where Q(x) is a real-valued, sufficiently smooth periodic potential
r
a Q(x+1) = Q(x). (1.2)
The spectrum, spec(H ), of the Schrödinger operator is continuous and is the union of closed
Q
intervals called spectral bands [25]. The complement of the spectrum is a union of open intervals
calledspectral gaps. Thespectrumisdeterminedbythefamilyofself-adjointeigenvalueproblems
parameterized by the quasi-momentum k∈(−1/2,1/2]:
H u(x;k) = E u(x;k) , (1.3)
Q
u(x+1;k) = e2πik u(x;k) . (1.4)
1
Thatis,weseekk−pseudo-periodicsolutionsoftheeigenvalueequation. Foreachk∈(−1/2,1/2],
the self-adjoint eigenvalue problem (1.3)-(1.4) has discrete eigenvalue-spectrum (listed with mul-
tiplicity):
E (k)≤E (k)≤···≤E (k)≤... (1.5)
0 1 b
with corresponding k− pseudo-periodic eigenfunctions:
u (x;k) = e2πikx p (x;k), p (x+1;k) = p (x;k), b≥0. (1.6)
b b b b
The bth spectral band is given by
(cid:91)
B = E (k). (1.7)
b b
k∈(−1/2,1/2]
The spectrum of H is given by:
Q
(cid:91) (cid:91) (cid:91)
spec(H )= B = E (k). (1.8)
Q b b
b≥0 b≥0 k∈(−1/2,1/2]
Since the boundary condition (1.4) is invariant with respect to k (cid:55)→ k+1, the functions E (k)
b
can be extended to all R as periodic functions of k. The minima and maxima of E (k) occur at
b
k = k ∈ {0,1/2}; see Figure 1. At such values of k , ∂2E is either strictly positive or strictly
∗ ∗ k b
negative; see Lemma 2.2.
Consider now the perturbed operator H , where V(x) is spatially localized. By Weyl’s
Q+V
theorem on the stability of the essential spectrum, one has spec (H ) = spec (H ) [25].
cont Q+V cont Q
The effect of a localized perturbation is to possibly introduce discrete eigenvalues into the open
spectral gaps. Note that H does not have discrete eigenvalues embedded in its continuous
Q+V
spectrum; see [26, 14].
Inthispaperwepresentadetailedstudyofthebifurcationoflocalizedboundstatesintogaps
of the continuous spectrum induced by a small and localized perturbation of H :
Q
H ≡ −∂2+Q(x)+λV(x) . (1.9)
Q+λV x
HereQ(x)isasufficientlysmooth1−periodicfunctiondefinedonRandV(x)isspatiallylocalized.
The parameter λ is assumed to be positive and will be taken to be sufficiently small. We next
turn to a summary of our results. See Theorem 3.1 and Theorem 3.4 for detailed statements.
Let E =E (k ), k ∈{0,1/2}, denote an endpoint (uppermost or lowermost) of the (b )th
(cid:63) b∗ ∗ ∗ ∗
spectral band, bordering a spectral gap. We show that under the condition:
ˆ
1
∂2E (k ) × |u (x;k )|2 V(x) dx < 0 , (1.10)
k b∗ ∗ b∗ ∗
0
the following holds: There exists a positive number, λ , such that for all 0<λ<λ , H has
0 0 Q+λV
a simple discrete eigenvalue
E(λ) = E + λ2µ + O(λ2+α), α>0 . (1.11)
(cid:63)
which bifurcates from the edge, E =E (k ), of B into a spectral gap.
´ (cid:63) b∗ ∗ b∗
1. If ∂2E (k ) > 0 and 1 |u (x;k )|2 V(x) dx < 0, then µ < 0 and E(λ) lies near the
k b∗ ∗ 0 b∗ ∗
lowermost edge of B ; see the center panel of Figure 1.
b∗ ´
2. If ∂2E (k ) < 0 and 1 |u (x;k )|2 V(x) dx > 0, then µ > 0 and E(λ) lies near the
k b∗ ∗ 0 b∗ ∗
uppermost edge of B ; see the right panel of Figure 1.
b∗
2
For0<λ<λ ,ψλ(x),theeigenstatecorrespondingtotheeigenvalue,E(λ),iswell-approximated
0
in L∞ by, g (λx), where g (y) denotes the unique eigenstate of the homogenized operator
0 0
d d
H = − A + B δ(y) , (1.12)
b∗,eff dy b∗,eff dy b∗,eff
with constant effective parameters A and B . Here,
b∗,eff b∗,eff
1
A = ∂2E (k ) (1.13)
b∗,eff 8π2 k b∗ ∗
is the inverse effective mass associated the the spectral edge, E =E (k ),
(cid:63) b∗ ∗
ˆ
1
B = |u (x;k )|2 V(x) dx , (1.14)
b∗,eff b∗ ∗
0
and δ(y) denotes the Dirac delta mass at y=0.
