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arXiv:1110.6807v1 [math.SP] 31 Oct 2011

ON THE GROUND STATE OF QUANTUM LAYERS

ZHIQIN LU AND JULIE ROWLETT

1. Introduction

Mesoscopic physics describes length scales from one atom to micrometers.

At these scales, the behavior of particles is no longer described by classi-

cal physics: quantum effects are observed. Numerous phenomena such as

quantum dots and wells occur at the scale of mesoscopic physics, and all

nanotechnology is on the mesoscopic scale.

Consider, for example, electrons trapped between two semi-conducting

materials, or more generally, quantum particles trapped between hard walls.

Mathematically, such situations are described using a quantum layer.

Let p : Σ → R3be an embedded surface in R3. We will always make the

following assumptions on Σ.

1.0.1. Hypotheses.

(1) Σ is a C2smooth surface;

(2) Σ is orientable, complete, but non-compact;

(3) Σ is not totally geodesic;

(4) Σ is asymptotically flat in the sense that the second fundamental

form tends to zero at infinity.

A quantum layer over Σ is an oriented differentiable manifold Ω∼= Σ ×

[−a,a] for some (small) positive number a. Let?N be the unit normal vector

of Σ in R3. Define

˜ p : Ω → R3

by

˜ p(x,t) = p(x) + t?Nx.

If a is small, then ˜ p is clearly an embedding. The Riemannian metric ds2

is defined as the pull-back of the Euclidean metric via ˜ p. The Riemannian

manifold (Ω,ds2

Ω) is called the quantum layer. Physically, the quantum par-

ticles are trapped between two copies of the same semi-conducting material

Σ at a uniform distance of 2a apart, where a is of mesoscopic scale. A

natural question is:

Under what conditions on the geometry of the semi-conducting

material Σ do bound states exist?

Mathematically, we would like to understand the following.

Does there exist a geometric condition on Σ which guarantees

the existence of bound states?

1

Ω

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2 ZHIQIN LU AND JULIE ROWLETT

Let us formulate this precisely. Let ∆ = ∆Ωbe the Laplacian with re-

spect to the Riemannian metric ds2

Ω, and assume the Dirichlet boundary

condition.Since Ω is a smooth, complete manifold with boundary, the

Dirichlet Laplacian is the Friedrichs extension of the Laplacian acting on

C∞

0(Ω) and is self-adjoint. The spectrum consists of two parts: discrete,

isolated eigenvalues of finite multiplicity and essential spectrum. We dis-

tinguish the eigenvalues which are disjoint from the essential spectrum and

refer to these as the discrete spectrum, since there may also be embedded

eigenvalues within the essential spectrum. The ground state is the smallest

discrete eigenvalue. Physically, the quantum particles are governed by the

Dirichlet Laplacian as Hamiltonian, and the existence of discrete spectrum

is equivalent to the existence of the ground state. Therefore:

The existence of the ground state is equivalent to the exis-

tence of bound states in the physical model.

Let κ be the Gauss curvature of Σ throughout this paper. We first make

the following conjecture.

Conjecture 1. Under the preceding assumptions (1.0.1) on Σ, if

(1.1)

?

Σ

|κ|dΣ < +∞,

then there exists an α = α(Σ) such that for all a ∈ (0,α), the ground state

of the quantum layer over Σ of width 2a exists.

Remark 1.1. By a theorem of Huber [4], if (1.1) is valid, then Σ is conformal

to a compact Riemann surface with finitely many points removed. Moreover,

White [10] proved that if

(1.2)

?

Σ

κ−dΣ < +∞,

where

κ =

?

κ+

−κ−

κ ≥ 0

κ < 0

,

then

?

Σ

|κ|dΣ < +∞.

Thus (1.1) can be weakened to (1.2).

Conjecture 1 was proven under the condition

(1.3)

?

Σ

κdΣ ≤ 0

through the work of Duclos, Exner and Krejˇ ciˇ r´ ık [2] and Carron, Exner, and

Krejˇ ciˇ r´ ık [1].

Our work focuses on the remaining case:

?

Σ

κdΣ > 0.

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SPECTRUM OF QUANTUM LAYERS3

In this case, by a theorem of Hartman [3], we have

(1.4)

?

Σ

κdΣ = 2π

?

χ(Σ) −

?

λi

?

where χ(Σ) is the Euler characteristic number of Σ, and λiare the isoperi-

metric constants at each end defined by

λi= lim

r→∞

vol(B(r))

πr2

.

The existence of the limit follows from the integrability of the Gauss curva-

ture (1.1).

Moreover, it follows from Huber [4] that

Σ = Σ\{p1,··· ,ps}

for a compact Riemann surface Σ. Thus we have

χ(Σ) ≤ χ(Σ) − s = 2 − 2g(Σ) − s < 2.

This together with (1.4) implies that χ(Σ) = 1, and hence the surface is

differomorphic to R2. Consequently,

?

