SEMICLASSICAL BOUNDS

IN MAGNETIC BOTTLES

Diana Barseghyana,b, Pavel Exnera,c, Hynek Kovaˇr´ıkd,

Timo Weidle

a) Department of Theoretical Physics, Nuclear Physics Institute ASCR,

25068 ˇ

Reˇz near Prague, Czech Republic

b) Department of Mathematics, Faculty of Science, University

of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic

c) Doppler Institute for Mathematical Physics and Applied

Mathematics, Czech Technical University, Bˇrehov´a 7, 11519 Prague

d) Dicatam, Sezione di Matematica, Universit`a degli Studi

di Brescia, via Branze 38, 25123 Brescia, Italy

e) Fakult¨at f¨ur Mathematik und Physik Institut f¨ur Analysis,

Dynamik und Modellierung, Universit¨at Stuttgart, Pfaﬀenwaldring 57,

70569 Stuttgart, Germany

dianabar@ujf.cas.cz, diana.barseghyan@osu.cz, exner@ujf.cas.cz,

hynek.kovarik@unibs.it, weidl@mathematik.uni-stuttgart.de

Abstract. The aim of the paper is to derive spectral estimates into several

classes of magnetic systems. They include three-dimensional regions with

Dirichlet boundary as well as a particle in R3conﬁned by a local change of

the magnetic ﬁeld. We establish two-dimensional Berezin-Li-Yau and Lieb-

Thirring-type bounds in the presence of magnetic ﬁelds and, using them, get

three-dimensional estimates for the eigenvalue moments of the corresponding

magnetic Laplacians.

1 Introduction

Let −∆Ωbe the Dirichlet Laplacian corresponding to an open bounded do-

main Ω ⊂Rd, deﬁned in the quadratic form sense on H1

0(Ω). The operator

is obviously non-negative and since the embedding H1

0→L2(Ω) is compact,

its spectrum is purely discrete accumulating at inﬁnity only. It is well known

1

that for d= 3, up to a choice of the scale, the eigenvalues describe energies

of a spinless quantum particle conﬁned to such a hard-wall ‘bottle’.

Motivated by this physical problem, we consider in the present work a

magnetic version of the mentioned Dirichlet Laplacian, that is, the operator

HΩ(A)=(i∇+A(x))2associated with the closed quadratic form

k(i∇+A)uk2

L2(Ω) , u ∈ H1

0(Ω) ,

where the real-valued and suﬃciently smooth function Ais a vector potential.

The magnetic Sobolev norm on the bounded domain Ω is equivalent to the

non-magnetic one and the operator HΩ(A) has a purely discrete spectrum as

well. We shall denote the eigenvalues by λk=λk(Ω, A), assuming that they

repeat according to their multiplicities.

One of the objects of our interest in this paper will be bounds of the

eigenvalue moments of such operators. For starters, recall that for non-

magnetic Dirichlet Laplacians the following bound was proved in the work

of Berezin, Li and Yau [Be72a, Be72b, LY83],

X

k

(Λ −λk(Ω,0))σ

+≤Lcl

σ,d |Ω|Λσ+d

2for any σ≥1 and Λ >0,(1.1)

where |Ω|is the volume of Ω and the constant on the right-hand side,

Lcl

σ,d =Γ(σ+ 1)

(4π)d

2Γ(σ+1+d/2) ,

is optimal. Furthermore, the bound (1.1) holds true for 0 ≤σ < 1 as well,

but with another, probably non-sharp constant on the right-hand side,

X

k

(Λ −λk(Ω,0))σ

+≤2σ

σ+ 1σ

Lcl

σ,d |Ω|Λσ+d

2,0≤σ < 1.(1.2)

see [La97]. In the particular case σ= 1 the inequality (1.1) is equivalent, via

Legendre transformation, to the lower bound

N

X

j=1

λj(Ω,0) ≥Cd|Ω|−2

dN1+ 2

d, Cd=4πd

d+ 2Γ(d/2 + 1) 2

d.(1.3)

Turning next to the magnetic case, we note ﬁrst that the pointwise dia-

magnetic inequality [LL01], namely

|∇|u(x)|| ≤ |(i∇+A)u(x)|for a.a. x∈Ω,

2

implies λ1(Ω, A)≥λ1(Ω,0), however, the estimate λj(Ω, A)≥λj(Ω,0) fails

in general if j≥2. Nevertheless, momentum estimates are still valid for

some values of the parameters. In particular, it was shown [LW00] that the

sharp bound (1.1) holds true for arbitrary magnetic ﬁelds provided σ≥3

2,

and the same sharp bound holds true for constant magnetic ﬁelds if σ≥1,

see [ELV00]. Furthermore, in the dimension d= 2 the bound (1.2) holds true

for constant magnetic ﬁelds if 0 ≤σ < 1 and the constant on its right-hand

side cannot be improved [FLW09].

Our main aim in the present work is to derive suﬃciently precise two-

dimensional Berezin-type estimates for quantum systems exposed to a mag-

netic ﬁeld and to apply them to the three-dimensional case. We are going

to address two questions, one concerning eigenvalue moments estimates for

magnetic Laplacians on three dimensional domains having a bounded cross

section in a ﬁxed direction, and the other about similar estimates for mag-

netic Laplacians deﬁned on whole R3.

Let us review the paper content in more details. In Sec. 2 we will describe

the dimensional-reduction technique [LW00] which allows us to derive the

sought spectral estimates for three-dimensional magnetic ‘bottles’ using two-

dimensional ones. Our next aim is to derive a two-dimensional version of

the Li-Yau inequality (1.3) in presence of a constant magnetic ﬁeld giving

rise to an extra term on the right-hand side. The result will be stated and

proved in ﬁrst part of Sec. 3. This in turn will imply, by means of Legendre

transformation, a magnetic version of the Berezin inequality which we are

going to present in second part of Sec. 3. It has to be added that the question

of semiclassical spectral bounds for such systems has been addressed before,

in particular, another version of the magnetic Berezin inequality was derived

by two of us [KW13]. In ﬁnal part of Sec. 3 we are going to compare the

two results and show that the one derived here becomes substantially better

when the magnetic ﬁeld is strong.

In some cases the eigenvalues of the magnetic Dirichlet Laplacian with a

constant magnetic ﬁeld can be computed exactly in terms of suitable special

functions. In the ﬁrst part of Sec. 4 we are present such an example con-

sidering the magnetic Dirichlet Laplacian on a two-dimensional disc with a

constant magnetic ﬁeld. Its eigenvalues will be expressed in terms of Kummer

function zeros. Next, in the second part, we are going to consider again the

magnetic Dirichlet Laplacian on a two-dimensional disc, now in a more gen-

eral situation when the magnetic ﬁeld is no longer homogeneous but retains

the radial symmetry; we will derive the Berezin inequality for the eigenvalue

3

moments. In Sec. 5 we shall return to our original motivation and use the

mentioned reduction technique to derive Berezin-type spectral estimates for

a class of three-dimensional magnetic ‘bottles’ characterized by a bounded

cross section in the x3direction.

