THE ELECTRONIC STRUCTURE OF SMOOTHLY DEFORMED CRYSTALS …jianfeng/paper/tightbinding.pdf ·...

THE ELECTRONIC STRUCTURE OF SMOOTHLY DEFORMED

CRYSTALS: CAUCHY-BORN RULE FOR THE NONLINEAR

TIGHT-BINDING MODEL

WEINAN E AND JIANFENG LU

Abstract. The electronic structure of a smoothly deformed crystal is ana-

lyzed using a minimalist model in quantum many-body theory, the nonlinear

tight-binding model. An extension of the classical Cauchy-Born rule for crys-

tal lattices is established for the electronic structure under sharp stability

conditions. A nonlinear elasticity model is rigorously derived. The onset of

instability is briefly examined.

1. Introduction

This is the second of a series of papers that are devoted to the study of the elec-

tronic structure of smoothly or elastically deformed crystals, by analyzing various

quantum mechanics models at different levels of complexity, including the Kohn-

Sham density functional theory, Thomas-Fermi type of models and tight-binding

models. Our overall objective is to establish the microscopic foundation of the

nonlinear elasticity theory in terms of quantum mechanics and to examine the

boundary where elasticity theory breaks down. In this regard, our objective is a lot

like the one in [6, 7], except that [6] and [7] considered only classical models, and

in this series we consider quantum mechanics models and focus on the behavior of

the electrons and spins. In particular, in analogy with the stability condition for

elastic waves and phonons established in [6,7], we will establish stability conditions

for charge density waves and spin waves.

In the previous paper [4], we considered the linear model and we examined

the structure of the subspace spanned by the wave functions. In particular, we

extended the construction of Wannier functions for perfect crystals to smoothly

deformed crystals. We also established an analog of the Cauchy-Born rule for this

construction. Roughly speaking, it asserts that locally, the electronic structure of

a smoothly deformed crystal is well approximated by the electronic structure of

a homogeneously deformed crystal with the same deformation gradient. This is

a linearization principle, or more generally, a locality principle: It states that the

electronic structure depends only on the local behavior of the deformation. It is

Date: Oct 12, 2009; Revised: Feb 4, 2010.

1

2 WEINAN E AND JIANFENG LU

in the same spirit as the Cauchy-Born rule for crystal lattices (atoms instead of

electrons) established in [6, 7].

In this paper, we continue this study using the nonlinear tight binding model.

This is a rather popular minimalist type of model in quantum many-body theory.

It can be considered as a very crude discretization of the Kohn-Sham density func-

tional theory, which has become a basic tool in material science and chemistry. The

relative simplicity of the tight-binding model allows us to illustrate more clearly the

main issues in the subject matter.

As in the previous work, our reference point is the electronic structure of perfect

crystals, i.e. perfect lattices, particularly the equilibrium lattice. The solution

we are interested in is a continuation of the electronic structure of the perfect

crystal. This works as long as certain stability conditions are satisfied. One main

objective is to identify these stability conditions. The overall strategy is quite

simple. However, a substantial amount of machinery has to be developed in order

to carry out this program. Related to the topic discussed in this paper is the

work in [3] which establishes the same type of Cauchy-Born approximation for the

(linear) tight-binding model, for which the potential is given. Matrix perturbation

theory was used in [3] which enabled us to establish the validity of the leading

order approximation. However, it does not give us the correct convergence rate

or higher order approximations. More importantly, the stability condition, which

is vital for the work discussed in this paper, is not relevant in [3] since the model

considered there was linear. We should also mention the related work [1] where the

continuum limit for Thomas-Fermi-von Weizsacker model is studied. There, the

stability condition is automatically satisfied since the model is convex. We should

remark that the strategy followed in this paper is quite different from those in [1,3].

For the most part, we will focus on one-dimensional systems. Extension to

high dimensional systems is straightforward, except that the notations are more

complicated. We will discuss this briefly in the last section. We will only be

interested in the electronic structure, so the positions of the atoms are fixed.

The paper is organized as follows. In section 2, we start with some general

discussions of the tight binding model and the problem we are interested in. Section

3 discusses some basic background material for the electronic structure of perfect

crystals. A crucial component of the analysis is the stability condition. To get some

intuitive understanding of this, we give some very simple examples of stability and

instability in section 4. In rough terms, our objective is to describe the electronic

structure of smoothly deformed crystals in terms of the electronic structure of

homogeneously deformed crystals. Therefore we devote section 5 to a discussion of

the electronic structure of homogeneously deformed crystals. The analysis of the

electronic structure of smoothly deformed crystals begins in section 6 where the

main results are presented. There are two main components in the proof of the

CAUCHY-BORN RULE FOR THE NONLINEAR TIGHT-BINDING MODEL 3

main results: Construction of the approximate solution and the convergence of the

Newton-Raphson iteration. These are presented in sections 7 and 8 respectively.

The extension to three dimension will be presented in section 9. We also put in the

appendix the construction of higher order approximate solutions.

Before ending this introduction, we make a remark about notation. We will be

dealing with the electronic structure of undeformed, homogeneously deformed and

smoothly deformed crystals. We will use different subscripts to distinguish objects

in the different cases. For example, we denote the lattice Schrodinger operator as

He for the undeformed case, HA for the homogeneously deformed crystals with de-

formation gradient A, and Hετ for the smoothly deformed crystals where τ describes

the deformed state. Here the subscript e stands for “equilibrium”, the superscript

ε stands for system with lattice constant ε. In addition, we will use both uj and

u(j) to denote the j-th component of the vector u.

2. The tight binding model

We begin with the simplest setting. Assume that there are N atoms on the

line R located at Y1, Y2, . . . , YN respectively and there are Ne electrons. We are

interested in the electronic structure of the whole system. The tight binding model

describes such a system in terms of a discrete Hamiltonian matrix H [8]. The

diagonal matrix elements model the “on-site” self-interaction of the orbitals and

the off-diagonal terms describe the interaction between different sites, known as the

“hopping term”. In the simplest situation, only the nearest neighbor interaction

is taken into account, therefore we obtain a tri-diagonal matrix. The value of the

matrix elements depends on the kind of atoms involved in the system and the

positions of the atoms. In practice, they are often described by empirical functions

which have been calibrated using experimental results as well as results from first

principle calculations.

The electronic structure of the system is obtained by assigning the electrons to

the eigen-states with lowest eigenvalues, subject to the Pauli exclusion principle.

If we neglect the degree of freedom associated with the spins, the Pauli exclusion

principle states that each eigen-state can accommodate at most one electron. The

electron density is then given by:

(1) ρj =∑

k

nk|ψk,j |2.

Here ψk,j is the j-th component of the k-th eigenvector, nk is the occupation number

of the k-th eigen-state, i.e., the number of electrons occupying that state.

Imagine that the system we are interested in is a macroscopic crystal. There are

two length scales in the system: The size of the crystal and the lattice parameter

which is roughly the distance between neighboring atoms. Their ratio, or more

precisely the reciprocal of their ratio, will be denoted by ε. We are interested in


the limit as ε→ 0. Specifically, we are interested in the electronic structure of the

system when the system is the result of a perfect periodic crystal after a smooth

deformation. We will see that as a byproduct, we obtain a quantum mechanical

foundation for the nonlinear elasticity theory for such materials.

To set up the problem, we need to describe the boundary condition. We will

consider the situation when the macroscopic system is also periodic. Two remarks

are in order. First of all, this periodicity is different from the microscopic peri-

odicity associated with the electronic structure of perfect crystals. Secondly, this

does represent a significant loss of generality: By assuming macroscopic periodic

boundary condition, we neglected interesting surface electronic states. We intend

to pursue this last issue in our future work.

This simple model can be extended in several directions. The atoms do not

have to reside on the line, they can be anywhere in any set in Rd where d is the

space dimension. Interaction between distant neighbors with finite range can be

accommodated. We will not consider these generalizations since they complicate

significantly the notations without bringing in any new insight into the problem

we are interested in. However, there is one generalization that we will consider,

namely the nonlinear tight binding model, also known as the self-consistent tight

binding model. In the spirit of the density functional theory [9], the value of the

Hamiltonian matrix elements can depend on the local density, making the model

nonlinear. As we will see later, the nonlinear model brings in the interesting issue

of the stability of the electronic structure.

To summarize, the tight binding model that we will consider is the following:

(2) (H [ρ])ij =

ai + v(ρi), j = i;

b(Yi+1 − Yi), j = i+ 1;

b(Yi − Yi−1), j = i− 1;

0, otherwise.

Here v and b are smooth functions on (0,∞). Macroscopic periodicity will be

enforced through the positions of the atoms {Yj}. The precise setting is discussed

in section 6. The operator H [ρ] may also be represented in the second quantization

form as

(3) H [ρ] =∑

i

(ai + v(ρi))c†i ci +

∑

i

(b(Yi+1 − Yi)c†i ci+1 + h.c.),

where ci and c†i are the annihilation and creation operator associated with the i-th

site respectively.

Given the lattice Schrodinger operator, what do we mean by a solution to the

problem? For finite systems, this is straightforward, as we just explained. There are

no general answers for general infinite systems. For infinite but periodic systems,

we will explain this issue in the next section.


3. The electronic structure of perfect crystals

In some sense, our main result is a sharp continuation theorem for the electronic

structure of the equilibrium state of a crystal: Knowing the electronic structure of

the equilibrium state, we ask how it changes when the crystal is deformed smoothly.

To understand this, we have to understand first the electronic structure of the

equilibrium state.