Although the topology of the surface is completely known, this is the only

remaining case in which the conjecture has not yet been proven.

We briefly recall the main results of [1,2].

(1.5)0 <

Σ

κdΣ ≤ 2π.

Theorem 1.1 (Duclos, Exner and Krejˇ ciˇ r´ ık). Let Σ be a C2-smooth com-

plete simply connected non-compact surface with a pole embedded in R3. Let

the layer Ω∼= Σ × [−a,a] built over the surface be not self-intersecting. If

the surface is not a plane, but it is asymptotically planar,1then if a satisfies

condition (1.7) below, each of the following implies Conjecture 1.

(1) The Gauss curvature satisfies (1.1) and (1.3);

(2) Σ is C3smooth, and a is sufficiently small;

(3) Σ is C3smooth, the Gauss curvature is integrable, the gradient of the

mean curvature ∇gH is L2integrable, and the total mean curvature

is infinite;

(4) the Gauss curvature is integrable and Σ is cylindrically symmetric.

In [1], Carron, Exner and Krejˇ ciˇ r´ ık proved that the conjecture holds under

more general conditions; in particular, they no longer required the surface

to have a pole.

Theorem 1.2 (Carron, Exner and Krejˇ ciˇ r´ ık). Let Σ be a complete asymp-

totically planar, noncompact connected surface of class C2embedded in R3

and such that the Gauss curvature satisfies (1.1). Let the layer Ω of width 2a

1Asymptotically planar is equivalent to: the second fundamental form tends to zero at

infinity.

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4ZHIQIN LU AND JULIE ROWLETT

be defined so that Ω does not overlap, and a satisfies condition (1.7) below.

Then, any of the following imply Conjecture 1.

(1) The Gauss curvature satisfies (1.3);

(2) a is small enough, and the gradient of the mean curvature ∇gH is

locally L2integrable;

(3) the gradient of the mean curvature ∇gH is L2integrable, and the

total mean curvature is infinite;

(4) Σ contains a cylindrically symmetric end with a positive total Gauss

curvature.

The general method used in both [1,2]: first compute the infimum of the

essential spectrum, next construct appropriate test functions, and finally ap-

ply the variational principle to prove that if one of the conditions is satisfied,

then there must be an eigenvalue strictly less than the essential spectrum.

The existence of the ground state immediately follows. The pole and sym-

metry assumption (4) were necessary in [2] because their test functions are

radially symmetric.

The first main result of the present paper generalizes [1,2] by demonstrat-

ing that Conjecture 1 holds if the surface is weakly κ-parabolic2.

Theorem 1.3. Let Σ be a complete surface in R3which satisfies the hy-

pothesis (1.0.1), and assume that Σ is weakly κ-parabolic. Then, there exists

α > 0 such that for all a ∈ (0,α), the ground state of the quantum layer

over Σ of width 2a exists.

Although the proof of Theorem 1.3 is based on the same principles used

in [1,2], our theorem not only generalizes their results, but also shows that

their argument fits nicely into the notion of weak κ-parabolicity, which pro-

vides a geometric abstraction of their arguments.

A question raised in [2] which remained unanswered in [1] is:

Is it possible to build quantum layers which have strictly posi-

tive total Gauss curvature without assuming cylindrical sym-

metry or L2integrability of the mean curvature?

The first author and C. Lin proved in [7, Theorem 1.1] that Conjecture 1

holds when Σ can be represented as the graph of a convex function satisfying

certain conditions. Our next main result generalizes [7] and demonstrates

that Conjecture 1 holds if the Gauss curvature is non-negative thereby giving

an affirmative answer to the above question.

Theorem 1.4. Let Σ be a complete surface in R3which satisfies the hy-

pothesis (1.0.1). Assume that the Gauss curvature of Σ is non-negative and

2We refer to § 3 for the definition of weak κ-parabolicity.

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SPECTRUM OF QUANTUM LAYERS5

satisfies (1.1). Then for all a ∈ (0,B−1

(1.6)

∞), where

B∞:= sup

p∈Σ||B(p)||,

the ground state of the quantum layer over Σ of width 2a exists.

Remark 1.2. The condition

(1.7)a ∈?0,B−1

∞

?

is merely a technicality to ensure that Ω embeds into R3. This is equivalent

to the non-overlapping assumption made in [1,2].

The proof of Theorem 1.4 is more subtle. Test functions similar to those

used in [1,2] rely on the weak κ-parabolicity of the surfaces, but a surface

with non-negative Gauss curvature will not be weakly κ-parabolic. The main

idea is to work on annuli, rather than on disks. In general, the integration

of the mean curvature outside a compact set may be quite small since the

surface is asymptotically flat. But using a result of White [10], we actually

know that the total mean curvature is at least of linear growth. This estimate

plays a crucial role in the proof.