Turning to the second one of the indicated questions, from Sec. 6 on, we

shall be concerned with magnetic Laplacians in L2(R3) associated with the

magnetic ﬁeld B:R3→R3which is as a local perturbation of a constant

magnetic ﬁeld of intensity B0>0. Again, as before, we ﬁrst derive suitable

two-dimensional estimates; this will be done in Sec. 6. In the last two sections

we apply this result to the three-dimensional case. In Sec. 7 we show that the

essential spectrum of the magnetic Laplacian with corresponding perturbed

magnetic ﬁeld coincides with [B0,∞). The Sec. 7.1 we then prove Lieb-

Thirring-type inequalities for the moments of eigenvalues below the threshold

of the essential spectrum for several types of magnetic ‘holes’.

2 Dimensional reduction

As indicated our question concerns estimating eigenvalues due to conﬁne-

ment in a three-dimensional ‘bottle’ by using two-dimensional Berezin type

estimates. In such situation one can use the dimension-reduction tech-

nique [LW00]. In particular, let −∆Ωbe the Dirichlet Laplacian on an open

domain Ω ⊆R3, then for any σ≥3

2the inequality

tr (Λ −(−∆Ω))σ

+≤Lcl

1,σ ZR

tr Λ−(−∆ω(x3))σ+1

2

+dx3(2.1)

is valid, where −∆ω(x3)is the Dirichlet Laplacian on the section

ω(x3) = x0= (x1, x2)∈R2|x= (x0, x3) = (x1, x2, x3)∈Ω,

see [LW00], and also [ELM04, Wei08]. The integral at the right-hand side of

(2.1), in fact restricted to those x3for which inf spec(−∆ω(x3))<Λ, yields the

classical phase space volume. Note that in this way one can obtain estimates

also in some unbounded domains [GW11] as well as remainder terms [Wei08].

A similar technique can be used also in the magnetic case. To describe

it, consider a suﬃciently smooth magnetic vector potential A(·) : Ω →R3

generating the magnetic ﬁeld

B(x) = (B1(x), B2(x), B3(x)) = rot A(x).

4

For the sake of deﬁniteness, the shall use the gauge with A3(x) = 0. Fur-

thermore, we consider the magnetic Dirichlet Laplacians

HΩ(A) = (i∇x−A(x))2on L2(Ω)

and e

Hω(x3)(e

A)=(i∇x0−e

A(x))2on L2(ω(x3)) ,

where e

A(x) := (A1(x), A2(x)). Note that for the ﬁxed x3the two-dimensional

vector potential e

A(x0, x3) corresponds to the magnetic ﬁeld

˜

B(x0, x3) = B3(x) = ∂A2

∂x1−∂A1

∂x2

.

Referring to [LW00, Sec. 3.2] one can then claim that for a σ≥3

2we have

tr(Λ − HΩ(A))σ

+≤Lcl

1,σ ZR

tr(Λ −e

Hω(x3)(e

A))σ+1/2

+dx3.(2.2)

3 Berezin-Li-Yau inequality with a constant

magnetic ﬁeld

Suppose that the motion is conﬁned to a planar domain ωbeing exposed to

inﬂuence of a constant magnetic ﬁeld of intensity B0perpendicular to the

plane, and let A:R2→R2be a vector potential generating this ﬁeld. We

denote by Hω(A) the corresponding magnetic Dirichlet Laplacian on ωand

µj(A) will be its eigenvalues arranged in the ascending with repetition ac-

cording to their multiplicity. Our ﬁrst aim is to extend the Li-Yau inequality

(1.3) to this situation with an additional term on the right-hand side depend-

ing on B0only. This will be then used to derive the Berezin-type inequality.

Conventionally we denote by Nthe set of natural numbers, while the set of

integers will be denoted by Z.

The following result is not new. Indeed, it can be recovered from [ELV00,

Sec. 2], however, for the sake of completeness we include a proof.

3.1 Li-Yau estimate

Theorem 3.1. Assume that ω⊂R2is open and ﬁnite. Then the inequality

X

j≤N

µj(A)≥2πN 2

|ω|+B2

0

2π|ω|m(1 −m) (3.1)

5

holds, where m:= n2πN

B0|ω|ois the fractional part of 2πN

B0|ω|.

Proof. Without loss of generality we may assume that B0>0. Let Pkbe the

orthogonal projection onto the k-th Landau level, B0(2k−1), of the Landau

Hamiltonian (i∇+A(x))2in L2(R2) which is an integral operator with the

kernel Pk(x, y) – see [KW13]. Note that we have

Pk(x, x) = 1

2πB0,(3.2)

ZR2Zω|Pk(y, x)|2dxdy=ZωZR2

Pk(y, x)Pk(x, y) dydx

=Zω

Pk(x, x) dx=B0

2π|ω|.(3.3)

Let φjbe a normalized eigenfunction corresponding to the eigenvalue µj(A).

We put fk,j(y) := RωPk(y, x)φj(x) dx, where y∈R2, and furthermore

FN(k) := X

j≤Nkfk,j k2

L2(R2).

We have the following identity,

X

j≤N

µj(A) = X

j≤Nk(i∇ − A)φjk2

L2(ω)

=X

j≤NX

k∈Nk(i∇ − A)fk,j k2

L2(R2)

=X

k∈N

B0(2k−1) X

j≤Nkfk,j k2

L2(R2)

=X

k∈N

B0(2k−1)FN(k) =: J[FN].

Moreover, the normalization of the functions φjimplies

X

k∈N

FN(k) = X

j≤NX

k∈Nkfk,j k2

L2(R2)=X

j≤Nkφjk2

L2(ω)=N . (3.4)

6

Finally, in view of Bessel’s inequality the following estimate holds true,

FN(k) = X

j≤Nkfk,j k2

L2(R2)=ZR2X

j≤NZω

Pk(y, x)φj(x) dx

2

dy

≤ZR2Zω|Pk(y, x)|2dxdy=B0

2π|ω|.(3.5)

Let us now minimize the functional J[FN] under the constraints (3.4) and

(3.5). To this aim, recall ﬁrst the bathtub principle [LL01]:

Given a σ-ﬁnite measure space (Ω,Σ, µ), let fbe a real-valued measurable

function on Ω such that µ{x:f(x)< t}is ﬁnite for all t∈R. Fix further a

number G > 0 and deﬁne a class of measurable functions on Ω by

C=g: 0 ≤g(x)≤1 for all xand ZΩ

g(x)µ(dx) = G.

Then the minimization problem of the functional

I= inf

g∈C ZΩ

f(x)g(x)µ(dx)

is solved by

g(x) = χ{f<s}(x) + cχ{f=s}(x),(3.6)

giving rise to the minimum value

I=Z{f<s}

f(x)µ(dx) + csµ{x:f(x) = s},

where

s= sup{t:µ{x:f(x)< t} ≤ G}

and

cµ{x:f(x) = s}=G−µ{x:f(x)< s}.

Moreover, the minimizer given by (3.6) is unique if G=µ{x:f(x)< s}or

if G=µ{x:f(x)≤s}.

Applying this result to the functional J[FN] with the constraints (3.4) and

(3.5) we ﬁnd that the corresponding minimizers are

FN(k) = B0

2π|ω|, k = 1,2, . . . , M ,

7

FN(M+ 1) = B0

2π|ω|m ,

FN(k) = 0, k > M + 1,

where M=h2πN

B0|ω|iis the entire part and m=n2πN

B0|ω|o, so that M+m=2πN

B0|ω|.