Assume that the equilibrium state of the crystal is given by a complex lattice

L ⊂ R. Denote the underlying Bravais lattice as L, which we will assume identical

to Z and also denote the shift vectors as p1, p2, . . . , pM ∈ Γ where Γ is the unit cell

of L. Therefore, the complex lattice L is the union of M shifted simple lattice:

L =

M⋃

i=1

(L + pi).

Let us denote the vectors in the lattice as Xi so that L = {Xi}.The lattice Schrodinger operator for the undeformed case is given by

(4) (He[ρ])ij =

ai + v(ρi), j = i;

b(Xi+1 −Xi), j = i+ 1;

b(Xi −Xi−1), j = i− 1;

0, otherwise.

Here ai is M -periodic: ai = ai+M . Notice that the operator He depends on the

electron density ρ through the potential term v(ρi). The operator He[ρ] may also

be represented in the second quantization form as

(5) He[ρ] =∑

i


∑

i

(b(Xi+1 −Xi)c†i ci+1 + h.c.),

The electron density of the equilibrium state is a M -periodic function: ρi =

ρi+M . Hence the operator He[ρ] commutes with translation operator τM . There-

fore, one may apply Bloch-Floquet theory [10] to the operator. Let Γ∗ = [0, 2π)

and ΓM = {1, 2, . . . ,M}. Define the unitary operator from l2(Z) to L2(Γ∗, l2(ΓM ))

(6) (Uf)ξ(j) =∑

k∈Z

f(j + kM)e−ikξ.

Its inverse is given by

(7) (U∗fξ)(j + kM) =

∫

Γ∗

fξ(j)eikξ dξ.

The Bloch decomposition of the Schrodinger operator is given by

(8) He[ρ] =

∫

Γ∗

Hξ[ρ] dξ,


where for M = 2,

(9) Hξ[ρ] =

(a1 + v(ρ1) b(D+X1) + e−iξb(D+X2)

b(D+X1) + eiξb(D+X2) a2 + v(ρ2)

),

and for M > 2,

(10)

Hξ[ρ] =

a1 + v(ρ1) b(D+X1) e−iξb(D+XM )

b(D+X1) a2 + v(ρ2) b(D+X2). . .

. . .. . .

b(D+XM−2) aM−1 + v(ρM−1) b(D+XM−1)

eiξb(D+XM ) b(D+XM−1) aM + v(ρM )

.

Here we have introduced the shorthand notation D+Xi = Xi+1−Xi. Indeed, given

f ∈ l2(Z), one can immediately check that

(UHe[ρ]f)ξ = Hξ[ρ](Uf)ξ.

For each ξ ∈ Γ∗, Hξ is a M × M Hermitian matrix. Denote the eigenvalues

and eigenvectors of Hξ as En(ξ) (with the ordering E1(ξ) ≤ E2(ξ) ≤ · · · ) and ψn,ξ

respectively, we have

(11) Hξ =

M∑

n=1

En(ξ)|ψn,ξ〉〈ψn,ξ|.

The spectrum of He[ρ] has a band structure:

(12) s(He[ρ]) =

M⋃

n=1

En(Γ∗).

Assume that there are Z electrons for each unit cell, so that the first Z bands

are occupied according to Pauli’s exclusion principle. We call a system insulator, if

there is a gap between the first Z bands and the rest of the spectrum:

(13) Eg = dist(sZ , s(He[ρ])\sZ) > 0,

where sZ = ∪n≤ZEn(Γ∗).

Suppose for a given ρ, He[ρ] is an insulator, we choose a compact contour C in

r(He[ρ]) that separates the first Z bands from the rest of the spectrum, such that

(14) dist(C , s(H [ρ])) = Eg/2.

We remark that the particular number Eg/2 is not important (however Eg/2 is the

largest possible value), the main requirement is that the contour is away from the

spectrum. Define the map

(15) Fe(ρ)(j) =1

2πi

∫

C

1

λ−He[ρ]dλ(j, j).


Fe(ρ) is the diagonal of the kernel of the spectral projection onto the occupied space

of He[ρ]. Physically, it is the electron density corresponding to the Hamiltonian

He[ρ].

Using Bloch-Floquet theory, it is not hard to see that the map Fe is well-defined.

Indeed, let P [ρ] be the projection operator on the right-hand side of (15),

P [ρ] =1

2πi

∫

C

1

λ−He[ρ]dλ,

then the Bloch decomposition of P [ρ] is given by

P [ρ] =

∫

Γ∗

Pξ[ρ] dξ =

∫

Γ∗

dξ1

2πi

∫

C

1

λ−Hξ[ρ]dλ.

For each ξ, the operator Pξ[ρ] acts on a finite dimensional space, and hence the

diagonal of the kernel is well-defined. Moreover, we have for any k ∈ Z,

(16) Fe(ρ)(j + kM) = P [ρ](j + kM, j + kM) =

∫

Γ∗

dξPξ[ρ](j, j).

Therefore, Fe(ρ) ∈ l∞(Z) and is also M -periodic.

With the help of the map Fe, we can formulate the electronic structure problem

as a self-consistent equation for the electron density:

(17) ρ(j) =1

2πi

∫

C

1

λ−He[ρ]dλ(j, j)

i.e. the electron density associated with the effective lattice Schrodinger operator

He[ρ] is equal to ρ itself.

We will assume that for the equilibrium state there exists a M -periodic density

ρe ∈ l∞(Z) that satisfies the following conditions:

• ρe is positive and bounded below away from zero: There exists a constant

µ > 0, such that ρe(j) ≥ µ for any j.

• The Hamiltonian He[ρe] corresponds to an insulator, with gap given by Eg.

• ρe is a self-consistent solution of (17) for any smooth compact contour C

in the resolvent set r(He[ρe]) that encloses the first Z bands and separates

sZ from the rest of the spectrum.

ρe will serve as our reference point, we will study the change of the electron

density after the system is deformed.

4. Stability of the electronic structure

Consider the linearization of the map Fe at the equilibrium density ρe, given by

(18) (Lew)(j) = (Le,ρew)(j) =1

2πi

∫

C

1

λ−He[ρe]v′(ρe)w

1

λ −He[ρe]dλ(j, j),

where v′(ρe)w is understood as a multiplicative operator.


Given N , an integer multiple of M , define the periodic lp space as

(19) lpN (Z) = {f N -periodic |N∑

j=1

|fj|p <∞},

endowed with its natural norm. We also define l∞N (Z) similarly as the space of

N -periodic bounded functions.

Proposition 1. The operator Le is a bounded operator on lpN(Z) for 1 ≤ p ≤ ∞.

The bound is uniform in N .

This proposition follows from the following technical Lemma.

Lemma 2. Assume T1 and T2 are bounded operators on l2(Z) that commute with

the translation operator τM . For any n ∈ N, the operator defined by

(Gf)(j) = (T1fT2)(j, j),

is a bounded operator on lpnM for 1 ≤ p ≤ ∞, and

‖G‖L (lpN ) ≤ ‖T1‖L (l2)‖T2‖L (l2).

Here (T1fT2)(j, j) is the kernel of the operator, where f is understood as a multi-

plicative operator.

Proof. Since τMT1 = T1τM , the kernel of T1 satisfies

T1(j, k +M) = T1(j −M,k),

and similarly for T2.

Define

K(j, k) = T1(j, k)T2(k, j),

then G can be written as

(Gf)(j) =∑

k∈Z

K(j, k)f(k).

Let us denote N = nM . If f is N -periodic, Gf is also N -periodic, since

(Gf)(j +N) =∑

k∈Z

K(j +N, k)f(k) =∑

k∈Z

K(j, k −N)f(k)

=∑

k∈Z

K(j, k −N)f(k −N) =∑

k∈Z

K(j, k)f(k) = (Gf)(j).


It remains to estimate the bound of the operator G. Since f is N -periodic, we

have(Gf)(j) =

∑

k∈Z

K(j, k)f(k)

=∑

m∈Z

N∑

k=1

K(j, k +mN)f(k +mN)

=

N∑

k=1

f(k)∑

m∈Z

K(j, k +mN).

Hence, the lpN norm of G is the matrix p-norm of the matrix Qjk =∑

mK(j, k+

mN), 1 ≤ j, k ≤ N . Using interpolation, we have

‖Q‖L (lPN ) ≤ ‖Q‖1/p

L (l1N )‖Q‖1/p′

L (l∞N )

where p′ is the dual index of p. Therefore, it suffices to control the 1-norm and

∞-norm of the matrix Q. The argument for both norms are analogous, and let us

focus on controlling 1-norm.

Notice that

‖Q‖L (l1N ) = supk‖Q(·, k)‖l1N

≤ supk‖K(·, k)‖l1,

we also have

‖K(·, k)‖l1 =∑

j∈Z

∣∣K(j, k)∣∣ =

∑

j∈Z

∣∣T1(j, k)T2(k, j)∣∣ ≤ ‖T1(·, k)‖l2‖T2(k, ·)‖l2 .

Let ek be the vector in l2(Z) that ek(j) = δjk, then

‖T1(·, k)‖l2 = ‖T1ek‖l2 ≤ ‖T1‖L (l2).

Applying the same argument to T ∗2 , we get

‖T2(k, ·)‖l2 ≤ ‖T2‖L (l2).

Therefore,

(20) ‖Gf‖lpN≤ ‖T1‖L (l2)‖T2‖L (l2)‖f‖lpN

.

�

Now we are ready to prove the Proposition.

Proof of Proposition 1. Since the contour C is compact, it suffices to prove the

boundedness of the operator

w 7→ 1

λ−He[ρe]v′(ρe)w

1

λ −He[ρe](j, j),

uniformly for each λ ∈ C .