This note is organized as follows. In section § 2, we recall the variational

principles for the essential spectrum and the ground state, and we determine

the infimum of the essential spectrum. In § 3, we introduce the notion of

κ-parabolicity and prove Theorem 1.3. The proof of Theorem 1.4 comprises

§ 4. We conclude in § 5 with a discussion of further generalizations.

Acknowledgements

The first author is partially supported by NSF grant DMS-09-04653. The

second author gratefully acknowledges the support of the Max Planck Insti-

tut f¨ ur Mathematik in Bonn, Germany.

2. Variational principles and the infimum of the essential

spectrum

It is well known that

(2.1)σ0= inf

0(Ω)

f∈C∞

?

Ω|∇f|2dΩ

?

?

Ωf2dΩ

is the infimum of the Laplacian, and

(2.2)σess= sup

K

inf

0(Ω\K)

f∈C∞

Ω|∇f|2dΩ

?

Ωf2dΩ

is the infimum of the essential spectrum, where K runs over all compact

subsets of Ω. Since Ω = Σ × [−a,a], it is not hard to see that

(2.3)σess= sup

K⊂Σ

inf

f∈C∞

0(Ω\K×[−a,a])

?

Ω|∇f|2dΩ

?

Ωf2dΩ

,

where K runs over all compact subsets of Σ.

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6ZHIQIN LU AND JULIE ROWLETT

It follows that σ0 ≤ σess. Moreover, we have the following well-known

result.

Proposition 2.1. If σ0< σess, then the ground state exists and is equal to

σ0.

?

Let (x1,x2) be a local coordinate system of Σ. Then (x1,x2,t) defines

a local coordinate system of Ω. Such a local coordinate system is called a

Fermi coordinate system. Let x3= t, and let ds2

have

is equal to the spectral threshold of the planar quantum layer of width 2a,

namely, π2/4a2.

We make the following definitions. For a smooth function f on Ω, let

|∇f|2dΩ −π2

?

?

where

2

?

is the square of the norm of the horizontal differential.

Obviously, we have

Ω= Gijdxidxj. Then we

(2.4)Gij=

(p + t?N)xi(p + t?N)xj

0

1

1 ≤ i,j ≤ 2;

i = 3,or j = 3, but i ?= j;

i = j = 3.

We will demonstrate below that the infimum of the essential spectrum

Q(f,f) =

?

Ω

4a2

?

Ω

f2dΩ; (2.5)

Q1(f,f) =

Ω

|∇′f|2dΩ;(2.6)

Q2(f,f) =

Ω

?∂f

∂t

?2

dΩ −π2

4a2

?

Ω

f2dΩ,(2.7)

|∇′f|2=

i,j=1

Gij∂f

∂xi

∂f

∂xj

Q(f,f) = Q1(f,f) + Q2(f,f),

and

?

Ω

|∇f|2dΩ =

?

Ω

|∇′f|2dΩ +

?

Ω

?∂f

∂t

?2

dΩ.

Clearly, we have

?

Ω

|∇f|2dΩ ≥

?

Ω

?∂f

∂t

?2

dΩ.

Let ds2

coordinates (x1,x2). We shall compare the matrices (Gij)1≤i,j≤2and (gij)

outside a big compact set of Σ. By (2.4), we have

Σ= gijdxidxjbe the Riemannian metric of Σ with respect to the

(2.8)Gij= gij+ tpxi?Nxj+ tpxj?Nxi+ t2?Nxi?Nxj,

where we note that gij= pipj.

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SPECTRUM OF QUANTUM LAYERS7

Using (2.8), we have

(2.9) det(Gij) = det(gij)(1 − Ht + κt2),

where H is the mean curvature, defined to be the trace of the second fun-

damental form.

We assume that at the point x, local coordinates (x1,x2) are chosen such

that gij= δij. We have the estimate

|Gij− δij| ≤ a||B||,

where B is the second fundamental form of the surface Σ. This leads to the

following conclusions.

Proposition 2.2. Let Σ be an embedded surface in R3which satisfies hy-

potheses (1.0.1). Then, for any

a ∈?0,B−1

∞

?,

the quantum layer Ω∼= Σ×[−a,a] is an embedded submanifold of R3, where

B∞is defined in (1.6). Moreover, for any ε > 0, there is a compact set K

of Σ such that on Σ \ K we have

?g11

In particular, we have

(1 − ε)

g12

g22

g21

?

≤

?G11

G12

G22

G21

?

≤ (1 + ε)

?g11

g12

g22

g21

?

.

(1 − ε)2dΣdt ≤ dΩ ≤ (1 + ε)2dΣdt.

On the other hand, there exists α = α(Σ,ε) > 0 such that for all a ∈ (0,α),

the above inequalities hold at any point of Σ.

?

For a compact set E ⊂ Σ, we shall use the notation

f ∈ C∞

0(Σ) whose support lies in Σ \ E.

0(Σ \ E)

to denote a function f ∈ C∞

Based on the preceding proposition and the variational principle, we are

able to determine σess. The following lemma is originally due to [1,2], but

we include a short proof for completeness.