Remark1.1. Inthecasewhere,Q≡0,thenH =−∂2anditsspectrumconsistsofasemi-infinite
0 x
interval, spec(H )=[0,∞), the union of touching bands with no finite length gaps. Furthermore,
0
p (x;k)≡1, for all |k|≤1/2 and b≥0. The only band-edge band edge is located at E =E (0)=
b (cid:63) 0
0, where we have: k =0, E (k)=4π2k2 and ∂2E (k )=8π2.
∗ 0 k 0 ∗
Inthiscase,ourresultsdescribethebifurcationofaneigenvaluefromtheedgeofthecontinuous
s´pectrumofH0inducedbyasmallandlocalizedperturbation: HλV =−∂x2+λV,underthecondition
V <0; see the discussion below of [27].
R
E (k) E (k) E (k)
2 2 2 2
B
1
B E (k) E (k) E (k)
1 1 1
E (k) E (k) E (k)
B0 0 0 0
k k k
-1/2 0 1/2 -1/2 0 1/2 -1/2 0 1/2
Figure1: Sketchofspectra. Eigenvalues,E (k),k∈(−1/2,1/2],b=0,1,2,...,aredisplayedingreen. Thecontinuous
b
spectrum,isinblue,anddiscreteeigenvaluesareindicatedthroughcrossmarkers. Leftpanelcorrespondsspec(HQ),Q
periodic. Thecenterpanelcorrespondsspec(H ),whereλV issmall,localizednegative. Therightpanelcorresponds
Q+λV
tospec(H ),whereλV issmall,localizedandpositive.
Q+λV
1.1 Previous related work
ThecaseQ≡0,whereH =−∆+V(x)wasconsideredbySimon[27]inoneandtwospatial
Q+λV
dimensions (in spatial dimensions n≥3, it is well known that for sufficiently small λ, −∆+λV
doesnothavediscretespectrumfora´llλsufficientlysmall). Inonedimension, itisprovedthatif
V is sufficiently localized and −∞< V <0, then H has a small negative eigenvalue E(λ) of
R λV
orderλ2;seetheCorollary3.3andthediscussionfollowingit. BorisovandGadyl’shin[1]studythe
spectrum of perturbations of one-dimensional periodic Schrödinger operators and obtain results
3
closely related to the current work. Their proofs rely on ODE techniques. In contrast, we use
frequency space arguments, which can be extended to higher dimensions. Deift & Hempel [5]
obtained results on the existence and number of eigenstates in spectral gaps for operators of the
general type H −λW, where H has a band spectrum and W is bounded. Figotin & Klein [12,
13] studied localized defect modes in context of acoustic and electromagnetic waves. A formal
asymptotic study, in terms of a Floquet-Bloch decomposition, in one and two spatial dimensions
was given in Wang et. al. [28]. Johnson et. al. [23] formulate a variational principle for defect
modes with frequencies in spectral gaps. They use formal trial function arguments to show the
existenceofsuchdefectmodesinspatialdimensionsoneandtwo. Byformalasymptoticarguments,
they obtain the condition (1.10), for the case of the first spectral gap. Results on bound states
and scattering resonances of one-dimensional Schrödinger operators with compactly supported
potentials appear in work of Bronski & Rapti [2] and Korotyaev [19, 20]. Bifurcations of defect
modes into spectral gaps was considered in dimensions d=1,2 and 3 by Hoefer & Weinstein [15]
for operators of the form −∆+Q(x)+ε2V(εx), where Q is periodic on Rd and V is spatially
localized. This scaling was motivated by work of Ilan & Weinstein [16] on the bifurcation of
nonlinear bound states from continuous spectra for the nonlinear Schrödinger / Gross-Pitaevskii
equation. The works [16, 15] employ the Lyapunov-Schmidt reduction procedure of the present
work; see also [24, 7, 6].