Let

T1 =1

λ−He[ρe]v′(ρe), T2 =

1

λ−He[ρe].

T1 and T2 commutes with τM , as a result of the periodicity of ρe.


By the choice of the contour, we have

‖T2‖L (l2) ≤ 1/ dist(λ, s(He[ρe])) ≤ 2/Eg.

Moreover,

‖T1‖L (l2) ≤ ‖v′(ρe)‖l∞2/Eg.

We conclude using Lemma 2 that we can bound Le using Eg and ‖v′(ρe)‖l∞ . In

particular, Le is bounded independent of N .

�

We now introduce the crucial stability condition.

Definition 3. The electronic structure of the system is stable, if for every n ∈ N,

I − Le as an operator on l2nM (Z) is invertible and the inverse operator is bounded

independent of n:

(21) ‖(I − Le)−1‖L (l2nM (Z)) ≤ C.

An equivalent condition can be given using the Bloch-Floquet theory. The Bloch

decomposition of operator Le is given by

(22) Le =

∫

Γ∗

Lξ dξ,

where Lξ is an operator on l2(ΓM ) defined by

(23)

(Lξw)(j) =

∫

Γ∗

dη∑

n≤Z

∑

m>Z

(ψn,η(j)ψ∗

m,η−ξ(j)

Em(η − ξ) − En(η)

⟨ψn,η, ψm,η−ξv

′(ρe)w⟩

+ψm,η(j)ψ∗

n,η−ξ(j)

Em(η) − En(η − ξ)

⟨ψm,η, ψn,η−ξv

′(ρe)w⟩).

Here

(24) Hξ[ρe] =M∑

n=1

En(ξ)|ψn,ξ〉〈ψn,ξ|.

For simplicity, we suppress the dependence on ρe in the notation on the right-hand

side.

The following lemma gives an alternative condition of stability:

Lemma 4. The electronic structure associated with the density ρe is stable if and

only if the following condition holds: For every ξ ∈ Γ∗, I − Lξ as an operator on

l2(ΓM ) is invertible, and the norm of (I − Lξ)−1 is bounded uniformly in ξ.

Proof. Given n ∈ N and f ∈ l2nM (Z). Define for l = 0, 1, . . . , n− 1, ξl = 2πl/n. Let

fξl(j) =

n∑

k=1

f(j + kM)e−iξlk.


We then have for j ∈ ΓM , k ∈ Z,

f(j + kM) =1

n

n−1∑

l=0

fξl(j)eiξlk.

Applying (I − Le)−1 on both sides, we obtain

(25) ((I − Le)−1f)(j + kM) =

1

n

n−1∑

l=0

(I − Lξl)−1fξl

(j)eiξlk.

Consider the functions

gl(j + kM) = (I − Lξl)−1fξl

(j)eiξlk, j ∈ ΓM , k = 0, 1, . . . , n− 1.

As functions in l2nM (Z), the gl’s are orthogonal to each other. Indeed, we have

nM∑

j=1

gl(j)gl′(j) =

M∑

j=1

n−1∑

k=0

gl(j + kM)gl′(j + kM)

=

M∑

j=1

n−1∑

k=0

(I − Lξl)−1fξl

(j)(I − Lξl′)−1fξl′

(j)ei(ξl−ξl′ )k

= nδll′M∑

j=1

∣∣(I − Lξl)−1fξl

(j)∣∣2.

Therefore,

‖(I − Le)−1f‖l2nM

=1

n

n−1∑

l=0

‖gl‖l2nM=

1√n

n−1∑

l=0

‖(I − Lξl)−1fξl

‖l2M

≤ supξ∈Γ∗

‖(I − Lξl)−1‖L (l2M )

1√n

n−1∑

l=0

‖fξl‖l2M

= supξ∈Γ∗

‖(I − Lξl)−1‖L (l2M )‖f‖l2nM

.

Conversely, since Lξ depends on ξ continuously, it suffices to show that (I−Lξ)−1

is uniformly bounded for ξ = 2πl/n for some l < n. Given function g ∈ l2M (Z),

define for j ∈ ΓM , k ∈ Z, f(j + kM) = 1ng(j)e

iξk, then f is a function in l2nM (Z).

The boundedness of (I − Lξ)−1 then follows from

‖(I − Lξ)−1g‖l2M

=√n‖(I − Le)

−1f‖l2nM

≤ ‖(I − Le)−1‖L (l2nM )

√n‖f‖l2nM

≤ ‖(I − Le)−1‖L (l2nM )‖g‖l2M

.

�

From Lemma 4, stability can be checked by analyzing the invertibility of I −Lξ

for every ξ ∈ Γ∗. Notice that Lξ is an operator acting on l2(ΓM ). Hence, it is

a M ×M matrix. Therefore, we only need to check the invertibility for matrices


corresponding to each ξ in Γ∗. This is very similar in spirit to phonon or band

structure analysis for perfect crystals.

Let us consider a particular example in which the system loses stability. Consider

a system with M = 6 and Z = 2, with pi = (i − 1)/6, i = 1, . . . , 6, so that the

atoms are equally spaced. Take

a(1 : 6) = (0.25, 0.1, 0, 0, 0.1, 0.25)

and extend the vector a by periodicity. Let v be given by

v(ρ) =3

2ρ−1/3 so that v′(ρ) = −1

2ρ−4/3.

Note that v′(ρe) ≤ 0 since ρe is greater than 0. (If v′(ρe) ≥ 0, it can be shown that

the system is stable). We will consider only homogeneous deformation, so that the

atoms will be equally spaced. Notice that in this setup, the value b(D+Xi) is the

same for every site. So we will use a single parameter b = b(D+Xi).

We solve the system numerically with periodic boundary conditions on a super-

cell containing 32 unit cells. Once we have the electron density in equilibrium for

a given b, we check whether the system is stable or not. It is easy to see that Lξ as

a 6 × 6 matrix is diagonalizable with positive eigenvalues (Lξ is actually similar to

a Hermitian matrix). Let us call these eigenvalues λn(ξ) with n = 1, . . . , 6, ordered

by magnitude. We can then plot λn(·) as a function of ξ. Whether the system is

stable or not is determined by whether these functions intersect with λ = 1.

We start with b very negative with the equilibrium density periodic with period

6 (as shown in the left panel of Figure 4). If b is very negative, the electronic

structure is stable. For example, in Figure 1, the spectrum of Lξ for b = −1.04 is

shown. It is clear that (I − Lξ)−1 is uniformly bounded for ξ ∈ Γ∗. Hence, the

electronic structure is stable.

0 2 4 60

0.2

0.4

0.6

0.8

1

ξ

λ

Figure 1. The spectrum of Lξ for ξ ∈ Γ∗ for the equilibrium

density at b = −1.04.


When we increase the value of b (e.g., as we stretch uniformly the system), the

solution of the electron density will be a smooth continuation of the periodic density

profile we started with. However, when b becomes large enough, the system will lose

its stability. The spectrum of Lξ is shown in Figure 2 for the case when b = −1.015.

It can be seen that the spectrum of Lξ touches λ = 1 for certain values of ξ.

0 2 4 60

0.2

0.4

0.6

0.8

1

ξ

λ



If we further increase the value of b, then the electronic structure obtained by the

continuation procedure becomes unstable according to our definition. For example,

the spectrum of Lξ is shown in Figure 3 for b = −0.96.

0 2 4 60

0.2

0.4

0.6

0.8

1

ξ

λ



To see how the behavior of the electron density changes, we look for self-consistent

solutions of (17) with period 12 instead of 6. We start the iteration with the elec-

tronic structure obtained from the continuation of the period-6 solution with a

small perturbation. We should recover the period-6 solution if it was stable. In-

stead, we obtain a new period-12 solution which does satisfy our stability condition,


as shown in Figure 4. Further study reveals that this is a subcritical period-doubling

bifurcation. There are other self-consistent solutions.

20 40 600

0.2

0.4

0.6

0.8

(a)

20 40 600

0.2

0.4

0.6

0.8

(b)

Figure 4. Left: The density profile of the self-consistent solution

at b = −1.04, only ten periods are shown; Right: The density

profile of the self-consistent solution at b = −0.96, only five periods

are shown.

5. Electronic structure of homogeneously deformed crystals

Next we consider the case when the system is deformed under a homogeneous de-

formation u(x) = Ax, where the deformation gradient A ∈ R, so that the positions

of the perturbed nuclei become

(26) Yi = (I +A)Xi, i ∈ Z.

For a M -periodic electron density ρ, the Hamiltonian then becomes

(27) (HA[ρ])ij =

ai + v(ρi), j = i;

b((I +A)D+Xi), j = i+ 1;

b((I +A)D+Xi−1), j = i− 1;

0, otherwise,

or written in the second quantization form,

(28) HA[ρ] =∑

i


∑

i

(b((I +A)D+Xi)c†i ci+1 + h.c.).

When the deformation gradient A is small and ρ is close to the equilibrium elec-

tron density ρe, the contour C defined for the equilibrium case lies in the resolvent

set of HA[ρ], and hence the map FA is well defined in this case as

(29) FA(ρ)(j) =1

2πi

∫

C

1

λ−HA[ρ]dλ(j, j).


Lemma 5. There exist constants a0 > 0 and 0 < δ0 < µ/2, such that for |A| ≤ a0

and ‖ρ− ρe‖l∞M≤ δ0, we have

(30) dist(C , s(HA[ρ])) ≥ Eg/4.