Consequently, we have the lower bound

J[FN]≥B0

2π|ω|

M

X

k=1

(2k−1)B0+B0

2π|ω|m(2M+ 1)B0

=B0

2π|ω|(M2+ 2Mm +m)

=B2

0

2π|ω|(M+m)2+B2

0

2π|ω|(m−m2)

which implies X

j≤N

µj(A)≥2πN 2

|ω|+B2

0

2π|ω|m(1 −m).

This is the claim we have set out to prove.

Since 0 ≤m < 1 by deﬁnition the last term can regarded as a non-

negative remainder term, which is periodic with respect to N

Φ, where Φ = B0|ω|

2π

is the magnetic ﬂux, i.e. the number of ﬂux quanta through ω. Note that for

N < Φ the right-hand side equals NB and for large enough B0this estimate

is better than the lower bound in terms of the phase-space volume.

3.2 A magnetic Berezin-type inequality

The result obtained in the previous subsection allows us to derive an exten-

sion of the Berezin inequality to the magnetic case. We keep the notation

introduced above, in particular, Hω(A) is the magnetic Dirichlet Laplacian

on ωcorresponding to a constant magnetic ﬁeld B0and µj(A) are the respec-

tive eigenvalues. Without loss of generality we assume again that B0>0.

Then we can make the following claim.

Theorem 3.2. Let ω⊂R2be open and ﬁnite, then for any Λ> B0we have

N

X

j=1

(Λ −µj(A)) ≤(Λ2−B2

0)|ω|

8π+(Λ −B0)B0|ω|

4πΛ + B0

2B0.(3.1)

8

Proof. Subtracting NΛ from both sides of inequality (3.1), we get

N

X

j=1

(Λ −µj(A)) ≤NΛ−2πN 2

|ω|−B2

0

2π|ω|m(1 −m),(3.2)

and consequently

N

X

j=1

(Λ −µj(A))+≤NΛ−2πN 2

|ω|−B2

0

2π|ω|m(1 −m)+

.

We are going to investigate the function f:R+→R,

f(z) := zΛ−2πz2

|ω|−B2

0|ω|

2π2πz

B0|ω|1−2πz

B0|ω|,

on the intervals

B0|ω|k

2π≤z < B0|ω|(k+ 1)

2π, k = 0,1,2,...,

looking for an upper bound. It is easy to check that

f0(z) = Λ −4π

|ω|z−B2

0|ω|

2π

2π

B0|ω|+2B2

0|ω|

2π2πz

B0|ω|2π

B0|ω|

= Λ −4π

|ω|z−B0+ 2B02πz

B0|ω|,

thus the extremum of fis achieved at the point z0such that

Λ−B0−4π

|ω|z0+ 2B02πz0

B0|ω|= 0 .(3.3)

Denoting x0:= 2πz0

B0|ω|, the condition reads Λ −2B0x0−B0+ 2B0{x0}= 0

giving

x0=Λ−B0+ 2B0{x0}

2B0

.

9

It yields the value of function fat z0, namely

f(z0) = ΛB0|ω|

2π

(Λ −B0+ 2B0{x0})

2B0−B2

0|ω|

2πΛ−B0+ 2B0{x0}

2B02

−B2

0|ω|

2π{x0}(1 − {x0})

=Λ|ω|

4π(Λ −B0+ 2B0{x0})−|ω|

8π(Λ −B0+ 2B0{x0})2

−B2

0|ω|

2π{x0}(1 − {x0})

=|ω|

4πΛ(Λ −B0+ 2B0{x0})−(Λ −B0+ 2B0{x0})2

2

−2B2

0{x0}(1 − {x0})

=|ω|

4πΛ2−ΛB0+ 2ΛB0{x0} − Λ2

2+ ΛB0−B2

0

2−2ΛB0{x0}

+2B2

0{x0} − 2B2

0{x0}2−2B2

0{x0}+ 2B2

0{x0}2

=|ω|(Λ2−B2

0)

8π.(3.4)

Furthermore, the values of fat the endpoints B0k|ω|

2π, k = 0,1,2, . . . , equal

fB0k|ω|

2π=B0Λk|ω|

2π−2π

|ω|

B2

0k2|ω|2

4π2=B0k|ω|

2π(Λ −kB0).

10

Consider now an integer msatisfying 1 ≤m≤hΛ+B0

2B0i, then

fB0|ω|

2πΛ + B0

2B0−m

=B0|ω|

2πΛ + B0

2B0−mΛ−Λ + B0

2B0−mB0

≤B0|ω|

2πΛ + B0

2B0−mΛ−Λ + B0

2B0−mB0+Λ + B0

2B0B0

=(Λ −(2m−1)B0)|ω|

4πΛ + (2m−1)B0

2+Λ + B0

2B0B0

=(Λ2−(2m−1)2B2

0)|ω|

8π+(Λ −(2m−1)B0)B0|ω|

4πΛ + B0

2B0

≤(Λ2−B2

0)|ω|

8π+(Λ −B0)B0|ω|

4πΛ + B0

2B0.(3.5)

On the other hand, for integers satisfying k≥hΛ+B0

2B0ione can check easily

that

4B2

0k2−4B0Λk+ Λ2−B2

0≥0,

which means B0k|ω|

2π(Λ −B0k)≤(Λ2−B2

0)|ω|

8π.(3.6)

Combining inequalities (3.5) and (3.6) we conclude that at the interval end-

points, z=B0k|ω|

2π, k = 0,1,2, . . . , the value of function fdoes not exceed

(Λ2−B2

0)|ω|

8π+(Λ−B0)B0|ω|

4πnΛ+B0

2B0o. Hence in view of (3.4) we have

f(z)≤(Λ2−B2

0)|ω|

8π+(Λ −B0)B0|ω|

4πΛ + B0

2B0

for any z≥0. Combining this inequality above with the bound (3.2), we

arrive at the desired conclusion.

Remark 3.3. Using the Aizenman-Lieb procedure [AL78] and the fact that

inf σ(Hω(A)) ≥B0we can get also bound for other eigenvalue moments.

11

Speciﬁcally, for any σ≥3/2 Theorem 3.2 implies

N

X

j=1

(Λ −µj(A))σ+1/2

+=Γ(σ+ 3/2)

Γ(σ−1/2)Γ(2) Z∞

0

(Λ −t)σ−3/2

+

N

X

j=1

(t−µj(A))+dt

≤Γ(σ+ 3/2)

Γ(σ−1/2) Z∞

0

(Λ −t)σ−3/2

+(t2−B2

0)+|ω|

8π

+(t−B0)+B0|ω|

4πΛ + B0

2B0dt

≤Γ(σ+ 3/2)|ω|

Γ(σ−1/2) (Λ2−B2

0)+

8π

+(Λ −B0)+B0

4πΛ + B0

2B0Z∞

0

(Λ −t)σ−3/2

+dt

=Γ(σ+ 3/2)Λσ−1/2|ω|

Γ(σ−1/2)(2σ−1) (Λ2−B2

0)+

4π+(Λ −B0)+B0

2πΛ + B0

2B0.