Theaboveresultsdescribeperturbationsofspectrainthecasewheretheperturbingpotential
tends to zero in the strong sense. What of the case where the perturbing potential converges
weakly? Thiscorrespondstothequestionoftheeffectivebehaviorofhigh-contrastmicrostructures.
In [8], the authors consider a class of problems, depending on a small parameter, ε, including the
casewherethepotential,q(ε)(x)=q(x,x/ε),convergesweaklyasεtendstozero. Inparticular,we
consideredthesmallεlimitofthespectral,scatteringandtime-evolutionpropertiesforoperators
oftheformH(ε) =−∂2+q(x,x/ε),foralargeclassofpotentialsforwhichy(cid:55)→q(·,y)isoscillatory
x
(including periodic and certain almost periodic case´s) and x (cid:55)→ q(x,·) is spatially localized. An
importantsubtletyarisesinthecasewhereq (x)= q(x,y)dy≡0,i.e. qε tendstozeroweakly.
av R
Inthiscase,classicalhomogenizationtheorybreaksdownatlowenergies. Indeed,thehomogenized
operator, obtained by averaging the potential over its fast variations, is H = −∂2, which does
0 x
not capture key spectral and scattering information. Among these are the low energy behavior
ofthetransmissioncoefficient(relatedtothespectralmeasure)andtheexistenceofabifurcating
bound state at a very small negative energy. We show that the correct behavior is captured by
an effective Hamiltonian with effective potential well: H(ε) = −∂2 −ε2Λ (y), Λ (y) > 0.
eff y eff eff
UsingTheorem3.1andtheresultsof[27],weconcludethatH(ε) hasaboundstatewithnegative
energy of the order ε4, with a precise expansion for ε small. This analysis is then used to derive
a multi-scale local energy time-decay estimate, for localized initial conditions orthogonal to the
bound state, in which the different dispersive time-dynamics on different time-scales is explicit.
These results extend to q (x) belonging to a class of non-generic potentials (including q ≡ 0),
av av
characterizedbythepropertythatq (x)hasazeroenergyresonance;see[8]andcitationstherein.
av
The case where q (x) is a generic (therefore non-zero) localized potential is treated in [9].
av
1.2 Outline, remarks on the proof and future directions
In section 2 we present background material concerning spectral properties of Schrödinger opera-
torswithperiodicpotentialsdefinedonR. Insection3wegiveprecisetechnicalstatementsofour
main results: Theorem 3.1 and Theorem 3.4.
Thestrategyofourproofistotransformtheeigenvalueproblem,usingtheappropriatespectral
transform (Fourier or Floquet-Bloch), to a formulation in frequency (momentum) space. Antici-
pating a bifurcation from the spectral edge, we break the problem into coupled equations for the
frequencycomponentslocatednearthebandedgeandthosewhicharefarfromthebandedge. The
4
precisecutoffdependsonthesmallparameter,λ. WeemployaLyapunov-Schmidttypeapproach
in which we solve for the far components as a functional of the near components and thereby
obtaina(nonlocal)reductiontoabifurcationequationforthenear-frequencycomponents. Unlike
classical Lyapunov-Schmidt applications [22], our reduced equation is infinite dimensional. For λ
small, in an appropriate scaled limit, the bifurcation equation is asymptotically exactly solvable.
In section 4, we prove a general technical lemma, crucial to the analyses of sections 5 and 6,
covering the kinds of bifurcation equations which arise.
Finally, there are several appendices. In appendices A and B, we present results on regularity
and other properties of the Floquet-Bloch eigensolutions E (k), u (x;k) = e2πikxp (x;k); see
b b b
also[25]. InappendixC,wegiveproofs,byabootstrapmethod,ofCorollary3.3andCorollary3.6
which contain more detailed expansions and sharper error terms for the bifurcating eigenstates
than those in Theorem 3.1 and Theorem 3.4.
We conclude this section with several possible extensions of the present work.
1. We believe that our methods can be extended to give detailed properties of the spectral
measure of −∂2+Q+λV near the band edges. Such information could be used to derive
x
the detailed dispersive time-decay, of the sort proved in [8], but in the present case, for
initial conditions, which are spectrally localized near band edges. Initial data with spectral
componentsawayfromthebandedgecansamplearegimewherethedispersionrelationhas
higher degeneracy, yielding different (slower) dispersive time-decay [4].