Lemma 2.1. Let Σ be an embedded surface in R3which satisfies the hy-

potheses (1.0.1), and assume the Gauss curvature satisfies (1.1). Then, for

any quantum layer Ω built over Σ of width 2a > 0, where a satisfies (1.7),

σess=

π2

4a2.

Proof. We first prove that

σess≥π2

4a2.

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8 ZHIQIN LU AND JULIE ROWLETT

Let ε > 0 be given, and let K be a compact set of Σ as in Proposition 2.2.

Let˜ K ⊂ Ω be defined as

˜K∼= K × [−a,a].

For f ∈ C∞

?

where the last inequality follows from the 1-dimensional Poincar´ e inequality.

By Proposition 2.2 again, we have

|∇f|2dΩ ≥(1 − ε)2

(1 + ε)2

which by the variational principle for σessimplies

σess≥(1 − ε)2

(1 + ε)2

Letting ε → 0 completes the proof of the first inequality.

To complete the proof of the lemma, we demonstrate the estimate

σess≤π2

Since the Gauss curvature tends to zero, it is well known that the infimum

of the essential spectrum of Σ is zero. Therefore, for any compact set K

and any ε > 0, there exists a smooth function ϕ ∈ C∞

?

Using Proposition 2.2, for sufficiently large K, we have

?

We let

χ(t) = cos(πt/2a)

and consider the function ϕχ on Ω. Thus by (2.7) and (2.9), since χ2and

(χ′)2are even functions, we have

?a

?

We compute?a

?a

0(Ω \˜K), by Proposition 2.2,

dΩ ≥ (1−ε)2

Ω

?∂f

∂t

?2

?

Σ

?a

−a

?∂f

∂t

?2

dtdΣ ≥ (1−ε)2π2

4a2

?

Σ

?a

−a

f2dtdΣ,

?

Ω

π2

4a2

?

Ω

f2dtdΣ,

π2

4a2.

(2.10)

4a2.

0(Σ \ K) such that

Σ

|∇ϕ|2dΣ ≤ ε

?

Σ

ϕ2dΣ.

Ω

|∇′ϕ|2dΩ ≤ 2(1 + ε)a

?

Σ

|∇ϕ|2dΣ ≤ 2ε(1 + ε)a

?

Σ

ϕ2dΣ.

Q2(ϕχ,ϕχ) =

?

Σ

ϕ2dΣ

−a

??χ′(t)?2−π2

4a2χ(t)2

4a2χ(t)2

?

?

dt

+

Σ

ϕ2κdΣ

?a

−a

t2

??χ′(t)?2−π2

dt.

−a

?χ′(t)?2dt =π2

t2(χ′)2dt =a(π2+ 6)

4a2

?a

−a

χ2dt;

?a

t2χ2dt =a3(π2− 6)

−a

χ2dt = a;

−a

12

;

?a

−a

3π2

.

(2.11)

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SPECTRUM OF QUANTUM LAYERS9

Thus we have

(2.12)Q2(ϕχ,ϕχ) = a

?

Σ

ϕ2κdΣ,

and hence

Q(ϕχ,ϕχ) ≤ 2aε(1 + ε)

?

Σ

ϕ2dΣ + a

?

Σ

ϕ2κdΣ.

By Proposition 2.2 and (1.1), there exists an ε1 sufficiently small such

that

?

and

?

By the variational principle for σessand the definition of Q, we have

Ω

(ϕχ)2dΩ ≥ 2(1 − ε1)a

?

Σ

ϕ2dΣ,

Σ

ϕ2κdΣ ≤ ε1

?

Σ

ϕ2dΣ.

σess−π2

4a2≤Q(ϕχ,ϕχ)

?

Ω(ϕχ)2dΩ≤2ε(1 + ε) + ε1

2(1 − ε1)

This proves the lemma.

?

3. κ-parabolicity

We refer to [5] for the following definition and basic properties of parabolic

manifolds.

Definition 3.1. A complete manifold is parabolic if it does not admit a

positive Green’s function. Otherwise it is non-parabolic.

We first establish the following well-known result.

Proposition 3.1. Assume that κ ∈ L1(Σ). Then there exists a positive

constant c1such that the volume growth of Σ satisfies

?

B(R)

dΣ < c1R2.

Proof. By the results of Huber [4] and Hartman [3], Σ has only finitely

many ends, and at each end,

λ = lim

r→∞

vol(B(r))

πr2

exists. This proves the proposition.

?

By the above proposition, a surface whose Gauss curvature κ ∈ L1is a

parabolic manifold (cf. [5]).

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10ZHIQIN LU AND JULIE ROWLETT

The definition of parabolicity is equivalent to: the capacity of any ball of

radius R is zero. That is, for any positive R and ε, there exists a smooth

function ϕ which satisfies

ϕ ∈ C∞

ϕ ≡ 1 on B(R);

0 ≤ ϕ ≤ 1;

?