3.3 Comparison to earlier results

Given a set ω⊂R2and a point x∈ω, we denote by

δ(x) = dist(x, ∂ω) = min

y∈∂ω |x−y|

the distance of xto the boundary, then

R(ω) = sup

x∈ω

δ(x)

is the in-radius of ω. Furthermore, given a β > 0 we introduce

ωβ={x∈ω:δ(x)< β}, β > 0,

and deﬁne the quantity

σ(ω) := inf

0<β<R(ω)|ωβ|

β.(3.7)

Using these notions and the symbols introduced above we can state the fol-

lowing result obtained in the work of two of us [KW13]:

12

Theorem 3.4. Let ω⊂R2be an open convex domain, then for any Λ> B0

we have

N

X

j=1

(Λ −µj(A)) ≤Λ2|ω|

8π−1

512π

σ2(ω)

|ω|Λ (3.8)

−B2

01

2−Λ + B0

2B02|ω|

2π−1

128π

σ2(ω)

|ω|Λ.

To make a comparison to the conclusions of the previous section, let us make

both B0and Λ large keeping their ratio ﬁxed. Speciﬁcally, we choose a Λ

from the interval (B0,2B0) writing it as Λ = B0(1 + α) with an α∈(0,1).

The second term on the right-hand side of (3.1) is then α2B2

0|ω|

8π, and we want

to show that the diﬀerence between the bounds (3.8) and (3.1) tends to plus

inﬁnity as B0→ ∞. To this aim, we write Λ = B0(1 + α) with an α∈(0,1),

then

(Λ2−B2

0)|ω|

8π+(Λ −B0)B0|ω|

4πΛ + B0

2B0=B2

0|ω|

4πα(1 + α).(3.9)

On the other hand, a short calculation shows that for our choice of B0and

Λ the right-hand side of the bound (3.8) becomes

=Λ2|ω|

8π−B2

0|ω|

2π1

2−α

22

+Λ

512π

σ2(ω)

|ω|−1 + (1 −α)2

(1 + α)2,

in particular, after another easy manipulation we ﬁnd that for large B0this

expression behaves as B2

0|ω|

2πα+O(B0). Comparing the two bounds we see

that

rhs of (3.8) −rhs of (3.1) = B2

0|ω|

4πα(1 −α) + O(B0) (3.10)

tending to plus inﬁnity as B0→ ∞. At the same time,

rhs of (3.8)

rhs of (3.1) =2

1 + α+O(B−1

0) (3.11)

illustrating that the improvement represented by Theorem 3.2 is most pro-

nounced for eigenvalues near the spectral threshold.

13

4 Examples: a two-dimensional disc

Spectral analysis simpliﬁes if the domain ωallows for a separation of vari-

ables. In this section we will discuss two such situations.

4.1 Constant magnetic ﬁeld

We suppose that ωis a disc and the applied magnetic ﬁeld is homogeneous.

As usual in cases of a radial symmetry, the problem can be reduced to de-

generate hypergeometric functions. Speciﬁcally, we will employ the Kummer

equation

rd2ω

dr2+ (b−r)dω

dr−aω = 0 (4.1)

with real valued parameters aand bwhich has two linearly independent

solutions M(a, b, r) and U(a, b, r), the second one of which has a singularity

at zero [AS64].

Given an α > 0, we denote by ak

|m|,αk∈Nthe set of the ﬁrst parameter

values such that M(ak

|m|,α,|m|+ 1, α) = 0. Since for any a, b ≥0 the function

M(a, b, r) has no positive zeros [AS64], all the ak

|m|,α are negative. Then the

following claim is valid.

Theorem 4.1. Let Hω(A)be the magnetic Dirichlet Laplacian corresponding

to a constant magnetic ﬁeld B0and ωbeing the two dimensional disc with

center at the origin and radius r0>0. The eigenvalues of Hω(A)coincides

with nB0+B0|m| − m−2ak

|m|,√B0r0/√2om∈Z, k∈N.

Proof. We employ the standard partial wave decomposition – see, e.g., [Er96]

L2(ω) = ∞

M

m=−∞

L2((0, r0),2πr dr),(4.2)

and Hω(A) = L∞

m=−∞ hm, where

hm:= −d2

dr2−1

r

d

dr+m

r−B0r

22

.(4.3)

The last named operator diﬀers by mB0from the operator

˜

hm=−d2

dr2−1

r

d

dr+m2

r2+B2

0r2

4(4.4)

14

on the interval (0, r0) with Dirichlet boundary condition at the endpoint r0.

Looking for solutions to the eigenvalue equation

˜

hmu=λu (4.5)

we employ the Ansatz

u(r) = r|m|e−B0r2/4v(r),

where v∈L2((0, r0), rdr). Computing the ﬁrst two derivatives we get

˜

hmu=−v00 −2|m|+ 1

rv0+B0(|m|+ 1)v(r) + B0rv0r|m|e−B0r2/4,

hence the equation (4.5) can rewritten as

v00 +2|m|+ 1

r−B0rv0−(B0(|m|+ 1) −λ)v= 0 .(4.6)

Using the standard substitution we pass to the function g(r) = v√2r

√B0

belonging to L2(0, B0r2

0/2). Expressing the derivatives of vin terms of those

of g, one can rewrite equation (4.6) as

rg00(r)+(|m|+ 1 −r)g0−((|m|+ 1)B0−λ)

2B0

g(r) = 0 ,

which is the Kummer equation with b=|m|+ 1 and a=(|m|+1)B0−λ

2B0. The

mentioned singularity of its solution U(a, b, r) for small r, namely [AS64]

U(a, b, r) = Γ(b−1)

Γ(a)r1−b+O(rb−2) for b > 2

and

U(a, 2, z) = 1

Γ(a)

1

r+O(ln r), U(a, 1, r) = −1

Γ(a)ln r+O(1)

means that u(r) = r|m|e−B0r2/4U(|m|+1)B0−λ

2B0,|m|+ 1,B0r2

2does not belong

to H1

0((0, r0), rdr). Consequently, the sought solution of (4.5) has the form

r|m|e−B0r2/4M(|m|+ 1)B0−λ

2B0

,|m|+ 1,B0r2

2,

15

and in view of the Dirichlet boundary conditions at r0we arrive at the spec-

tral condition

M(|m|+ 1)B0−λ

2B0

,|m|+ 1,B0r2

0

2= 0 .

which gives n(|m|+ 1)B0−2B0ak

|m|,√B0r0/√2om∈Z, k∈Nas the eigenvalue set;

returning to the original operator hmwe get the claim of the theorem.

4.2 Radial magnetic ﬁeld

If the magnetic ﬁeld is non-constant but still radially symmetric, in general

one cannot ﬁnd the eigenvalues explicitly but it possible to ﬁnd a bound to

the eigenvalue moments in terms of an appropriate radial two-dimensional

Schr¨odinger operator.