2. An interesting direction is the extension of the above-discussed results for potentials, qε,
which converge weakly to a non-generic localized potential, e.g. q = 0, to the case where
av
qε converges weakly to a nontrivial periodic potential, Q(x).
3. Finally,anotherinterestingdirectionistheextensionofthemethodsofthecurrentpaperto
thestudyofbifurcationsofeigenvaluesformultiplicativelysmall(see[27])orweaklyconver-
gent spatially localized perturbations of the two-dimensional periodic Schrödinger operator.
1.3 Definitions and notation
WedenotebyC aconstant,whichdoesnotdependonthesmallparameter,λ. Itmaydependon
norms of Q(x) and V(x), which are assumed finite. C(ζ ,ζ ,...) is a constant depending on the
1 2
parameters ζ , ζ , .... We write A(cid:46)B if A≤C B, and A≈B if A(cid:46)B and B (cid:46)A.
1 2
The methods of this paper employ spectral localization relative to the background operator
−∂2+Q(x),whereQ(x)is1−periodic. Forthecase,Q≡0,weusetheclassicalFouriertransform
x
andforQ(x)anon-trivialperiodicpotential,weusethespectraldecompositionofL2(R)interms
of Floquet-Bloch states; see section 1 and section 2 below. The notations and conventions we use
are similar to those used in [15].
1. For f,g∈L2(R), the Fourier transform and its inverse are given by
ˆ ˆ
F{f}(ξ)≡f(cid:98)(ξ)= e−2πixξf(x)dx, F−1{g}(x)≡gˇ(x)= e2πixξg(ξ)dξ.
R R
2. χ and χ are the characteristic functions defined by
(cid:26)
1, |ξ|<δ
χ (ξ)=χ(|ξ|<δ)≡ , χ (ξ)=χ(|ξ|<δ)≡1−χ(|ξ|<δ) . (1.15)
δ 0, |ξ|≥δ δ
3. T and T−1 denote the Gelfand-Bloch transform and its inverse; see section 2.
4. Lp,s(R)isthespaceoffunctionsF :R→Rsuchthat(1+|·|2)s/2F ∈Lp(R),endowedwith
the norm
(cid:107)F(cid:107)Lp,s(R) ≡(cid:107)(1+|·|2)s/2F(cid:107)Lp(R) <∞, 1≤p≤∞. (1.16)
5
5. Wk,p(R)isthespaceoffunctionsF :R→Rsuchthat∂jF ∈Lp(R)for0≤j ≤k, endowed
x
with the norm
k
(cid:107)F(cid:107)Wk,p(R) ≡(cid:88)(cid:107)∂xjF(cid:107)Lp(R) <∞, 1≤p≤∞.
j=0
Acknowledgements: I.V. and M.I.W. acknowledge the partial support of U.S. National Science
FoundationundergrantsNSFEMSW21-RTG:NumericalMathematicsforScientificComputing,
U.S.NSFGrantDMS-10-08855,andtheColumbiaOpticsandQuantumElectronicsIGERTNSF
grant DGE-1069420.
2 Mathematical preliminaries
In this section we summarize basic results on the spectral theory of Schrödinger operators with
periodic potentials defined on R. For a detailed discussion, see for example, [10, 25, 21].
2.1 Floquet-Bloch states
We seek solutions of the k− pseudo-periodic eigenvalue problem
(−∂2+Q(x))u(x)=Eu(x) , u(x+1)=e2πiku(x) , (2.1)
x
for k ∈ (−1/2,1/2], the Brillouin zone. Setting u(x;k) = e2πikxp(x;k), we equivalently seek
eigenfunction solutions of the periodic elliptic boundary value problem:
(cid:0)−(∂ +2πik)2+Q(x)(cid:1)p(x;k)=E(k)p(x;k), p(x+1;k)=p(x;k) (2.2)
x
for each k∈(−1/2,1/2].