Proof. Comparing the Hamiltonians, we have

(HA[ρ] −He[ρe])ij =

v(ρ(i)) − v(ρe(i)), j = i;

b((I +A)D+Xi) − b(D+Xi), j = i+ 1;

b((I +A)D+Xi−1) − b(D+Xi−1), j = i− 1;

0, otherwise.

Therefore,

(31) ‖HA[ρ] −He[ρe]‖L (l1) = max1≤i≤M

(∣∣v(ρ(i)) − v(ρe(i))∣∣

+∣∣b((I +A)D+Xi) − b(D+Xi)

∣∣+∣∣b((I +A)D+Xi−1) − b(D+Xi−1)

∣∣).

For the first term in the bracket, we have∣∣v(ρ(i)) − v(ρe(i))

∣∣ ≤ C∣∣ρ(i) − ρe(i)

∣∣ ≤ C‖ρ− ρe‖l∞M.

The other two terms in the bracket can be bounded by C|A|. Moreover, it is not

hard to see that ‖HA[ρ]−He[ρe]‖L (l1) = ‖HA[ρ]−He[ρe]‖L (l∞). We arrive at the

estimate

(32) ‖HA[ρ] −He[ρe]‖L (l2) ≤ ‖HA[ρ] −He[ρe]‖L (l1) ≤ C(‖ρ− ρe‖l∞M+ |A|).

Hence, using the assumption that ‖(λ−H [ρe])−1‖L (l2) ≤ 2/Eg, we obtain

(33) ‖(HA[ρ] −He[ρe])(λ −He[ρe])−1‖L (l2) ≤ 2C(‖ρ− ρe‖l∞M

+ |A|)/Eg.

Since

(λ−HA[ρ])−1 = (λ−He[ρe])−1(I − (HA[ρ] −He[ρe])(λ−He[ρe])

−1)−1

,

by choosing a0 and δ0 sufficiently small such that the right hand side of (33) is

bounded by 1/2 uniformly with respect to λ ∈ C (C is a compact set), we get

(34) ‖(λ−HA[ρ])−1‖L (l2) ≤ 2‖(λ−He[ρe])−1‖L (l2) ≤ 4/Eg,

for any λ ∈ C . Hence we have dist(C , s(HA[ρ])) ≥ Eg/4. �

This result guarantees that the map

(35) F (ρ,A)(j) =1

2πi

∫

C

1

λ−HA[ρ]dλ(j, j),

from the domain Dρ ×DA to l∞M (Z) is well-defined, where

Dρ ={ρ M -periodic | ‖ρ− ρe‖∞ ≤ δ0

},(36)

DA ={A ∈ R | |A| ≤ a0

}.(37)


Using the smoothness of v and the resolvent expansion, it immediately follows

that

Lemma 6. The map F from Dρ ×DA to l∞M (Z) is C∞.

Using the implicit function theorem, it follows that when the deformation gra-

dient A is small, a self-consistent solution to the tight-binding model ρA can be

found in a small neighborhood around ρe.

Theorem 7. Assume that at the equilibrium state u = 0, the electronic structure

given by ρe is stable according to Definition 3. There exist constants a ∈ (0, a0)

and δ ∈ (0, δ0), such that if |A| ≤ a, the equation

ρA(j) = FA(ρA)(j) =1

2πi

∫

C

1

λ−HA[ρA]dλ(j, j).

has a unique M -periodic density solution ρA for ‖ρA − ρe‖l∞M≤ δ. Moreover, the

map from A to ρA is C∞.

We call this solution the Cauchy-Born electron density 1, and we denote it by

ρCB(A, ·). This is the ingredient that enters in the Cauchy-Born rule approximation

for general smooth displacement field.

Proof. Define the map T by

(38) T (ρ,A) = ρ− F (ρ,A)

from Dρ ×DA to l∞M (Z). T is a C∞ map, and we have

(39) T (ρe, 0) = ρe − F (ρe, 0) = 0.

Notice that

(40)δ

δρT (ρ,A)|ρ=ρe,A=0 = I − Le

is invertible as an operator on l2(ΓM ), due to the stability condition. Applying

the implicit function theorem on T and using Lemma 6, we arrive at the desired

conclusions.

�

6. The electronic structure of deformed crystals

We now turn to the continuum limit when the ratio between the atomistic length

scale (the lattice constant) and the length scale of the elastic deformation tends to

zero. We will fix the deformation (the displacement field), and set the lattice

constant to be ε, which is the small parameter we will use to study the continuum

1We remark that the terminology Cauchy-Born rule originally refers to the displacement of

atoms in the classical setting. Here, we use the same term for the deformation of the electronic

structure, since the underlying principle is a direct generalization.


limit. We will assume the macroscopic displacement field to be periodic, and ε is

the reciprocal of an integer.

Given a smooth Z-periodic displacement field u (so that the length scale is O(1)),

the nuclei positions after the deformation become

(41) Y εi = Xε

i + u(Xεi ) ≡ τ(Xε

i ), Xεi ∈ εL.

Here Xεi is the equilibrium position of the i-th nucleus, Xε

i = εXi, as the lattice

constant becomes ε.

Let us consider the following lattice Schrodinger operator

(42) (Hετ [ρ])ij =

ai + v(ρi), j = i;

b((Y εi+1 − Y ε

i )/ε), j = i+ 1;

b((Y εi − Y ε

i−1)/ε), j = i− 1;

0, otherwise,

or more concisely

(43) Hετ [ρ] =

∑

i


∑

i

(b(D+Y εi /ε)c

†ici+1 + h.c.).

One period of the macroscopic system has M/ε atoms. Therefore we will consider

the M/ε-periodic solutions to the tight-binding model.

The map F is given by

(44) F ετ (ρ)(j) =

1

2πi

∫

C

1

λ−Hετ [ρ]

dλ(j, j).

The right hand side denotes the diagonal of the kernel of the (orthogonal) projection

operator to the spectrum enclosed by C . Of course, for F ετ to be well defined, the

contour C should be in the resolvent set of Hετ [ρ].

Proposition 8. Assume that the contour C satisfies dist(C , s(H)) = d > 0, then

(45)

∣∣∣∣1

2πi

∫

C

1

λ−Hdλ(j, j)

∣∣∣∣ ≤ 1.

Proof. Denote

P =1

2πi

∫

C

1

λ−Hdλ

the spectral projection on the spectrum enclosed by C . Using the idempotency

P = P 2, we have

‖P‖L (l2) ≤ 1,

Let ej be a vector in l2(Z) such that ej(k) = δjk, then

|P (j, j)| = |〈ej , P ej〉| ≤ ‖P‖L (l2) ≤ 1.

�


Similar to Lemma 5, the contour C lies in the resolvent set of Hετ [ρ] if the

displacement is small and ρ is close to ρe. We omit the proof of the following

Lemma since it is similar to the proof of Lemma 5.

Lemma 9. There exist constants a0 > 0 and 0 < δ0 < µ/2, such that if u satisfies

‖∇u‖L∞ ≤ a0 and if ‖ρ− ρe‖l∞ ≤ δ0, then

(46) dist(C , s(Hετ [ρ])) ≥ Eg/4.

To solve the nonlinear tight-binding model, one looks for the self-consistent so-

lution of the tight-binding model

(47) ρ = F ετ (ρ).

Our objective is to show that the electronic structure around a point x is approx-

imated by the corresponding electronic structure of a homogeneously deformed

system with deformation gradient ∇u(x). As we have discussed in the last section,

using Lagrangian coordinates, the electron density for the system with homoge-

neous deformation gradient ∇u(x) is given by ρCB(∇u(x), ·). In analogy with the

context of crystal lattices, we also refer to this as the Cauchy-Born rule. The

electron density constructed by the Cauchy-Born rule is

(48) ρ0(j) = ρCB(∇u(Xεj ), j)

The following result says that this is indeed the case if the stability condition is

satisfied. In this case, one can find a self-consistent solution to the tight-binding

model near the state given by the Cauchy-Born rule.


given by ρe is stable according to Definition 3. There exist constants a ∈ (0, a0), ε0,

and C, such that if u is smooth (C∞) and u satisfies ‖∇u‖L∞ ≤ a, and if ε ≤ ε0,

then there exists a unique ρε ∈ l∞(Z) that satisfies

(1) ρε is M/ε-periodic and ‖ρε(j) − ρCB(∇u(Xεj ), j)‖l2

M/ε≤ Cε1/2.

(2) ρε is a self-consistent solution to the equation:

(49) ρε(j) = F ετ (ρε)(j) =

1

2πi

∫

C

1

λ−Hετ [ρε]

dλ(j, j).

Once we have the results for density, it is straightforward to apply the similar

analysis as in [4] to obtain results for the occupied space (Wannier functions) and

the energy. For example, let us consider the band energy corresponding to the

lattice Schrodinger operator given by ρε here. Since Hετ [ρε] commutes with τM/ε,

its band structure can be analyzed using Bloch decomposition, given by

(50) Hετ =

∫

Γ∗

(Hετ )ξ dξ,


where (Hετ )ξ is a M/ε by M/ε Hermitian matrix. Denote its eigenvalues by Eε

n,τ (ξ)

ordered as Eε1,τ (ξ) ≤ Eε

2,τ (ξ) ≤ · · · . Then the average band energy of the system

is given by

(51) E(τ, ε) = ε

Z/ε∑

n=1

∫

Γ∗

Eεn,τ (ξ) dξ.

Similarly, the energy per unit cell of the homogeneous deformed system with de-

formation gradient A is given by

(52) ECB(A) =

Z∑

n=1

∫

Γ∗

En,A(ξ) dξ,

where En,A(ξ) is the n-th eigenvalue for the Bloch fiber operator (HA[ρA])ξ.