0(Σ);

Σ

|∇ϕ|2dΣ < ε.

For the rest of the paper, we shall repeatedly use the above equivalent

capacity definition of parabolicity.

Parallel to the above, we make the following definition of weak κ-parabolicity.

Definition 3.2. The κ-capacity of a subset E ⊂ Σ is defined to be the

infimum of

?

where ϕ is a smooth function on Σ with compact support, and ϕ ≡ 1 in a

neighborhood of E.

We say Σ is weakly κ-parabolic, if either Σ is a minimal surface, or there

exists p ∈ Σ such that H(p) ?= 0, and the κ-capacity of a neighborhood of p

is non-positive.

Σ

|∇ϕ|2+ κϕ2,

Proof of Theorem 1.3. If Σ is a minimal surface, then κ ≤ 0. Con-

sequently the total Gauss curvature is nonpositive, and the theorem follows

from the results in [1,2].

By Lemma 2.1 and the variational principles, it suffices to prove

σ0< σess

for the remaining cases.

Let p ∈ Σ be a point such that H(p) ?= 0. We assume that there is

a constant ε1 > 0 such that |H(p)| > ε1on the ball of radius δ centered

at p, which is denoted Bp(δ), with δ a fixed positive constant. For any

ε2> 0, by the definition of weakly κ-parabolic (and choosing δ > 0 smaller

if necessary), there exists a smooth function ϕ with compact support such

that ϕ ≡ 1 on Bp(δ), and

?

Let j be a smooth function such that the support of j is contained in

Bp(δ), and

????

(3.1)

Σ

|∇ϕ|2+ κϕ2< ε2.

?

Σ

HjdΣ

????> ε3> 0

Page 11

SPECTRUM OF QUANTUM LAYERS11

for some positive constant ε3> 0. For a suitable choice of orientation, we

may assume that

?

By (2.11), we have

Σ

HjdΣ > ε3> 0.

Q2(ϕχ,ϕχ) = a

?

Σ

ϕ2κdΣ.

By Proposition 2.2, for any ε4> 0, there exists α = α(ε4) > 0 such that

for all a ∈ (0,α),

Q1(ϕχ,ϕχ) =

Ω

Since Q = Q1+ Q2, and since the inequality (3.1) is strict, using (2.12), we

can choose ε4sufficiently small and a corresponding α such that

?

Since j assumes values in [0,1] and depends only on the choice of δ which

is fixed, there is a constant c1such that

Q(jχ(t)t,jχ(t)t) ≤ c1a2.

The support of j is contained in {ϕ ≡ 1}, so by (2.6) we have

Q1(ϕχ(t),jχ(t)t) = 0.

?

|∇′ϕ|2χ2dΩ ≤ (1 + ε4)a

?

Σ

|∇ϕ|2dΣ.

Q(ϕχ,ϕχ) ≤

Σ

a(1 + ε4)|∇ϕ|2+ aκϕ2≤ aε2.

By (2.9), and since ϕ ≡ 1 on the support of j, we have

?

We compute that

χ′(t)(χ(t)t)′−π2

Q2(ϕχ(t),jχ(t)t) =

Ω

χ′(t)(χ(t)t)′jdΩ −π2

4a2

?

Ω

jχ2(t)tdΩ.

(3.2)

4a2χ2(t)t = −π

4asin

?πt

a

?

−π2t

4a2cos

?πt

a

?

is an odd function, and thus

Q2(ϕχ(t),jχ(t)t) = −

?

Σ

HjdΣ ·

?a

−a

(χ′(t)(χ(t)t)′−π2

4a2χ2(t)t)tdt.

But

(3.3)

?a

−a

(χ′(t)(χ(t)t)′−π2

4a2χ2(t)t)tdt = −

?a

−a

χ′(t)χ(t)tdt =a

2

by a straightforward computation. Therefore,

Q(ϕχ(t),jχ(t)t) = −a

Let ε > 0. Using all the above estimates, we have

Q(ϕχ(t) + εjχ(t)t,ϕχ(t) + εjχ(t)t) < aε2− aε3ε + c1a2ε2.

(3.4)

2

?

Σ

HjdΣ.

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12 ZHIQIN LU AND JULIE ROWLETT

Since ε2can be made arbitrary small, and ε3> 0 is fixed, for a suitable

choice of ε, we have

Q(ϕχ(t) + εjχ(t)t,ϕχ(t) + εjχ(t)t) < 0,

for all a ∈ (0,α). The conjecture then follows from the definition of Q, the

variational principle and Lemma 2.1.

?

The following sufficient condition of κ-parabolicity implies that Theorem

1.3 is indeed a generalization of both [1,2].

Lemma 3.1. Let Σ be a complete surface such that

?

Σ

|κ|dΣ < ∞,

and

(3.5)

?

Σ

κdΣ ≤ 0.

Then Σ is weakly κ-parabolic.