Theorem 4.2. Let Hω(A)be the magnetic Dirichlet Laplacian Hω(A)on a

disc ωof radius r0>0centered at the origin with a radial magnetic ﬁeld

B(x) = B(|x|). Assume that

α:= Zr0

0

sB(s) ds < 1

2.(4.7)

Then for any Λ, σ ≥0, the following inequality holds true

tr(Λ −Hω(A))σ

+≤1

√1−2α+ sup

n∈Nn

√1−2α (4.8)

×tr

Λ−

−∆ω

D+1

x2+y2 Z√x2+y2

0

sB(s) ds!2

σ

+

.

In particular, the estimate (4.8) implies

inf σ(Hω(A)) ≥inf σ

−∆ω

D+1

x2+y2 Z√x2+y2

0

sB(s) ds!2

.

Proof. Let us again employ the partial-wave decomposition (4.2), with the

angular component (4.3) replaced by

hm:= −d2

dr2−1

r

d

dr+m

r−1

rZr

0

sB(s) ds2

,(4.9)

16

and inspect the eigenvalues of this operator. Obviously, for m≤0 we have

hm≥ − d2

dr2−1

r

d

dr+m2

r2+1

r2Zr

0

sB(s) ds2

,(4.10)

while for any m > 0 we can use the inequality

2|m|

r2Zr

0

sB(s) ds≤2m2

r2Zr

0

sB(s) ds

which in view of the assumption (4.7) yields

hm≥ − d2

dr2−1

r

d

dr+ (1 −2α)m2

r2+1

r2Zr

0

sB(s) ds2

.

Next we divide the set of natural numbers into groups such that for all the

elements of any ﬁxed group the entire part √1−2α mis the same, and we

estimate the operator hmfrom below by

hm≥ − d2

dr2−1

r

d

dr+√1−2α m2

r2+1

r2Zr

0

sB(s) ds2

.(4.11)

Since the number of elements in each group is bounded from above by the

sum 1

√1−2α+ supn∈Nnn

√1−2αo, using (4.10) and (4.11) one infers that

tr(Λ −Hω(A))σ

+≤1

√1−2α+ sup

n∈Nn

√1−2α

×∞

X

m=−∞

tr Λ− −d2

dr2−1

r

d

dr+m2

r2+1

r2Zr

0

sB(s) ds2!!σ

+

=1

√1−2α+ sup

n∈Nn

√1−2α

×tr Λ−∞

M

m=−∞ −d2

dr2−1

r

d

dr+m2

r2+1

r2Zr

0

sB(s) ds2!!σ

+

with any σ, Λ≥0. However, the direct sum in the last expression is nothing

else than a partial-wave decomposition of the two-dimensional Schr¨odinger

operator with the radial potential V(r) = 1

r2Rr

0sB(s) ds2and the Dirichlet

condition at the boundary of the disc; this yields the desired claim.

17

5 Application to the three-dimensional case

Let us return now to our original motivation of estimating eigenvalues due to

conﬁnement in a three-dimensional ‘bottle’. One can employ inequality (2.2)

in combination with the results of the previous sections to improve in some

cases the spectral bound by taking the magnetic ﬁeld into account instead

of just dropping it.

Let Ω ⊂R3with the bounded x3cross sections. The class of ﬁelds to

consider are those of the form B(x) = (B1(x), B2(x), B3(x3)), that is, those

for which the component B3perpendicular to the cross section depends on

the variable x3only. Such ﬁelds certainly exist, for instance, one can think of

the situation when the ‘bottle’ is placed into a homogeneous magnetic ﬁeld.

The ﬁeld is induced by an appropriate vector potential A(·):Ω→R3,

B(x) = (B1(x), B2(x), B3(x3)) = rot A(x),

and we consider the magnetic Dirichlet Laplacians

HΩ(A) = (i∇x−A(x))2on L2(Ω).

We use the notion introduced in Sec. 2. In view of the variational principle

we know that the ground-state eigenvalue of e

Hω(x3)(e

A) cannot fall below the

ﬁrst Landau level B3(x3). Consequently, integrating with respect to x3in the

formula (2.2) one can drop for all the x3for which B3(x3)≥Λ. Combining

this observation with Remark 3.3 we get

tr(Λ − HΩ(A))σ

+≤Γ(σ+ 3/2)Λσ−1/2

4π(2σ−1)Γ(σ−1/2) Lcl

1,σ Z{x3:B3(x3)<Λ}|ω(x3)|

×Λ2−B3(x3)2+ 2B3Λ−B3(x3)Λ + B3

2B3dx3

for any σ≥3/2.

Example 5.1. (circular cross section) Let Ω be a three-dimensional cusp

with a circular cross section ω(x3) of radius r(x3) such that r(x3)→0 as

x3→ ∞. Then the above formula in combination with Theorem 4.1 yields

tr(Λ − HΩ(A))σ

+≤Lcl

1,σ X

m∈Z, k∈NZRΛ−B3(x3)

−B3(x3)|m| − m−2ak

|m|,√B3(x3)r0(x3)/√2σ+1/2

+

dx3

18

for any σ≥3/2. The particular case B(x) = {0,0, B}applies to a cusp-

shaped region placed to a homogeneous ﬁeld parallel to the cusp axis.

Example 5.2. (radial magnetic ﬁeld) Consider the same cusp-shaped region

Ω in the more general situation when the third ﬁeld component can depend

on the radial variable, B(x)=(B1(x), B2(x), B3(x2

1+x2

2, x3)), assuming that

sup

x3∈R

α(x3) = sup

x3∈RZr0(x3)

0

sB3(s, x3) ds < 1

2.

Then the dimensional reduction in view of Theorem 4.2 gives

tr(Λ − HΩ(A))σ

+≤Lcl

1,σ ZR 1

p1−2α(x3)+ sup

n∈N(n

p1−2α(x3))!

×tr

Λ−

−∆ω(x3)

D+1

x2

1+x2

2 Z√x2

1+x2

2

0

sB3(s, x3) ds!2

σ+1/2

+

for any σ≥3/2.

6 Spectral estimates for eigenvalues from

perturbed magnetic ﬁeld

Now we change the topic and consider situations when the discrete spectrum

comes from the magnetic ﬁeld alone. We are going to demonstrate a Berezin-

type estimate for the magnetic Laplacian on R2with the ﬁeld which is a radial

and local perturbation of a homogeneous one. We consider the operator

H(B) in L2(R2) deﬁned as follows,

H(B) = −∂2

x+ (i∂y+A2)2, A =0, B0x−f(x, y),(6.1)

with fgiven by

f(x, y) = −Z∞

x

g(pt2+y2) dt .

with g:R+→R+; the operator H(B) is then associated with the magnetic

ﬁeld

B=B(x, y) = B0−g(px2+y2).

19

Since have chosen the vector potential in such a way that the unperturbed

part corresponds to the Landau gauge, we have

H(B0) = −∂2

x+ (i∂y+B0x)2.

Using a partial Fourier transformation, it is easy to conclude from here that

the corresponding spectrum consists of identically spaced eigenvalues of inﬁ-

nite multiplicity, the Landau levels,

σ(H(B0)) = {(2n−1)B0, n ∈N}.(6.2)

It is well known that inf σess(H(B)−B) = 0, hence the relative compactness

of B0−Bwith respect to H(B)−B0in L2(R2) implies

inf σess(H(B)) = B0.