Theeigenvalueproblem(2.2)hasadiscretesetofeigenpairs{p (x;k),E (k)} satisfying(1.5)
b b b≥0
and (1.6). The functions p (x;k) may be chosen so that {p (x;k)} is, for each fixed k ∈
b b b≥0
(−1/2,1/2], a complete orthonormal set in L2 ([0,1]). It can be shown that the set of Floquet-
per
Blochstates{u (x;k)≡e2πikxp (x;k), b∈N, −1/2<k≤1/2}iscompleteinL2(R),i.e. forany
b b
f ∈L2(R),
ˆ
(cid:88) 1/2
f(x) − (cid:104)u (·,k),f(cid:105) u (x;k) dk→ 0
b L2(R) b
0≤b≤N −1/2
in L2(R) as N ↑∞.
Recall further that the spectrum of −∂2+Q(x) is continuous, and equal to the union of the
x
closed intervals, the spectral bands; see (1.7), (1.8).
Definition 2.1. We say there is a spectral gap between the bth and (b+1)st bands if
sup |E (k)| < inf |E (k)| .
b b+1
|k|≤1/2 |k|≤1/2
Our study of eigenvalue bifurcation from the band edge E ≡ E (k ) into a spectral gap,
(cid:63) b∗ ∗
requires regularity and detailed properties of E (k) near its edges. These are summarized in the
b
followingresults(seeasketchofE (k)inFigure1,leftpanel). Proofsandreferencestoproofsare
b
given in Appendices A and B; for a detailed discussion see [10, 21, 17, 25].
Lemma 2.2. Assume E (k ) is an endpoint of a spectral band of −∂2+Q(x), which borders on
b ∗ x
a spectral gap. Then k ∈{0,1/2} and the following results hold:
∗
6
1. b even: E (0) corresponds to the left (lowermost) end point of the band,
b
E (1/2) corresponds to the right (uppermost) end point.
b
b odd: E (0) corresponds to the right (uppermost) end point of the band,
b
E (1/2) corresponds to the left (lowermost) end point.
b
2. ∂ E (k )=0;
k b ∗
3. b even: ∂2E (0)>0, ∂2E (1/2)<0;
k b k b
b odd: ∂2E (0)<0, ∂2E (1/2)>0;
k b k b
4. ∂3E (k )=0;
k b ∗
5. E (k ) is a simple eigenvalue of the eigenvalue problem (2.1).
b ∗
Lemma 2.3. Let E (k ) denote a simple eigenvalue; thus k =k as above applies. Then, there
b 1 1 ∗
are analytic mappings k(cid:55)→E (k), k(cid:55)→u (x;k), with u normalized, defined for k in a sufficiently
b b b
small complex neighborhood of k . Moreover, for k real and in this neighborhood (E (k),u (x;k))
1 b b
are Floquet-Bloch eigenpairs.
2.2 The Gelfand-Bloch transform
Given f ∈L2(R), we introduce the transform T and its inverse as follows
ˆ
T{f(·)}=f(cid:101)(x;k)=(cid:88)e2πinxf(cid:98)(k+n), T−1{f(cid:101)(x;·)}(x)= 1/2 e2πixkf(cid:101)(x;k)dk.
n∈Z −1/2
One can check that T−1T =Id. Let
u(x;k) = e2πikxp(x;k) (2.3)
denote a Floquet-Bloch mode. Then, by the Poisson summation formula, we have that
(cid:104)u(·,k),f(cid:105)L2(R) = (cid:104)p(·,k),f(cid:101)(·;k)(cid:105)L2per([0,1]) .
Define ˆ
1
Tb{f}(k)≡(cid:104)pb(·;k),f(cid:101)(·;k)(cid:105)L2per([0,1]) ≡ pb(x;k)f(cid:101)(x;k)dx. (2.4)
0
We recall that by completeness of the {p (x;k)} , the spectral decomposition of f ∈ L2(R) in
b b≥0
terms of Floquet-Bloch states is
ˆ
(cid:88) (cid:88) 1/2
f(cid:101)(x;k) = Tb{f}(k)pb(x;k) , f(x) = Tb{f}(k)ub(x;k)dk.
b≥0 b≥0 −1/2
Recall the Sobolev space, Hs, the space of functions with square integrable derivatives up to
order s. It is natural to construct the following Xs norm in terms of Floquet-Bloch states:
ˆ
(cid:107)φ(cid:101)(cid:107)2Xs ≡ 1/2 (cid:88)(cid:0)1+|b|2(cid:1)s|Tb{φ}(k)|2dk. (2.5)
−1/2b≥0
Proposition 2.4. Hs(R) is isomorphic to Xs for s≥0. Moreover, there exist positive constants
C1, C2 such that for all φ∈Hs(R), we have C1(cid:107)φ(cid:107)Hs(R) ≤(cid:107)φ(cid:101)(cid:107)Xs ≤C2(cid:107)φ(cid:107)Hs(R).