Proof. First, if the mean curvature H ≡ 0 on Σ, then Σ is a minimal

surface, and there is nothing to prove. So, we assume there exists some

p ∈ Σ such that H(p) ?= 0. Since κ is integrable, Σ is parabolic. That is,

for any ball B(R) of radius R centered at p, there exists a smooth function

0 ≤ ϕ ≤ 1 with compact support such that ϕ ≡ 1 on B(R) and

?

On the other hand, by the integrability of κ, for sufficiently large R,

Σ

|∇ϕ|2< ε,

?

Σ\B(R)

κϕ2dΣ < ε,

?

Σ\B(R)

|κ|dΣ < ε,

and therefore, by the assumption (3.5),

?

Σ

κϕ2dΣ < 2ε.

It follows that for any ε > 0 and R sufficiently large, there exists a function

ϕ ∈ C∞

?

This estimate proves the lemma.

0(Σ) such that ϕ ≡ 1 on B(R), and

Σ

|∇ϕ|2+ κϕ2dΣ < 3ε.

?

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SPECTRUM OF QUANTUM LAYERS13

4. Proof of Theorem 1.4

We proceed as in the proof of Theorem 1.3 by demonstrating that

σ0< σess.

By Proposition 2.1, this implies that the ground state exists.

First, if the Gauss curvature is identically zero, then by [8, Theorem 2],

the ground state exists.

Henceforth, we shall assume that there is at least one point of Σ at which

the Gauss curvature is positive. Then, by a theorem of Sacksteder [9], with

suitable choice of orientation, we can assume that the principle curvatures

of Σ are always nonnegative.

By (1.5), it follows from the results of White [10] (Theorem 1, p. 318),

that there exists an ε0> 0 such that for R ≫ 0,

?

where B is the second fundamental form of Σ. Since the principle curvatures

are nonnegative, we have

H ≥ ||B||.

Thus we have

?

provided that both R1and R2are large enough.

We follow the same general method of [1,2,6–8], but the new idea here is

to use the above estimate together with test functions supported in annuli

whose radii tend to infinity.

Let ϕ ∈ C∞

supp(ϕ) ⊂ B(5

ϕ ≡ 1 on B(2R) \ B(R);

0 ≤ ϕ ≤ 1;

?

where ε1→ 0 as R → ∞. The existence of such a function ϕ is guaranteed

by the parabolicity of Σ.

Let χ be defined as in the previous sections. By Proposition 2.2, there is

a constant c2such that

?

Hence

Q1(ϕχ,ϕχ) → 0 as R → ∞.

∂B(R)

||B|| > ε0,

(4.1)

B(R2)\B(R1)

H dΣ ≥ ε0(R2− R1)

0(Σ \ B(R

2)) be a smooth function such that

2R);

Σ

|∇ϕ|2dΣ < ε1,

(4.2)

Q1(ϕχ,ϕχ) ≤ c2a

Σ

|∇ϕ|2dΣ < c2aε1.

Page 14

14ZHIQIN LU AND JULIE ROWLETT

Next, we use the same calculations as in the preceding sections to compute

Q2(ϕχ,ϕχ) = a

?

Σ

ϕ2κdΣ.

Since the support of ϕ is contained in the annulus B(5

integrability assumption (1.1) on κ,

2R) \ B(R/2), by the

Q2(ϕχ,ϕχ) → 0 as R → ∞.

Since Q = Q1+ Q2, there exists ε3> 0 such that

Q(ϕχ,ϕχ) ≤ ε3,ε3→ 0 as R → ∞.

Now let’s consider a smooth function j on Σ with 0 ≤ j ≤ 1, such that

j ∈ C∞

j ≡ 1 on B(19

|∇j| < 2.

We consider the function jχ(t)t. Since the functions j, χ(t) and t and

their derivatives are bounded on Ω, it follows from the volume comparison

Proposition 3.1 that there is an absolute constant c1such that

0

?

B(5

3R) \ B(4

12R) \ B(17

3R)

?

;

12R);

Q(jχ(t)t,jχ(t)t) ≤ c1R2.

Next, let’s consider Q(ϕχ(t),jχ(t)t). Since the support of j is contained in

{ϕ ≡ 1}, by (2.6), Q1(ϕχ(t),jχ(t)t) = 0.

The same computation as in (3.4) shows that

Q2(ϕχ(t),jχ(t)t) = −a

2

?

Σ

HjdΣ.

Let ε > 0. By our preceding calculations

Q(ϕχ(t) + εjχ(t)t,ϕχ(t) + εjχ(t)t) < ε3− εa

?

Σ

HjdΣ + ε2c1R2.

By (4.1), we have

Q(ϕχ(t) + εjχ(t)t,ϕχ(t) + εjχ(t)t) < ε3−a

Since ε3→ 0 as R → ∞, we may first choose R sufficiently large and then

choose ε > 0 appropriately so that for all a ∈ (0,B−1

Q(ϕχ(t) + εjχ(t)t,ϕχ(t) + εjχ(t)t) < 0.

