Page 1

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

Page 2

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.

Page 3

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.

Page 4

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.

Page 5

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.