We have to specify the sense in which the magnetic perturbation is local. In

the following we will suppose that

(i) the function g∈L∞(R+) is non-negative and such that both fand

∂x2fbelong to L∞(R2), and

lim

x2

1+x2

2→∞ |∂x2f(x1, x2)|+|f(x1, x2)|= 0 .

(ii) kgk∞≤B0.

Let us next rewrite the vector potentials A0and Aassociated to B0and B

in the polar coordinates. Passing to the circular gauge we obtain

A0= (0, a0(r)) , A = (0, a(r)) ,(6.3)

with

a0(r) = B0r

2, a(r) = B0r

2−1

rZr

0

g(s)sds . (6.4)

Hence the operators H(B0) and H(B) are associated with the closures of the

quadratic forms in L2(R+, rdr) with the values

Q(B0)[u] = Z2π

0Z∞

0|∂ru|2+|ir−1∂θu+a0(r)u|2rdrdθ(6.5)

20

and

Q(B)[u] = Z2π

0Z∞

0|∂ru|2+|ir−1∂θu+a(r)u|2rdrdθ , (6.6)

respectively, both deﬁned on C∞

0(R+). Furthermore, for every k∈N0we

introduce the following auxiliary potential,

Vk(r) := 2k

r(a0(r)−a(r)) + a2(r)−a2

0(r),(6.7)

and the functions

ψk(r) = sB0

Γ(k+ 1) B0

2k/2

rkexp −B0r2

4.(6.8)

Finally let us denote by

α=Z∞

0

g(r)rdr(6.9)

the ﬂux associated with the perturbation; recall that in the rational units we

employ the ﬂux quantum value is 2π. Now we are ready to state the result.

Theorem 6.1. Let the assumptions (i) and (ii) be satisﬁed, and suppose

moreover that α≤1. Put

Λk=ψk,Vk(·)−ψkL2(R+,rdr).(6.10)

Then the inequality

tr(H(B)−B0)γ

−≤2γ∞

X

k=0

Λγ

k, γ ≥0,(6.11)

holds true whenever the right-hand side is ﬁnite.

Remark 6.2. For a detailed discussion of the asymptotic distribution of

eigenvalue of the operator H(B) we refer to [RT08].

Proof. We are going to employ the fact that both A0and Aare radial func-

tions, see (6.3), and note that by the partial-wave decomposition

tr (H(B)−B0)γ

−=X

k∈Z

tr (hk(B)−B0)γ

−,(6.12)

21

where the operators hk(B) in L2(R+, rdr) are associated with the closures of

the quadratic forms

Qk[u] = Z∞

0 |∂ru|2+

k

ru−a(r)u

2!rdr ,

deﬁned originally on C∞

0(R+), and acting on their domain as

hk(B) = −∂2

r−1

r∂r+k

r−a(r)2

.

In view of (6.7) it follows that

hk(B) = hk(B0) + Vk(r),

where

hk(B0) = −∂2

r−1

r∂r+k

r−a0(r)2

.

To proceed we need to recall some spectral properties of the two-dimensional

harmonic oscillator,

Hosc =−∆ + B2

0

4(x2+y2) in L2(R2).

It is well known that the spectrum of Hosc consists of identically spaced

eigenvalues of a ﬁnite multiplicity,

σHosc={nB0, n ∈N},(6.13)

where the ﬁrst eigenvalue B0is simple and has a radially symmetric eigen-

function. The latter corresponds to the term with k= 0 in the partial-wave

decomposition of Hosc, which implies

σHosc=[

k∈Z

σ−∂2

r−1

r∂r+k2

r2+B2

0r2

4,

where the operators in the brackets at the right-hand side act in L2(R+, rdr).

Hence in view of (6.13) we have

inf

k6=0 σ−∂2

r−1

r∂r+k2

r2+B2

0r2

4≥2B0.(6.14)

22

On the other hand, for k < 0 it follows from (ii), (6.7) and (6.9) that

Vk(r) = 2k

rZr

0

g(s)sds−B0Zr

0

g(s)sds+1

r2Zr

0

g(s)sds2

≥kB0−B0.

By (6.14) we thus obtain the following inequality which holds in the sense of

quadratic forms on C∞

0(R+) for any k < 0,

hk(B) = hk(B0) + Vk(r) = −∂2

r−1

r∂r+k2

r2+B2

0r2

4−kB0+Vk(r)

≥ −∂2

r−1

r∂r+k2

r2+B2

0r2

4−α B0

≥(2 −α)B0.

Since α≤1 holds by hypothesis, this implies that

tr (H(B)−B0)γ

−=X

k∈Z

tr (hk(B)−B0)γ

−=X

k≥0

tr (hk(B)−B0)γ

−,(6.15)

see (6.12). In order to estimate tr (hk(B)−B0)γ

−for k≥0 we employ

Πk= (·, ψk)L2(R+,rdr)ψk,

the projection onto the subspace spanned by ψk, and note that

ψk∈ker(hk(B0)−B0),kψkkL2(R+,rdr)= 1 ∀k∈N∪ {0}.(6.16)

Let Qk= 1 −Πk. From the positivity of Vk(·)−it follows that for any

u∈C∞

0(R+) it holds

u, ΠkVk(·)−Qk+QkVk(·)−Πku

≤u, ΠkVk(·)−Πku+u, QkVk(·)−Qku,(6.17)

where the scalar products are taken in L2(R+, rdr). From (6.17) we infer

that

hk(B)−B0= (Πk+Qk) (hk(B0)−B0+Vk(·)) (Πk+Qk)

≥(Πk+Qk)hk(B0)−B0−Vk(·)−(Πk+Qk)

≥Πkhk(B0)−B0−2Vk(·)−Πk

+Qkhk(B0)−B0−2Vk(·)−Qk.(6.18)

23

The operator hk(B0) has for each k∈N0discrete spectrum which consists

of simple eigenvalues. Moreover, from the partial-wave decomposition of the

operator H(B0) we obtain

σ(H(B0)) = {(2n−1)B0, n ∈N}=[

k∈Z

σ(hk(B0)) ,

see (6.2). It means that

∀k∈Z:σ(hk(B0)) ⊂ {(2n−1)B0, n ∈N},

and since ψkis an eigenfunction of hk(B0) associated to the simple eigenvalue

B0, see (6.16), it follows that

Qk(hk(B0)−B0)Qk≥2B0Qk,∀k∈N∪ {0}.(6.19)

On the other hand, by (6.7) and (6.9) we infer

sup

r>0Vk(r)−≤α B0∀k∈N∪ {0}.

The last two estimates thus imply that

Qkhk(B0)−B0−2Vk(·)−Qk≥Qk(2 B0(1 −α)) Qk≥0,

where we have used the assumption α≤1. With the help of (6.18) and the

variational principle we then conclude that

tr (hk(B)−B0)γ

−≤tr Πkhk(B0)−B0−2Vk(·)−Πkγ

−

= tr −2 ΠkVk(·)−Πkγ

−= 2γtr ΠkVk(·)−Πkγ

= 2γψk,Vk(·)−ψkγ

L2(R+,rdr)= 2γΛγ

k,

see (6.10). To complete the proof it now remains to apply equation (6.15).