6εR + ε2c1R2.

∞),

It then follows from the definition of Q, the variational principle and

Lemma 2.1 that

σ0< σess.

?

Page 15

SPECTRUM OF QUANTUM LAYERS15

5. Further Discussions

The proof of Theorem 1.4 can be generalized to demonstrate the following.

Theorem 5.1. Let Σ be a complete surface in R3which satisfies the hy-

potheses (1.0.1), and assume that the Gauss curvature of Σ satisfies (1.1).

Assume that there exists a positive constant ε0so that the following holds:

for any R > 0, there is a function j ∈ C∞

??

Then Conjecture 1 holds.

0(Σ \ B(R)) satisfying

(5.1)

Σ

HjdΣ

?2

> ε0

?

Σ

(|∇j|2+ j2)dΣ.

Proof. As noted previously, we may assume

?

Σ

κdΣ > 0.

Since Σ is parabolic, for any R > 0 sufficiently large there exists a smooth

function ϕ ∈ C∞

ϕ ≡ 1 on B(R′/2) \ B(2R);

?

0 ≤ ϕ ≤ 1;

?

where R′is a sufficient large number. It follows from the same proof as in

Theorem 1.4 that

0(B(R′) \ B(R)) such that

Σ

|∇ϕ|2dΣ < ε1;

Σ

κϕ2dΣ < ε1,

(5.2)

Q(ϕχ,ϕχ) < ε3,ε3→ 0 as R → ∞.

We choose R′large enough such that

supp(j) ⊂ B(R′/2) \ B(2R).

Without loss of generality, we assume that

?

Σ

HjdΣ > 0.

Then using the same method as in the proof of Theorem 1.4, letting ε > 0,

we have

Q(ϕχ + εjχ(t)t,ϕχ + εjχ(t)t)

≤ ε3− aε

?

Σ

HjdΣ + c1ε2a2

?

Σ

(|∇j|2+ j2)dΣ

for some constant c1which is independent of R. Since ε3→ 0 as R → ∞,

we first choose R sufficiently large to make ε3sufficiently small. Then, using

(5.1), there exists α > 0 such that for all sufficiently small ε > 0,

Q(ϕχ + εjχ(t)t,ϕχ + εjχ(t)t) < 0,∀a ∈ (0,α).

Page 16

16ZHIQIN LU AND JULIE ROWLETT

?

Based on the above theorem, we make the following purely Riemannian

geometric conjecture.

Conjecture 2. Let Σ be a complete surface in R3which satisfies the hy-

pothesis (1.0.1), and assume that the Gauss curvature κ of Σ satisfies (1.1).

If the total Gauss curvature is positive

?

Σ

κ > 0,

then there exists a positive constant ε0 such that the following holds: for

any R > 0 sufficiently large, there exists j ∈ C∞

holds.

0(Σ \ B(R)) such that (5.1)

Remark 5.1. By Theorem 5.1, Conjecture 2 implies Conjecture 1.

Remark 5.2. It is straightforward to verify that conditions (3) and (4) in

Theorem 1.2 are each implied by (5.1). For example, to prove that (3)

implies (5.1), j is replaced by jH, and (5.1) follows from a direct calculation.

Assuming (4), (5.1) follws from either [2, Lemma 6.1] or [1, page 783].

The following proposition uses the result of White [10] to demonstrate a

weaker version of the inequality of (5.1). Namely, we have

Proposition 5.1. Assume that Σ is a complete surface in R3which satisfies

the hypotheses (1.0.1), and that the Gauss curvature κ satisfies

?

Σ

|κ|dΣ < ∞,

?

Σ

κdΣ > 0.

Then for any R > 0 there exists a positive constant ε0and a function j ∈

C∞

0(Σ \ B(R)) which satisfies

??

Proof. By the result [10, Theorem 1, p. 318], for sufficiently large R,

(5.3)

Σ

j|H|dΣ

?2

> ε0

?

Σ

(|∇j|2+ j2)dΣ.

?

∂B(R)

||B|| > c1> 0

for some constant c1.

By the parabolicity of Σ and Proposition 3.1, we can find a function j

whose support is contained in B(5

3R) \ B(4

12R) \ B(17

3R), with

j ≡ 1 on B(19

?

?

12R);

Σ

|∇j|2dΣ < 1,

Σ

j2dΣ ≤ c1R2,

(5.4)

Page 17

SPECTRUM OF QUANTUM LAYERS17

where c1is a positive constant independent of R. Hence we have

?

Σ

||B||jdΣ ≥ c2R

for some constant c2.

On the other hand, since κ is integrable, for sufficiently large R, we have

?

Σ

j

?

|κ|dΣ ≤

??

Σ

jdΣ ·

??

Σ

jκdΣ ≤ ε3R,

for some small positive constant ε3when R is large.

Since |H| ≥ ||B|| −?2|κ|, the above inequalities show that

?