7
Proof. Since E (0)=infspec(−∂2+Q), then L ≡−∂2+Q−E (0) is a non-negative operator
0 x 0 x 0
andHs(R)hastheequivalentnormdefinedby(cid:107)φ(cid:107)Hs ∼(cid:107)(I+L0)s/2φ(cid:107)L2.Usingorthogonality,it
follows that
ˆ
(cid:107)φ(cid:107)2 ∼(cid:107)(I+L )s/2φ(cid:107)2 = (cid:88) 1/2 |T {φ}(k)|2|1+E (k)−E (0)|sdk
Hs 0 L2 b b 0
b≥0 −1/2
ˆ
≈(cid:88)(cid:0)1+|b|2(cid:1)s 1/2 |Tb{φ}(k)|2dk ≡(cid:107)φ(cid:101)(cid:107)2Xs.
b≥0 −1/2
The approximation in the last line follows from the Weyl asymptotics E (k)∼b2; see, for exam-
b
ple, [3]. This completes the proof of Proposition 2.4 .
We conclude this section with a Lemma, which gives various estimates on the Floquet-Bloch
states of H and the spectrum of H , for a class of periodic potentials, Q, and localized
Q Q+λV
potentials, V. These estimates are used within the proof of Theorem 3.4, in section 6.
Lemma 2.5. Assume that Q∈L∞ is 1-periodic, and V is such that (1+|·|)V(·)∈L1. Let Ω be
a small neighborhood of k a simple eigenvalue, such that Lemma 2.3 applies. Then one has:
1
(cid:12) (cid:12)
(a) k∈(s−up21,21](cid:13)(cid:13)pb(·;k)(cid:13)(cid:13)L∞ ≤ k∈(s−up12,12]n(cid:88)∈Z(cid:12)(cid:12)(cid:12)(cid:68)pb(·;k),e2πin·(cid:69)L2([0,1])(cid:12)(cid:12)(cid:12) < ∞, (2.6a)
(cid:12) (cid:12)
(b) ksu∈pΩ(cid:13)(cid:13)∂kpb(·;k)(cid:13)(cid:13)L∞ ≤ ksu∈pΩn(cid:88)∈Z(cid:12)(cid:12)(cid:12)(cid:68)∂kpb(·;k),e2πin·(cid:69)L2([0,1])(cid:12)(cid:12)(cid:12) < ∞. (2.6b)
ˆ ˆ
∞ ∞
(c) sup |pb∗(x;k)|2|xV(x)| dx + sup |pb∗(x;k)||V(x)| dx <∞. (2.6c)
k∈(−1,1] −∞ k∈(−1,1] −∞
2 2 2 2
Proof. We begin by proving that p (x;k) is uniformly bounded for x ∈ R and k ∈ (−1/2,1/2].
b
Since p (·;k) is 1−periodic, it will be bounded if its Fourier coefficients are summable. Thus we
b
study ˆ
(cid:88)(cid:12)(cid:12)(cid:104)pb(·;k),e2πin·(cid:105)L2([0,1])(cid:12)(cid:12) = (cid:88)(cid:12)(cid:12)(cid:12)(cid:12) 1pb(x;k)e−2πinx dx(cid:12)(cid:12)(cid:12)(cid:12).
n∈Z n∈Z 0
Since k ∈(−1/2,1/2], we can use integration by parts for n(cid:54)= 0, the Cauchy-Schwarz inequality
and eqn (2.2) for p (x;k) to obtain
b
(cid:88)(cid:12)(cid:12)(cid:104)pb(·;k),e2πin·(cid:105)L2([0,1])(cid:12)(cid:12)≤(cid:13)(cid:13)pb(x;k)(cid:13)(cid:13)L2([0,1])(cid:13)(cid:13)1(cid:13)(cid:13)L2([0,1])
n∈Z
(cid:12)ˆ (cid:12)
+ (cid:88) (cid:12)(cid:12) 1(Q(x)−E (k))p (x;k)(cid:18) 1 (cid:19)2e−2πinx dx(cid:12)(cid:12)
(cid:12) b b 2πi(n−k) (cid:12)
n∈Z\{0}(cid:12) 0 (cid:12)
(cid:88) 1 (cid:13) (cid:13)
≤1+ 4π2n2(cid:13)(Q(·)−Eb(k))pb(·;k)(cid:13)L2([0,1]),
n∈Z\{0}
Thus, supk∈(−1/2,1/2](cid:13)(cid:13)pb(·;k)(cid:13)(cid:13)L∞ ≤supk∈(−1/2,1/2](cid:80)n∈Z(cid:12)(cid:12)(cid:104)pb(·;k),e2πin·(cid:105)L2([0,1])(cid:12)(cid:12)<∞.