7 Three dimensions: a magnetic ‘hole’

Let us return to the three-dimensional situation and consider a magnetic

Hamiltonian H(B) in L2(R3) associated to the magnetic ﬁeld B:R3→R3re-

garded as a perturbation of a homogeneous magnetic ﬁeld of intensity B0>0

pointing in the x3-direction,

B(x1, x2, x3) = (0,0, B0)−b(x1, x2, x3),(7.1)

24

with the perturbation bof the form

b(x1, x2, x3) = −ω0(x3)f(x1, x2),0, ω(x3)gqx2

1+x2

2.

Here ω:R→R+,g:R+→R+and

f(x1, x2) = −Z∞

x1

gqt2+x2

2dt . (7.2)

The resulting ﬁeld Bthus has the component in the x3-direction given the

B0plus a perturbation which is a radial ﬁeld in the x1, x2−plane with a

x3−dependent amplitude ω(x3). The ﬁrst component of Bthen ensures that

∇ · B= 0, which is required by the Maxwell equations which include no

magnetic monopoles; it vanishes if the ﬁeld is x3-independent.

A vector potential generating this ﬁeld can be chosen in the form

A(x1, x2, x3) = (0, B0x1−ω(x3)f(x1, x2),0) ,

which reduces to Landau gauge in the unperturbed case, and consequently,

the operator H(B) acts on its domain as

H(B) = −∂2

x1+ (i∂x2+B0x1−ω(x3)f(x1, x2))2−∂2

x3.(7.3)

We have again to specify the local character of the perturbation: we will

suppose that

(i) the function g∈L∞(R+) is non-negative, such that fand ∂x2fbelong

to L∞(R2), and

lim

x2

1+x2

2→∞ |∂x2f(x1, x2)|+|f(x1, x2)|= 0 ,

(ii) ω≥0, ω∈L2(R)∩L∞(R), and

kωk∞kgk∞≤B0,lim

|x3|→∞ ω(x3) = 0 .

Lemma 7.1. The assumptions (i) and (ii) imply σess (H(B)) = [B0,∞).

25

Proof. We will show that the essential spectrum of H(B) coincides with the

essential spectrum of the operator

H(B0) = −∂2

x1+ (i∂x2+B0x1)2−∂2

x3,

which is easy to be found, we have σ(H(B0)) = σess (H(B0)) = [B0,∞). Let

T=H(B)− H(B0) = −2ωf (i∂x2+B0x1)−iω ∂x2f+ω2f2.

From assumption (i) in combination with [Da, Thm. 5.7.1] it follows that the

operator (ω ∂x2f+ω2f2)(−∆ + 1)−1is compact on L2(R3). The diamagnetic

inequality and [Pi79] thus imply that the sum iω ∂x2f+ω2f2is relatively

compact with respect to H(B0).

As for the ﬁrst term of the perturbation T, we note that since (i∂x2+B0x1)

commutes with H(B0), it holds

ωf (i∂x2+B0x1) (H(B0) + 1)−1

=ωf (H(B0) + 1)−1/2(i∂x2+B0x1) (H(B0) + 1)−1/2.(7.4)

In the same way as above, with the help of [Da, Thm.5.7.1], diamagnetic

inequality, and [Pi79], we conclude that ω f (H(B0) + 1)−1/2is compact on

L2(R3). On the other hand, (i∂x2+B0x1) (H(B0) + 1)−1/2is bounded on

L2(R3). As their product the operator (7.4) is compact; by Weyl’s theorem

we then have σess (H(B)) = σess(H(B0)) = [B0,∞).

7.1 Lieb-Thirring-type inequalities for H(B)

Now we are going to formulate Lieb-Thirring-type inequalities for the nega-

tive eigenvalues of H(B)−B0in three diﬀerent cases corresponding to diﬀer-

ent types of decay conditions on the function g. Let us start from a general

result. We denote by

α(x3) = ω(x3)Z∞

0

g(r)rdr

the magnetic ﬂux (up to the sign) through the plane {(x1, x2, x3):(x1, x2)∈

R2}associated with the perturbation. From Theorem 6.1 and inequality

(2.2) we make the following conclusion.

26

Theorem 7.2. Let assumptions (i) and (ii) be satisﬁed. Suppose, moreover,

that supx3α(x3)≤1and put

Λk(x3) = ψk,Vk(·;x3)−ψkL2(R+,rdr).(7.5)

Then the inequality

tr (H(B)−B0)σ

−≤Lcl

σ,12σ+1

2ZR

∞

X

k=0

Λk(x3)σ+1

2dx3, σ ≥3

2,(7.6)

holds true whenever the right-hand side is ﬁnite.

7.1.1 Perturbations with a power-like decay

Now we come to the three cases mentioned above, stating ﬁrst the results

and then presenting the proofs. We start from magnetic ﬁelds (7.1) with the

perturbation gwhich decays in a powerlike way. Speciﬁcally, we shall assume

that

0≤g(r)≤B0(1 + pB0r)−2β, β > 1.(7.7)

We have included the factor √B0on the right hand side of (7.7) having

in mind that B−1/2

0is the Landau magnetic length which deﬁnes a natural

length unit in our model.

For any β > 1 and γ > max n1

β−1,2owe deﬁne the number

K(β, γ ) = 2−γ+∞

X

k=1 Γ ((k+ 1 −β)+)

Γ(k)+1

2√2πk γ

,(7.8)

and recall also the classical Lieb-Thirring constants in one dimension,

Lcl

1,σ =Γ(σ+ 1)

2√πΓ(σ+ 3/2) , σ > 0.(7.9)

Theorem 7.3. Assume that gsatisﬁes (7.7) and that kωk∞≤2(β−1).

Then

tr (H(B)−B0)σ

−≤Lcl

1,σ Kβ, σ +1

22B0

β−1σ+1

2ZR

ω(x3)σ+1

2dx3

holds true for all

σ > max 3

2,3−β

2β−2.(7.10)

27

Remark 7.4. Since ω∈L∞(R)∩L2(R), it follows that ω∈Lσ+1

2(R) for

any σ≥3/2. Note also that by the Stirling formula we have

Γ (k+ 1 −β)

Γ(k)∼k1−βas k→ ∞.

Hence the constant Kβ, σ +1

2is ﬁnite for any σsatisfying (7.10).

7.1.2 Gaussian decay

Next we assume that the perturbation ghas a Gaussian decay, in other words

0≤g(r)≤B0e−εB0r2, ε > 0.(7.11)

Theorem 7.5. Assume that gsatisﬁes (7.11) and that kωk∞≤2ε. Then

for any σ > 3/2it holds

tr (H(B)−B0)σ

−≤Lcl

σ,1B0

εσ+1

2

G(ε, σ)ZR

ω(x3)σ+1

2dx3,

where

G(ε, σ) = 1 + ∞

X

k=1 (1 + 2ε)−k+1

2√2πk σ+1

2

.(7.12)

7.1.3 Perturbations with a compact support

Let Dbe a circle of radius Rcentered at the origin and put

g(r) = B0r≤R

0r > R .(7.13)

Theorem 7.6. Assume that gsatisﬁes (7.13) with Rsuch that B0R2≤2.