Theorem 11. Under same assumptions as Theorem 10, we have∣∣∣∣E(τ, ε) −

∫

Γ

ECB(∇u(x)) dx

∣∣∣∣ ≤ Cε1/2,

where the constant C is independent of ε.

The order ε1/2 comes from the fact that in Theorem 10, the density is approx-

imated by the Cauchy-Born construction to the same order. Improved estimates

(O(ε)) can be obtained if we go to next order in density, as discussed in the ap-

pendix. We also remark that since the model is nonlinear tight-binding (the lattice

Schrodinger operator depends on ρ), the band energy defined here is different from

the energy functional that gives the self-consistent equation as its Euler-Lagrange

equation. However, the difference between the two can be represented as an explicit

functional in ρ, which can be approximated again using Cauchy-Born rule, since ρ

is well approximated by the Cauchy-Born construction. We will come back to this

issue in the subsequent paper on Kohn-Sham type of models.

The proof of Theorem 11 follows the similar arguments as in [4], since we are now

considering a linear problem with potential given. For completeness, we provide an

alternative (and more direct) proof in Appendix B.

7. Asymptotics

We want to find a solution to the self-consistent equation

ρ(x) =1

2πi

∫

C

1

λ−Hετ [ρ]

dλ(x, x).

For this, let us start with the Cauchy-Born guess

(53) ρ0(i) = ρCB(∇u(Xεi ), i).

Here the superscript 0 indicates the approximate solution ρ0 will be the initial

iterate for the iteration set up in the next Section. Note that ρ0 actually depends


on ε. In this section, we will prove that the Cauchy-Born guess gives the self-

consistent solution of the tight-binding model to the leading order. Based on the

asymptotics, higher order corrections can also be found. This will be discussed in

the appendix.

7.1. Asymptotic analysis of lattice Schrodinger operator. We first consider

the operator Hετ [ρ0] given by ρ0 defined as (53). We will suppress the dependence

of ρ0 in the notation, since it is fixed. Expanding the deformation τ around Xεi ,

we have

(54)τ(x) = τ(Xε

i ) + τ ′(Xεi )(x−Xε

i )

+ 12τ

′′(Xεi )(x −Xε

i )2 + 16τ

′′′(Xεi )(x−Xε

i )3 + O(x −Xεi )4.

Therefore, we have

(55)

b(D+Y εi /ε) = b(τ ′(Xε

i )D+Xi) + ε 12b

′τ ′′(Xεi )(D+Xi)

2

+ ε2 18b

′′(τ ′′(Xεi )(D+Xi)

2)2 + ε2 16b

′τ ′′′(Xεi )(D+Xi)

3 + O(ε3)

= b0(Xεi , i) + εb1(X

εi , i) + ε2b2(X

εi , i) + O(ε3)

where b′ and b′′ are evaluated at τ ′(Xεi )D+Xi. Here we have introduced bk(x, i),

which are M -periodic functions with respect to the second variable:

b0(x, i) = b(τ ′(x)D+Xi);(56)

b1(x, i) = 12b

′(τ ′(x)D+Xi)τ′′(x)(D+Xi)

2;(57)

b2(x, i) = 18b

′′(τ ′(x)D+Xi)(τ′′(x)(D+Xi)

2)2(58)

+ 16b

′(τ ′(x)D+Xi)τ′′′(x)(D+Xi)

3.

Substituting the ansatz (53) into v(ρi), we obtain

(59) vi = v(ρi) = v(ρ0(Xεi , i)) = v0(X

εi , i),

where v0(x, i) = v(ρ0(x, i)).

Putting these together, we get the leading order Schrodinger operator:

(60) H0(x) =∑

i

(ai + v0(x, i))c†i ci +

∑

i

(b0(x, i)c†i ci+1 + h.c.),

with the correction given by

(61) Hετ −H0(x) =

∑

i

(v0(Xεi , i) − v0(x, i))c

†i ci

+∑

i

((b0(X

εi , i) − b0(x, i) + εb1(X

εi , i) + ε2b2(X

εi , i)

)c†ici+1 + h.c.

)+ O(ε3),

We will denote the difference as δH(x) = Hετ −H0(x). Note that H0(x) depends on

x as a parameter, it is understood as an operator-valued function on the domain.


7.2. Asymptotics for the map F ετ . Using the resolvent identity, the map F ε

τ can

be expanded as

(62)

F ετ (ρ0)(j) =

1

2πi

∫

C

1

λ−Hετ

dλ(j, j)

=1

2πi

∫

C

1

λ−H0(Xεj )

dλ(j, j)

+1

2πi

∫

C

1

λ−Hετ

δH(Xεj )

1

λ−H0(Xεj )

dλ(j, j).

Let us study the first term on the right-hand side. Denote

f(x, j) =1

2πi

∫

C

1

λ−H0(x)dλ(j, j).

Notice that by definition

H0(x) =∑

i

(ai + v0(x, i))c†i ci +

∑

i

(b0(x, i)c†i ci+1 + h.c.)

=∑

i

(ai + v(ρCB(∇u(x), i)))c†i ci +∑

i

(b(τ ′(x)D+Xi)c

†i ci+1 + h.c.

),

therefore we have

H0(x) = HA[ρCB(A, ·)]Here A = ∇u(x). Then, by the construction of the Cauchy-Born density ρCB, we

have

(63) f(x, j) =1

2πi

∫

C

1

λ−HA[ρCB(A, ·)] dλ(j, j) = ρCB(A, j).

Denote the remaining terms on the right-hand side of (62) by r(j), then F ετ (ρ0) =

ρ0 + r. We will show that r is of order ε, and hence the Cauchy-Born construction

ρ0 gives a leading order approximation to the solution.

For this, we need the following Combes-Thomas type of estimate [2] on the

exponential decay of the resolvent for λ outside the spectrum of H .

Lemma 12. Let H be a lattice Schrodinger operator in the form of

H =∑

i

aic†ici +

∑

i

(bic†ici+1 + h.c.)

with a, b ∈ l∞. Let λ ∈ r(H) with dist(λ, s(H)) = d > 0, there exist constants

γ > 0 and C, such that

(64)

∣∣∣∣1

λ−H(j, k)

∣∣∣∣ ≤ Ce−γ|j−k|, j, k ∈ Z.

Proof. For j0 and γ > 0, define B as a multiplicative operator given by

(Bv)(j) = eγ|j−j0|v(j).

We first show that B(λ−H)−1B−1 is a bounded operator on l2(Z).


From the definition, we have

(BHB−1)jk = eγ|j−j0|Hjke−γ|k−j0|

Hence

(BHB−1 −H)jk =

bj(eγ(|j−j0|−|j+1−j0|) − 1), k = j + 1;

bj−1(eγ(|j−j0|−|j−1−j0|) − 1), k = j − 1;

0, otherwise.

Therefore

‖BHB−1 −H‖L (l∞) ≤ supj

(|bj |∣∣eγ(|j−j0|−|j+1−j0|) − 1

∣∣

+ |bj−1|∣∣eγ(|j−j0|−|j−1−j0|) − 1)

∣∣)

≤ C‖b‖l∞γ,

when γ is sufficiently small so that∣∣e±γ − 1

∣∣ ≤ Cγ with some constant C > 1. The

same bound obviously holds for ‖BHB−1−H‖L (l1) too. Hence, using interpolation,

we get

‖BHB−1 −H‖L (l2) ≤ Cγ.

Notice that

B(λ−H)−1B−1 = (λ −BHB−1)−1

= (λ −H)−1(I − (BHB−1 −H)(λ−H)−1

)−1.

Since ‖(λ−H)−1‖L (l2) ≤ 1/d, we can take γ sufficiently small such that λ−BHB−1

is invertible, and

(65) ‖B(λ−H)−1B−1‖L (l2) ≤ 2/d.

Since ∣∣(B(λ −H)−1B−1)jk

∣∣ ≤ ‖B(λ−H)−1B−1‖L (l2) ≤ 2/d,

and notice that

(B(λ −H)−1B−1)jk = (λ−H)−1jk e

γ(|j−j0|−|k−j0|),

we obtain the estimate

|(λ −H)−1jk | ≤ Ce−γ(|j−j0|−|k−j0|).

In particular, take j0 = k, we get the desired exponential decay estimate.

�

We are now ready to control the norm ‖r‖l∞ .

Denote Rετ,λ = (λ−Hε

τ )−1 and R0,λ = (λ−H0(Xεj ))−1, then

(66) r(j) =1

2πi

∫

C

dλ∑

k,l∈Z

Rετ,λ(j, k)δH(Xε

j )(k, l)R0,λ(l, j).


Since the contour C is compact, it suffices to control for each λ the integrand of

(66). Notice that δH(k, l) = 0 if |k − l| > 1, we have

(67)

rλ(j) =∑

k,l∈Z


j )(k, l)R0,λ(l, j)

=∑

k∈Z


j )(k, k)R0,λ(k, j)

+∑

k∈Z


j )(k, k + 1)R0,λ(k + 1, j)

+∑

k∈Z


j )(k, k − 1)R0,λ(k − 1, j).

Consider the first term on the right-hand side, we have∣∣δH(Xε

j )(k, k)∣∣ =

∣∣v0(Xεk, k) − v0(X

εj , k)

∣∣

=∣∣v(ρ0(Xε

k, k)) − v(ρ0(Xεj , k))

∣∣

≤ C∣∣ρCB(∇u(Xε

k), k) − ρCB(∇u(Xεj ), k)

∣∣

≤ C∣∣τ ′(Xε

k) − τ ′(Xεj )∣∣ ≤ C

∣∣Xεk −Xε

j

∣∣.