Letusnowturntothestudyof∂ p (x;k)in(b). Deriving(2.2)withrespecttotheparameter
k b
k yields
(cid:0)−(∂ +2πik)2+Q(x)(cid:1)∂ p (x;k)=E (k)∂ p (x;k)+(∂ E (k)+4πi∂ −8π2k)p (x;k).
x k b b k b k b x b
8
Following the same method as above, one obtains
(cid:13)(cid:13)∂kpb(·;k)(cid:13)(cid:13)L∞ ≤(cid:88)(cid:12)(cid:12)(cid:104)∂kpb(·;k),e2πin·(cid:105)L2([0,1])(cid:12)(cid:12)
n∈Z
(cid:13) (cid:13) (cid:13) (cid:13)
≤C((cid:13)Q(cid:13)L∞,Eb(k))(cid:13)∂kpb(x;k)(cid:13)L2([0,1])+C(∂kEb(k)).
(cid:13) (cid:13)
The finiteness of (cid:13)∂kpb(x;k)(cid:13)L2([0,1]) and ∂kEb(k) for k∈Ω is a consequence of Lemma 2.3; thus
(b) follows. Finally, (c) is a straightforward consequence of (a), with (1+|·|)V(·)∈L1.
3 Bifurcation of defect states into gaps; main results
Consider the eigenvalue problem: (cid:0)−∂2+Q(x)+λV(x)(cid:1)ψλ = Eλψλ , ψ ∈L2(R), where Q(x)
x
is1−periodic,λissmall,andV(x)isspatiallylocalized. Ourfirstresultconcernsthatcasewhere
Q≡0:
´
Theorem 3.1 (Q≡´ 0). LetV besuchthatV(cid:98) ∈W1,∞(R); thus R(1+|x|)|V(x)|dx<∞suffices.
Assume V(cid:98)(0) = RV < 0. There exist positive constants λ0 and C(V,λ0), such that for all
0<λ<λ , there exists an eigenpair (Eλ,ψλ), solution of the eigenvalue problem
0
(cid:0)−∂2+λV(x)(cid:1)ψλ(x) = Eλψλ(x) (3.1)
x
with negative eigenvalue of the order λ2. Specifically,
(cid:12) ˆ (cid:12)
(cid:12) (cid:104) λ2 (cid:18) (cid:19)2(cid:105) (cid:12)
(cid:12)Eλ − − V (cid:12) ≤ Cλ5/2 . (3.2)
(cid:12) 4 (cid:12)
(cid:12) R (cid:12)
ˆ
(cid:12) (cid:18) (cid:18) (cid:19) (cid:19)(cid:12)
sup(cid:12)(cid:12)ψλ(x) − exp λ V |x| (cid:12)(cid:12) ≤ Cλ1/2. (3.3)
x∈R(cid:12) 2 R (cid:12)
The eigenvalue, Eλ, is unique in the neighborhood defined by (3.2), and the corresponding eigen-
function, ψ, is unique up to a multiplicative constant.
Remark 3.2. Theorem 3.1 shows, and is essentially proved by demonstrating, that for small
positive λ, the leading order behavior of the eigenstate (cid:0)Eλ,ψλ(x)(cid:1) is a scaling of the unique
eigenstate of the attractive Dirac delta potential:
(cid:16) Eλ , ψλ(x) (cid:17) ≈ (cid:0) λ2θ2 , g (λx) (cid:1) ,
0 0
´
where θ0 =−21 RV >0 and g0(y)=e−θ0|y| satisfy
ˆ
(cid:20) (cid:21)
−∂2 + V · δ(y) g (y)=−θ2 g (y). (3.4)
y 0 0 0
R
The error bounds in Theorem 3.1 are not optimal. However, the bootstrap argument of Ap-
pendixCcanbeusedtorecoverahigherorderexpansiononEλ, similartothatobtainedin[27].