Therefore, there exists ε > 0 such that for sufficiently large R,

Σ

j|H|dΣ ≥

?

Σ

||B||jdΣ −

?

Σ

j

?

2|κ|dΣ ≥ (c2−√2ε3)R.

?

Σ

j|H|dΣ > εR.

Finally, by definition of j, for sufficiently large R,

?

Σ

(|∇j|2+ j2)dΣ < 2c1R2.

Therefore, there exists a constant ε0> 0 such that for all R sufficiently

large,

??

Σ

j|H|dΣ

?2

> ε0

?

Σ

(|∇j|2+ j2)dΣ.

?

Note that the estimate (5.1) is weaker than (5.1), because the integration

is j|H| rather than jH. Although we are unable with our present meth-

ods to prove Conjecture 2, the proposition supports the conjecture since it

shows that if the mean curvature has fixed sign off some compact set, then

Conjecture 2 holds. However, by our methods, we cannot prove the con-

jecture when the mean curvature H continually oscillates between positive

and negative all the way to infinity. One would need a new and different

argument to prove Conjecture 2 in that case.

Our final theorem below shows that if the surface Σ satisfies certain

isoperimetric inequalities, this is sufficient for (5.1).

Theorem 5.2. Let Σ satisfy the hypotheses (1.0.1), and assume Σ also

satisfies the following.

(1) The isoperimetric inequality holds. That is, there is a positive con-

stant δ1such that if D is a domain in Σ, we have

(length(∂D))2≥ δ1Area(D).

Page 18

18ZHIQIN LU AND JULIE ROWLETT

(2) There is another positive constant δ2such that for any compact set

K ⊂ Σ, there is a curve C ⊂ Σ \ K such that if ? γ is the normal

vector of C in Σ, then there is a vector ? η in R3such that

?? γ,? η? ≥ δ2> 0.

(3) All such curves C are tamed. That is, let σ(t,x) be the geodesic flow

of ? γ. Then there exist constants δ3,A such that the following hold.

(3.1) σ(t,x) is defined up to |t| < δ3;

(3.2) the map C × (−δ3,δ3) → Σ is diffeomorphic onto its image;

(3.3) the derivatives of σ and its inverse are bounded by the fixed

constant A.

Then, if the Gauss curvature satisfies (1.1), (5.1) is valid.

Proof. To construct the required function j satisfying (5.1), we let ρ be

a smooth non-increasing cut-off function such that

?

Let C be the curve outside a compact set K, and let D be the compact

domain of Σ such that ∂D = C. Assume that ? γ is the outward norm of C

in Σ.

Define the cut-off function ˜ ρ on Σ as follows

?

where ε is a positive number to be determined later.

Let (x,y,z) be the standard coordinates of R3. Without loss of generality,

assume the vector ? η in the hypotheses of the theorem is the z-direction in

three-dimensional Euclidean space.

Let ? n be the normal vector of Σ ⊂ R3. Let nzbe the z-component of ? n.

Define the function j by

j = ˜ ρ · nz.

Since |ρnz|+|∇(ρnz)| ≤ ||B||∞+1, if we choose ε small enough, we have

?

for ε small, where the constant C depends on A.

On the other hand, since Hnz= ∆z, we have

????

holds.

ρ(t) =

1,

0,

t ≤ 1;

t ≥ 1.

˜ ρ =

1

ρ(ε−1dist(x,C))

x ∈ D

x ?∈ D

,

Σ

(|∇j|2+ j2)dΣ ≤ Cε−2Area(D)

?

Σ

H˜ ρnzdΣ

????=

????

?

Σ

∇z∇˜ ρ

????= (?? γ,∇z?+o(1))·length(C) >1

2δ2·length(C).

Therefore using the isoperimetric inequality, the conclusion of the theorem

?

The decay of the second fundamental form at infinity is a necessary condi-

tion for the existence of bound states, because in §7 of [2], they constructed a

quantum layer whose ground state does not exist. This example is, however,

Page 19

SPECTRUM OF QUANTUM LAYERS19

not asymptotically planar, so it falls outside the class of surfaces considered

here.

We conclude by making a slightly more general conjecture. The integra-

bility condition (1.1) on the Gauss curvature implies that Σ is parabolic.

However, the reverse implication need not hold. Therefore, the following is

more general than Conjecture 1.

Conjecture 3. Let Σ be a complete surface in R3which satisfies the hy-

pothesis (1.0.1). If Σ is parabolic, then there exists an α = α(Σ) such that

for all a ∈ (0,α), the ground state of the quantum layer over Σ of width 2a

exists.

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Department of Mathematics, University of California, Irvine, Irvine, CA

92697

E-mail address, Zhiqin Lu: zlu@uci.edu

Max Planck Institut f¨ ur Mathematik, Vivatgasse 7, D-53111 Bonn

E-mail address, Julie Rowlett: rowlett@mpim-bonn.mpg.de