It follows that∣∣∣∣∑

k∈Z


j )(k, k)R0,λ(k, j)

∣∣∣∣ ≤ Cε∑

k∈Z

∣∣∣Rετ,λ(j, k)R0,λ(k, j)|Xk −Xj|

∣∣∣.

The summation on the right-hand side is bounded due to the exponential decay of

the resolvent according to Lemma 12. We have obtained the bound for the first

term in (67).

Next we consider the second term on the right-hand side of (67). The argument

for the third term is the same, and hence we omit it. Recall that

δH(Xεj )(k, k + 1) = b0(X

εk, k) − b0(X

εj , k) + εb1(X

εk, k) + ε2b2(X

εk, k) + O(ε3).

It suffices to consider the term b0(Xεk, k) − b0(X

εj , k) since the other terms are

obviously O(ε). By definition, we have

∣∣∣b0(Xεk, k) − b0(X

εj , k)

∣∣∣ =∣∣∣b(τ ′(Xε

k)D+Xk) − b(τ ′(Xεj )D+Xk)

∣∣∣

≤ C∣∣τ ′(Xε

k) − τ ′(Xεj )∣∣ ≤ C|Xε

k −Xεj |.

The remaining steps are similar to the argument for the first term, using the expo-

nential decay of the resolvents. To sum up, we have proved the following

Proposition 13. Take ρ0(j) = ρCB(∇u(Xεj ), j). There exist constants a, ε0, and

C, such that if u is smooth and u satisfies ‖∇u‖L∞ ≤ a, and if ε ≤ ε0, we have

‖ρ0 − F ετ (ρ0)‖∞ ≤ Cε.


8. Continuum limit of density

We prove Theorem 10 in this section. To get the self-consistent solution in

the neighborhood of ρ0, we will use the Newton-Raphson iteration: Given the

approximate solution ρ0, we iterate this further using

(68) ρn+1 = ρn −(I − δF ε

τ

δρ0

)−1

(ρn − F ετ (ρn)).

Equivalently, for ρn, we need to solve for δρ from

(69) ρn + δρ = F ετ (ρn) +

δF ετ

δρ0(δρ) = F ε

τ (ρn) + Lε(δρ),

and then set ρn+1 = ρn + δρ. Here, to simplify notation, we denote

(70) Lε = Lετ,ρ0 =

δF ετ

δρ

∣∣∣∣ρ=ρ0

.

The linear operator Lε is given by

(71) (Lεw)(j) =1

2πi

∫

C

1

λ−Hετ [ρ0]

v′(ρ0)w1

λ −Hετ [ρ0]

dλ(j, j).

For ρ0, we take the approximate solution constructed using the Cauchy-Born

rule:

(72) ρ0(j) = ρCB(∇u(Xεj ), j)

We first prove that with this choice, Lε is well defined and (I − Lε) is invertible,

we then prove the convergence of the iteration (68).

8.1. Operator Lε. By the construction of ρ0 and Lemma 9, we have

(73) dist(C , s(Hετ [ρ0])) ≥ CEg

Hence the operator Lε is well defined as an operator on l2M/ε.

Next, we show that I − Lε is invertible and the inverse is bounded uniformly in

ε.

Proposition 14. Assume that the electronic structure of the equilibrium state with

density ρe is stable according to Definition 3. There exist constants a, ε0 and C,

such that if u is smooth and u satisfies ‖∇u‖L∞ ≤ a, and if ε ≤ ε0, then I − Lε is

invertible, and

(74) ‖(I − Lε)−1‖L (l2M/ε

) ≤ C.


Proof. Let us compare Lε and Le, the linearized operator at the equilibrium state:

(75)

(Lεw − Lew)(j)

=1

2πi

∫

C

Rετ,λ[ρ0]v′(ρ0)wRε

τ,λ[ρ0] dλ(j, j)

− 1

2πi

∫

C

Re,λ[ρe]v′(ρe)wRe,λ[ρe] dλ(j, j)

=1

2πi

∫

C

(Rε

τ,λ[ρ0] −Re,λ[ρe])v′(ρ0)wRε

τ,λ[ρ0] dλ(j, j)

+1

2πi

∫

C

Re,λ[ρe](v′(ρ0) − v′(ρe))wR

ετ,λ[ρ0] dλ(j, j)

+1

2πi

∫

C

Re,λ[ρe]v′(ρe)w

(Rε

τ,λ[ρ0] −Re,λ[ρe])dλ(j, j)

= I1 + I2 + I3.

Here we have used the notation

Rετ,λ[ρ0] = (λ−Hε

τ [ρ0])−1,

Re,λ[ρe] = (λ−He[ρe])−1.

For the first term on the right-hand side of (75), using the resolvent identity, we

have

Rετ,λ[ρ0] −Re,λ[ρe] = Rε

τ,λ[ρ0](Hετ [ρ0] −He[ρe])Re,λ[ρe].

Since

‖Hετ [ρ0] −He[ρe]‖L (l2) ≤ C(‖∇u‖L∞ + ‖ρ0 − ρe‖l∞),

using Lemma 2, we get

‖I1‖l2M/ε

≤ C(‖∇u‖L∞ + ‖ρ0 − ρe‖l∞)‖w‖l2M/ε

≤ C‖∇u‖L∞‖w‖l2M/ε

,

where we have used

|ρ0(j) − ρe(j)| = |ρCB(∇u(Xεj ), j) − ρCB(0, j)| ≤ C|∇u(Xε

j )|,

in the last inequality. The estimate of I2 and I3 follows similarly. Therefore, we

have

(76) ‖(Lε − Le)‖L (l2M/ε

) ≤ C‖∇u‖L∞ .

We have

(I − Lε)−1 = (I − Le)−1(I + (Le − Lε)(I − Le)

−1)−1

.

Because the electronic structure is stable according to Definition 3, (I − Le)−1 is

bounded uniformly in ε:

(77) ‖(I − Le)−1‖L (l2

M/ε) ≤ C1.


By choosing a sufficiently small, such that the right-hand side of (76) is less than

1/2C1, we obtain

‖(I − Lε)−1‖L (l2M/ε

) ≤ 2‖(I − Le)−1‖L (l2

M/ε) ≤ 2C1.

This concludes the proof of the proposition.

�

8.2. Convergence of Newton-Raphson iteration. We first prove a general re-

sult for the convergence of the Newton-Raphson iteration.

Lemma 15. Denote L = δρF |ρ=ρ0 and K = (I − L)−1. Assume that there exist

constants 0 < η ≤ 1 and M , and a norm ‖·‖, such that

(1) The initial iterate satisfies

(78) ‖K(ρ0 − F (ρ0))‖ ≤ ηδ,

(2) For ρ, ρ that ‖ρ− ρ0‖ ≤ δ and ‖ρ− ρ0‖ ≤ δ,

(79) ‖K(F (ρ) − F (ρ) − L(ρ− ρ))‖ ≤ (1 − η)‖ρ− ρ‖.

Then ρn defined in iteration (68) converges to a fixed point ρ∗ of F , and we have

(80) ‖ρ∗ − ρ0‖ ≤ δ.

The fixed point ρ∗ is unique in the δ-ball around ρ0.

Proof. Subtracting (68) by the same equation with n replaced by n− 1, we have

ρn+1 − ρn = ρn − ρn−1 −K(ρn − ρn−1 − (F (ρn) − F (ρn−1)))

= K(F (ρn) − F (ρn−1) − L(ρn − ρn−1)).

Let us first assume that the iterates ρn−1 and ρn lie in a δ-ball around ρ0, then by

(79),

(81) ‖ρn+1 − ρn‖ ≤ (1 − η)‖ρn − ρn−1‖.

Therefore,

(82) ‖ρn+1 − ρ0‖ ≤n∑

k=0

‖ρk+1 − ρk‖ ≤ 1/η‖ρ1 − ρ0‖.

Combined with (78),

(83) ‖ρ1 − ρ0‖ = ‖K(ρ0 − F (ρ0))‖ ≤ ηδ,

we have

(84) ‖ρn+1 − ρ0‖ ≤ δ.

Therefore, the contraction inequality (81) is proved by induction. The convergence

of the iteration to a fixed point follows directly.


For uniqueness, assume there are two fixed points of F , ρ and ρ in the M -ball

around ρ0, then by (79),

‖ρ− ρ‖ = ‖K(I − L)(ρ− ρ)‖= ‖K(ρ− ρ− L(ρ− ρ))‖= ‖K(F (ρ) − F (ρ) − L(ρ− ρ))‖≤ (1 − η)‖ρ− ρ‖,

which clearly gives a contradiction unless ρ = ρ. �

Proof of Theorem 10. We will use the l2 norm ‖·‖l2M/ε

.

By the construction of ρ0 and the asymptotic analysis, we have

‖ρ0 − F ετ (ρ0)‖l∞ ≤ Cε

Hence

‖ρ0 − F ετ (ρ0)‖l2

M/ε≤ Cε1/2.

The operator Kε = (I−Lε)−1 is bounded uniformly in ε. Therefore, we can choose

η between 0 and 1, and δ = O(ε1/2), such that the condition (78) is satisfied.