´
Corollary 3.3. Assume (1+|x|2)V ∈L1, and V(cid:98)(0) = V(z) dz < 0 . Then Eλ, as defined
R
in Theorem 3.1, satisfies the precise estimate:
ˆ ¨
1 1
Eλ = −λ2 [θ(λ)]2, with θ(λ) = − V − λ V(x)|x−y|V(y)dxdy + O(λ3/2). (3.5)
2 4
R R2
9
S´imon[27]andKlaus[18]proveexpansion(3.5),undertheconditions: (1+|x|)|V(x)∈L1(R)
and V ≤ 0, with the error term o(λ). Corollary 3.3 gives a sharper error term under a more
R
stringent decay condition on V. The proof of Corollary 3.3 is given in Appendix C .
Theorem 3.4 ( Q non-trivial, 1-periodic). Let Q ∈ L∞ be 1-periodic, and let V be such that
(1+|·|)V(·) ∈ L1 and V ∈ L∞. Let E : k ∈ (−1/2,1/2] → R denote the band dispersion
b∗
function associated with the bth band of the continuous spectrum of −∂2+Q(x). Fix a spectral
∗ x
band edge of the (b )th band; thus E =E (k ), where k =0 or k =1/2 (see Lemma 2.2).
∗ (cid:63) b∗ ∗ ∗ ∗
Assume either
ˆ
∞
∂2E (k )>0 and |u (x;k )|2V(x)dx< 0, (3.6)
k b∗ ∗ b∗ ∗
−∞
or ˆ
∞
∂2E (k )<0 and |u (x;k )|2V(x)dx> 0. (3.7)
k b∗ ∗ b∗ ∗
−∞
Then,thereexistpositiveconstants,λ andC =C(λ ,V,Q),suchthatforallλ<λ ,thefollowing
0 0 0
assertions hold:
1. There exists an eigenpair (cid:0)Eλ,ψλ(x)(cid:1) of the eigenvalue problem
(cid:0)−∂2+Q(x)+λV(x)(cid:1)ψλ(x) = Eλψλ(x), ψλ ∈L2(R) . (3.8)
x
2. Eλ and ψλ(x) satisfy the following approximations:
(cid:12) (cid:12)
(cid:12)Eλ − (cid:0) E (k ) + λ2E (cid:1) (cid:12) ≤ Cλ2+1/4 . (3.9)
(cid:12) b∗ ∗ 2 (cid:12)
(cid:12) (cid:12)
sup (cid:12) ψλ(x) − u (x;k )exp(λα |x|)(cid:12) ≤ Cλ1/4. (3.10)
(cid:12) b∗ ∗ 0 (cid:12)
x∈R
where
(cid:12)´ (cid:12)2 ´
E = −(cid:12)(cid:12) −∞∞|ub∗(x;k∗)|2V(x)dx(cid:12)(cid:12) , α ≡ −∞∞|ub∗(x;k∗)|2V(x)dx. (3.11)
2 1 ∂2E (k ) 0 1 ∂2E (k )
2π2 k b∗ ∗ 4π2 k b∗ ∗
By (3.6) and (3.7), we have that α <0. Note also that the direction of bifurcation of Eλ is
0
given by:
sgn(E )= −sgn(cid:0)∂2E (k )(cid:1) .
2 k b∗ ∗
3. The eigenvalue, Eλ, is unique in the neighborhood defined by (3.9), and the corresponding
eigenfunction, ψλ, is unique up to a multiplicative constant.
Remark3.5. ByTheorem3.4,thebifurcatingeigenvalueEλ liesinthespectralgapof−∂2+Q(x)
x
atadistanceO(λ2)nearthespectraledgeE ; seeFigure1. Moreover, E istheuniqueeigenvalue
∗ 2
and g0(y)=eα0|y| is the unique (up to multiplication by a constant) eigenfunction of the effective
(homogenized) Hamiltonian:
ˆ
d 1 d ∞
H = − ∂2E (k ) + |u (x;k )|2V(x)dx×δ(y).
eff dy 8π2 k b∗ ∗ dy b∗ ∗
−∞
ThefollowingrefinementofTheorem3.4canbeprovedviathebootstrapargumentpresented
in Appendix C.
10