We now check that the condition (79) is satisfied. Take two densities ρ1 and

ρ2 in the δ-ball around ρ0. It is easy to see that the map F and the linearized

operators at the two densities are well defined. Let us compare these two linearized

operators acting on w:

L1(w)(j) = Lετ,ρ1

(w)(j) =1

2πi

∫

C

1

λ−H1v′(ρ1)w

1

λ −H1dλ(j, j)

=1

2πi

∫

C

Rλ,1v′(ρ1)wRλ,1 dλ(j, j),

and

L2(w)(j) = Lετ,ρ2

(w)(j) =1

2πi

∫

C

1

λ−H2v′(ρ2)w

1

λ −H2dλ(j, j)

=1

2πi

∫

C

Rλ,2v′(ρ2)wRλ,2 dλ(j, j),

where H1 = Hετ [ρ1] and H2 = Hε

τ [ρ2]. Subtracting the two, we have

(85)

(L1 − L2)w(j) =1

2πi

∫

C

(Rλ,1 −Rλ,2)v′(ρ2)wRλ,2 dλ(j, j)

+1

2πi

∫

C

Rλ,1

(v′(ρ1) − v′(ρ2)

)wRλ,2 dλ(j, j)

+1

2πi

∫

C

Rλ,1v′(ρ1)w(Rλ,1 −Rλ,2) dλ(j, j)

= I1 + I2 + I3.


Consider the first term. Using resolvent identity, we have

I1 =1

2πi

∫

C

Rλ,1(H1 −H2)Rλ,2v′(ρ2)wRλ,2 dλ(j, j)

=1

2πi

∫

C

Rλ,1(v(ρ1) − v(ρ2))Rλ,2v′(ρ2)wRλ,2 dλ(j, j).

Since

‖v(ρ1) − v(ρ2)‖l∞ ≤ C‖ρ1 − ρ2‖l∞ ,

we obtain the estimate

‖Rλ,1(v(ρ1) − v(ρ2))Rλ,2v′(ρ2)‖L (l2) ≤ C‖ρ1 − ρ2‖l∞ ≤ C‖ρ1 − ρ2‖l2

M/ε.

Applying Lemma 2, we have

‖I1‖l2M/ε

≤ C‖ρ1 − ρ2‖l2M/ε

‖w‖l2M/ε

.

The argument for the third term is the same.

For the second term, we have

I2 =1

2πi

∫

C

Rλ,1

(v′(ρ1) − v′(ρ2)

)wRλ,2 dλ(j, j)

Since

‖v′(ρ1) − v′(ρ2)‖l∞ ≤ C‖ρ1 − ρ2‖l∞ ,

we have

‖Rλ,1

(v′(ρ1) − v′(ρ2)

)‖L (l2) ≤ C‖ρ1 − ρ2‖l∞ ≤ C‖ρ1 − ρ2‖l2

M/ε.

By Lemma 2, we have

‖I2‖l2M/ε

≤ C‖ρ1 − ρ2‖l2M/ε

‖w‖l2M/ε

.

To sum up, we have

(86) ‖(L1 − L2)w‖l2M/ε

≤ C‖ρ1 − ρ2‖l2M/ε

‖w‖l2M/ε

.

Back to the condition (79). For ρ and ρ in the δ-ball around ρ0, we know that

F ετ (ρ) − F ε

τ (ρ) = Lετ,eρ(ρ− ρ) +R(ρ, ρ)

for remainder R that

‖R(ρ, ρ)‖l2M/ε

≤ C‖ρ− ρ‖2l2M/ε

.

Therefore,

‖F ετ (ρ) − F ε

τ (ρ) − Lε(ρ− ρ)‖l2M/ε

≤ ‖(Lετ,eρ − Lε)(ρ− ρ)‖l2

M/ε+ ‖R(ρ, ρ)‖l2

M/ε

≤ C‖ρ− ρ0‖l2M/ε

‖ρ− ρ‖l2M/ε

+ C‖ρ− ρ‖2l2M/ε

≤ Cδ‖ρ− ρ‖l2M/ε

≤ Cε1/2‖ρ− ρ‖l2M/ε

As Kε is bounded uniformly in ε, condition (79) is satisfied by taking ε sufficiently

small.

�


9. Extensions

We have established the Cauchy-Born rule for density and also for band energy

for nonlinear (self-consistent) tight-binding model under the stability condition. We

have focused on one dimensional models with nearest-neighbor interaction. The

extension to the case with longer (but finite) range interactions is straightforward.

The lattice Schrodinger operator will then have more off-diagonal non-zero elements.

As long as the interaction range is finite, the same proof can be used with obvious

changes. The extension to higher dimensions is also easy. There is one point in the

proof that needs to be modified. When we built the approximate solution using

asymptotics, the difference is controlled in l∞ norm, however the periodic-l2 norm

is used for the convergence proof. Therefore, in higher dimension, we need better

approximate solutions for the self-consistent equation to initialize the iteration. To

be more specific, let us consider three dimension, we will use the Newton-Raphson

iteration, with initial iterate given by a higher order approximate solution to the

self-consistent equation

(87) ρ0(j) = ρCB(∇u(Xεj ), j) + ερ1(X

εj , j),

where ρCB is again the Cauchy-Born construction and ρ1 is the next order correction

given by the asymptotics in Appendix A.

We can then improve the estimate given in Proposition 13, the proof is similar.

Proposition 16. Take ρ0(j) given in (87). There exist constants a, ε0, and C,

such that if u is smooth and u satisfies ‖∇u‖L∞ ≤ a, and if ε ≤ ε0, we have

‖ρ0 − F ετ (ρ0)‖∞ ≤ Cε2.

Therefore, measure in l2 norm, we have

‖ρ0 − F ετ (ρ0)‖l2

Γ/ε≤ ε−3/2‖ρ0 − F ε

τ (ρ0)‖l∞Γ/ε

≤ Cε1/2,

where Γ is the unit cell in three dimension.

Hence, following the same argument as in the proof of Theorem 10, we can prove


given by ρe is stable according to Definition 3. There exist constants a ∈ (0, a0), ε0,

and C, such that if u is smooth (C∞) and u satisfies ‖∇u‖L∞ ≤ a, and if ε ≤ ε0,

then there exists a unique ρε ∈ l∞(Z3) that satisfies

(1) ρε is Γ/ε-periodic and ‖ρε − ρ0‖l2Γ/ε

≤ Cε1/2.

(2) ρε is a self-consistent solution to the equation:

ρε(j) = F ετ (ρε)(j) =

1

2πi

∫

C

1

λ−Hετ [ρε]

dλ(j, j), j ∈ Γ/ε.

The extension to Kohn-Sham density functional theory model will be considered

in subsequent papers. The overall strategy is similar to the discrete case, however, it

is more involved technically due to the continuous nature of the problem. Moreover,


since the Coulomb interaction is present, which has an infinite interaction range,

not only the mathematical difficulty is increased, the physical properties of the

system will also be changed. For instance, besides the analogous stability conditions

exhibits for the nonlinear tight-binding model, we will have an additional stability

condition arising from the Coulomb interaction. The details can be found in the

subsequent paper [5].

Appendix A. Higher order asymptotics of tight-binding model

In the appendix, we will discuss how to build up higher order asymptotics for the

self-consistent equation. We want to find a solution to the self-consistent equation

ρ(x) =1

2πi

∫

C

1

λ−Hετ [ρ]

dλ(x, x).

For this, let us assume the following ansatz for ρ:

(88) ρ(i) = ρ0(Xεi , i) + ερ1(X

εi , i) + ε2ρ2(X

εi , i) + · · ·

with ρk(x, i) M -periodic in the second variable. Let us build a higher order ap-

proximated solution to the self-consistent equation using asymptotics. The leading

order is the Cauchy-Born approximation which we have already shown.

A.1. Asymptotics for the Schrodinger operator. We first consider the Hamil-

tonian given by ρ with ansatz (88). Substituting the ansatz (88) into v(ρi), we

obtain similarly

(89)

vi = v(ρi) = v(ρ0(Xεi , i)) + εv′ρ1(X

εi , i)

+ ε2 12v

′′ρ1(Xεi , i)

2 + ε2v′ρ2(Xεi , i) + O(ε3)

= v0(Xεi , i) + εv1(X

εi , i) + ε2v2(X

εi , i) + O(ε3),

where v′ and v′′ are evaluated at ρ0(Xεi , i) and

v0(x, i) = v(ρ0(x, i));(90)

v1(x, i) = v′(ρ0(x, i))ρ1(x, i);(91)

v2(x, i) = 12v

′′(ρ0(x, i))ρ1(x, i)2 + v′(ρ0(x, i))ρ2(x, i).(92)

Putting these together, recalling the expansion of b from Section 7.1, we get the

leading order Hamiltonian as

(93) H0(x) =∑

i

(ai + v0(x, i))c†i ci +

∑

i

(b0(x, i)c†i ci+1 + h.c.),

with the correction given by

(94) Hετ −H0(x) =

∑

i

(v0(X

εi , i) − v0(x, i) + εv1(X

εi , i) + ε2v2(X

εi , i)

)c†i ci

+∑

i

((b0(X

εi , i) − b0(x, i) + εb1(X

εi , i) + ε2b2(X

εi , i)

)c†ici+1 + h.c.

)+ O(ε3),


Further expand the difference δH , we have for the potential terms

δvj = (δH)jj = (Hετ −H0(X

εi ))jj

= v0(Xεj , j) − v0(X

εi , j) + εv1(X

εj , j) + ε2v2(X

εj , j) + O(ε3)

= ε(Xj −Xi)∇xv0(Xεi , j) + εv1(X

εi , j)

+ ε2 12 (Xj −Xi)

2∇2xv0(X

εi , j) + ε2(Xj −Xi)∇xv1(X

εi , j)

+ ε2v2(Xεi , j) + O(ε3),

and also for the off-diagonal terms,

δbj = (δH)j,j+1 = (Hετ −H0(X

εi ))j,j+1

= b0(Xεj , j) − b0(X

εi , j) + εb1(X

εj , j) + ε2b2(X

εj , j) + O(ε3)

= ε(Xj −Xi)∇xb0(Xεi , j) + εb1(X

εi , j)

+ ε2 12 (Xj −Xi)

2∇2xb0(X

εi , j) + ε2(Xj −Xi)∇xb1(X

εi , j)

+ ε2b2(Xεi , j) + O(ε3).

Hence, δH can be decomposed into different orders

(95) δH = εδH1 + ε2δH2 + O(ε3).

A.2. Asymptotics for the map F ετ . Using the resolvent identity, the map

F ετ (ρ)(i) =

1

2πi

∫

C

1

λ−Hετ

dλ(i, i)

=1

2πi

∫

C

1

λ−H0(Xεi )

dλ(i, i)

+1

2πi

∫

C

1

λ−H0(Xεi )δH

1

λ−H0(Xεi )

dλ(i, i)

+1

2πi

∫

C

1

λ−H0(Xεi )δH

1

λ−H0(Xεi )δH

1

λ−H0(Xεi )

dλ(i, i)

+1

2πi

∫

C

1

λ−Hετ

(δH

1

λ−H0(Xεi )

)3

dλ(i, i).

Substitute in the expansion of δH , we obtain for the leading three orders

F ετ (ρ)(i) =

1

2πi

∫

C

1

λ−H0(Xεi )

dλ(i, i)

+ε

2πi

∫

C

1

λ−H0(Xεi )δH1

1

λ−H0(Xεi )

dλ(i, i)

+ε2

2πi

∫

C

1

λ−H0(Xεi )δH2

1

λ−H0(Xεi )

dλ(i, i)

+ε2

2πi

∫

C

1

λ−H0(Xεi )

(δH1

1

λ−H0(Xεi )

)2

dλ(i, i) + O(ε3).


Collecting orders of the self-consistent equation ρ = F ετ (ρ), we have for the

leading three orders

ρ0(x, j) =1

2πi

∫

C

1

λ−H0(x)dλ(j, j);(96)

ρ1(x, j) =1

2πi

∫

C

1

λ−H0(x)δH1(x)

1

λ −H0(x)dλ(j, j);(97)

ρ2(x, j) =1

2πi

∫

C

1

λ−H0(x)δH2(x)

1

λ −H0(x)dλ(j, j)(98)

+1

2πi

∫

C

1

λ−H0(x)

(δH1(x)

1

λ −H0(x)

)2

dλ(j, j).

The leading order equation (96) gives ρ0(x, j) = ρCB(A, j) and also H0(x) =

HA[ρCB(A, ·)] for A = u′(x).

We may rewrite the next order and third order equations in the forms

ρ1(x, j) =1

2πi

∫

C

1

λ−H0(x)δH1(x)

1

λ −H0(x)dλ(j, j)

=1

2πi

∫

C

1

λ−H0(x)(v′ρ1)(x, ·)

1


+1

2πi

∫

C

1

λ−H0(x)(δH1(x) − (v′ρ1)(x, ·))

1

λ −H0(x)dλ(j, j),

and

ρ2(x, j) =1

2πi

∫

C

1

λ−H0(x)δH2(x)

1


+1

2πi

∫

C

1

λ−H0(x)

(δH1(x)

1

λ −H0(x)

)2

dλ(j, j)

=1

2πi

∫

C

1

λ−H0(x)(v′ρ2)(x, ·)

1


+1

2πi

∫

C

1

λ−H0(x)(δH2(x) − (v′ρ2)(x, ·))

1


+1

2πi

∫

C

1

λ−H0(x)

(δH1(x)

1

λ −H0(x)

)2

dλ(j, j).

Using the linearized operator for homogeneous deformation

(LAw)(j) =1

2πi

∫

C

1

λ−HA[ρCB]v′(ρCB)w

1

λ −HA[ρCB]dλ(j, j).

We have

(I − LA)ρ1(x, j) =1

2πi

∫

C

1

λ−H0(x)(δH1(x) − (v′ρ1)(x, ·))

1

λ −H0(x)dλ(j, j),

and for the third order equation

(I − LA)ρ2(x, j) =1

2πi

∫

C

1

λ−H0(x)(δH2(x) − (v′ρ2)(x, ·))

1


+1

2πi

∫

C

1

λ−H0(x)

(δH1(x)

1

λ −H0(x)

)2

dλ(j, j),


where the right-hand sides do not depend on ρ2. By inverting (I − LA) (the in-

vertibility is guaranteed by the stability condition), we obtain higher order approx-

imations ρ1, ρ2, and so on. The approximation error can be controlled similarly as

what have been done for the leading order approximation in section 7.

Appendix B. Proof of Theorem 11

First notice that using resolvents, we may rewrite the band energy as

(99) E(τ, ε) = ε

M/ε∑

j=1

1

2πi

∫

C

Hετ [ρε]

1

λ−Hετ [ρε]

dλ(j, j).

Indeed, using Bloch decomposition and Cauchy formula, we have

(100)1

2πi

∫

C

Hετ [ρε]

1

λ−Hετ [ρε]

dλ =

Z/ε∑

n=1

∫

Γ∗

Eεn,τ (ξ)|ψε

n,ξ,τ 〉〈ψεn,ξ,τ | dξ,

where ψεn,ξ,τ is the eigenfunctions of Hε

τ [ρε] corresponding to Eεn,τ (ξ). Taking the

trace and using the normalization of eigenfunctions, we have (99).

Further rewrite (99) as

E(τ, ε) = ε

M/ε∑

j=1

1

2πi

∫

C

(λ

λ−Hετ [ρε]

− I

)dλ(j, j)

= ε

M/ε∑

j=1

1

2πi

∫

C

λ

λ−Hετ [ρε]

dλ(j, j).

Since C is a compact contour, it suffices to consider for each λ the term

M/ε∑

j=1

1

λ−Hετ [ρε]

(j, j) =

M/ε∑

j=1

1

λ−Hετ [ρ0]

(j, j)

+

M/ε∑

j=1

1

λ−Hετ [ρε]

(Hετ [ρε] −Hε

τ [ρ0])1

λ −Hετ [ρ0]

(j, j).

Since ‖ρε − ρ0‖l∞ ≤ Cε1/2, using Lemma 2, we obtain the estimate

∣∣∣∣M/ε∑

j=1

1

λ−Hετ [ρε]

(j, j) −M/ε∑

j=1

1

λ−Hετ [ρ0]

(j, j)

∣∣∣∣ ≤ Cε−1/2.

From the analysis in section 7, we have the estimate∣∣∣∣

1

λ−Hετ [ρ0]

(j, j) − 1

λ−HAj [ρCB(Aj , ·)](j, j)

∣∣∣∣ ≤ Cε,

with Aj = ∇u(Xεj ). Therefore

∣∣∣∣E(τ, ε) − ε

M/ε∑

j=1

1

2πi

∫

C

λ

λ−HAj [ρCB(Aj , ·)]dλ(j, j)

∣∣∣∣ ≤ Cε1/2.


The theorem follows from the smoothness of ∇u(x) and the identity

(101) ECB(A) =

M∑

j=1

1

2πi

∫

C

HA[ρCB(A, ·)] 1

λ−HA[ρCB(A, ·)] dλ(j, j).

Acknowledgement: The work of W. E and J. L. was supported in part by the

NSF grant DMS-0708026, the ONR grant N00014-01-1-0674 and the DOE grant

DE-FG02-03ER25587. J. L. was also supported in part by Porter Ogden Jacobus

Fellowship from Princeton University. We thank the anonymous reviewer for vari-

ous suggestions of improving the presentation of the paper.

References

[1] X. Blanc, C. Le Bris, and P.-L. Lions, From molecular models to continuum mechanics, Arch.

Ration. Mech. Anal. 164 (2002), 341–381.

[2] J. M. Combes and L. Thomas, Asymptotic behavior of eigenfunctions for multi-particle

Schrodinger operators, Commun. Math. Phys. 34 (1973), 251–270.

[3] W. E and J. Lu, The elastic continuum limit of the tight binding model, Chinese Ann. Math.

Ser. B. 28 (2007), 665–675.

[4] , The electronic structure of smoothly deformed crystals: Wannier functions and the

Cauchy-Born rule, 2009. submitted.

[5] , Continuum limit of solutions to the Kohn-Sham equation, 2010. in preparation.

[6] W. E and P. B. Ming, Cauchy-Born rule and the stability of crystalline solids: static problems,

Arch. Ration. Mech. Anal. 183 (2007), 241–297.

[7] , Cauchy-Born rule and the stability of the crystalline solids: dynamic problems, Acta

Math. Appl. Sin. Engl. Ser. 23 (2007), 529–550.

[8] W. A. Harrison, Elementary electronic structure, World Scientific, Singapore, 1999.

[9] R. Parr and W. Yang, Density-functional theory of atoms and molecules, International Series

of Monographs on Chemistry, Oxford University Press, New York, 1989.

[10] M. Reed and B. Simon, Methods of modern mathematical physics, Vol IV, Academic Press,

New York, 1980.

Department of Mathematics and Program in Applied and Computational Mathemat-

ics, Princeton University, Princeton, NJ 08544, [email protected]

Department of Mathematics, Courant Institute of Mathematical Sciences, New York

University, New York, NY 10012, [email protected]

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