Suppose moreover that kωk∞≤1. Then for any σ > 3/2it holds

tr (H(B)−B0)σ

−≤Lcl

σ,1JB0, σBσ+1

2

0ZR

ω(x3)σ+1

2dx3,(7.14)

where

J(B0, σ) = B0R2σ+1

2

1 + ∞

X

k=1 B0R2

2k+1 1

k!+1

2√2πk !σ+1

2

.

(7.15)

28

7.2 The proofs

Note that the assumptions of these theorems ensure that supx3α(x3)≤1,

hence in all the three cases we may apply Theorem 6.1 and, in particular,

the estimate (7.6). To this note it is useful to realize that by (6.4), (6.7) and

(6.9) we have

Vk(r;x3) = −α(x3)B0+2α(x3)k

r2−2k ω(x3)

r2Z∞

r

g(s)sds

+B0ω(x3)Z∞

r

g(s)sds+ω2(x3)

r2Zr

0

g(s)sds2

.(7.16)

Consequently, we obtain a simple upper bound on the negative part of Vk,

Vk(r;x3)−≤2k ω(x3)

r2Z∞

r

g(s)sds+α(x3)B0−2k

r2+

(7.17)

for all k∈N∪ {0}. For k= 0 we clearly we have

Λ0(x3)≤α(x3)B0,(7.18)

by (6.16). In order to estimate Λk(x3) with k≥1 we denote by λk(x3) the

contribution to Λk(x3) coming from the ﬁrst term on the right-hand side of

(7.17), i.e.

λk(x3) = 2 ω(x3)kZ∞

0

ψ2

k(r)Z∞

r

g(s)sdsr−1dr . (7.19)

Before coming to the proofs we need an auxiliary result.

Lemma 7.7. For any k∈Nit holds

Λk(x3)≤λk(x3) + α(x3)B0

√2πk .

Proof. In view of (7.5), (7.17), and (7.19) the claim will follow if we show

that Z∞

0

ψ2

k(r)B0−2k

r2+

rdr≤B0

√2πk .(7.20)

29

Let rk=q2k

B0. Using (6.8) and the substitution s=B0r2

2we then ﬁnd

Z∞

0

ψ2

k(r)B0−2k

r2+

rdr=B0Z∞

rk

ψ2

k(r)rdr−2kZ∞

rk

ψ2

k(r)r−1dr

=B0

Γ(k+ 1) Z∞

k

e−sskds−B0

Γ(k)Z∞

k

e−ssk−1ds .

Integration by parts gives

Z∞

k

e−sskds=e−kkk+kZ∞

k

e−ssk−1ds ,

hence Z∞

0

ψ2

k(r)B0−2k

r2+

rdr=e−kkkB0

Γ(k+ 1) ,

and inequality (7.20) follows from the Stirling-type estimate [AS64, Eq. 6.1.38]

Γ(k+ 1) = k!≥√2π k k+1

2e−k, k ∈N;

this concludes the proof.

Proof of Theorem 7.3. In view of (7.18) and Lemma 7.7 it suﬃces to

estimate λk(x3) in a suitable way from above for k≥1. Using (7.7) we ﬁnd

Z∞

0

g(r)rdr≤B0Z∞

0

(1 + pB0r)−2βrdr≤B0Z∞

0

(1 + pB0r)1−2βdr

=Z∞

0

(1 + s)1−2βds=1

2(β−1) ,

which implies

α(x3)≤ω(x3)

2(β−1) .(7.21)

Moreover, by virtue of (7.7)

Z∞

r

g(s)sds≤pB0Z∞

r

(1 + pB0s)1−2βds=1

2β−2(1 + pB0r)2−2β.

30

Assume ﬁrst that 1 ≤k≤β−1. In this case a combination of (6.8) and the

last equation gives

λk(x3)≤ω(x3)B0

(β−1) Γ(k)B0

2kZ∞

0

e−B0r2

2r2k−1(1 + pB0r)2−2βdr

=ω(x3)B0

(β−1) Γ(k)Z∞

0

e−ssk−1(1 + √2s)2−2βds

≤ω(x3)B0

(β−1) Γ(k)Z∞

0

e−sds=ω(x3)B0

(β−1) Γ(k),(7.22)

where we have used again the substitution s=B0r2

2.

On the other hand, for k > β −1 we have

λk(x3)≤ω(x3)B0

(β−1) Γ(k)B0

2kZ∞

0

e−B0r2

2r2k−1(1 + pB0r)2−2βdr

≤ω(x3)B0

(β−1) Γ(k)B0

2kZ∞

0

e−B0r2

2r2k−1(B0r2)1−βdr

≤ω(x3)B0

(β−1) Γ(k)Z∞

0

e−ssk−βds=ω(x3)B0Γ(k+ 1 −β)

(β−1) Γ(k).

This together with equations (7.21), (7.18), (7.22) and Lemma 7.7 shows that

∞

X

k=0

Λγ

k(x3)≤K(β, γ )B0

β−1γ

ω(x3)γ,

with the constant K(β, γ) given by (7.8). The claim now follows from (7.6)

upon setting γ=σ+1

2.

Proof of Theorem 7.5. We proceed as in the proof of Theorem 7.3 and

use equation (7.18) and Lemma 7.7. Since

α(x3)≤ω(x3)B0Z∞

0

B0e−εB0r2rdr=ω(x3)

2ε(7.23)

holds in view of (7.11), for k= 0 we get

Λ0(x3)≤α(x3)B0≤ω(x3)B0

2ε.

31

On the other hand,

Z∞

r

g(s)sds≤B0Z∞

r

e−εB0s2sds=1

2εe−εB0r2.

Hence using the substitution s=B0r2

2(1 + 2ε), we obtain

λk(z)≤ω(x3)B0

εΓ(k)B0

2kZ∞

0

e−B0r2

2(1+2ε)r2k−1dr

=ω(x3)B0

2ε

(1 + 2ε)−k

Γ(k)Z∞

0

e−ssk−1ds=ω(x3)B0

2ε(1 + 2ε)−k

for any k≥1. Summing up gives

∞

X

k=0

Λγ

k(x3)≤ω(x3)B0

2εγ 1 + ∞

X

k=1 (1 + 2ε)−k+1

2√2πk γ!.

Theorem 6.1 applied with γ=σ+1

2then completes the proof.

Proof of Theorem 7.6. In this case we have

α(x3) = ω(x3)B0R2

2.

Inequality (7.18) thus implies

Λ0(z)≤ω(z)B2

0R2

2.

For k≥1 we note that in view of (7.13)

Z∞

r

g(s)sds=

1

2(R2−r2)r≤R

0r > R

Hence from (6.8) and (7.19) we conclude that

λk(z)≤B2

0R2ω(x3)

Γ(k)B0

2kZR

0

e−B0r2

2r2k−1dr

≤B2

0R2ω(x3)

2Γ(k)ZB0R2

2

0

e−ssk−1ds

≤B2

0R2ω(z)

2kΓ(k)B0R2

2k

=B0ω(x3)

Γ(k+ 1) B0R2

2k+1

, k ∈N.

32

This in combination with the above estimate on Λ0(x3) and Lemma 7.7 im-

plies

∞

X

k=0

Λγ

k(x3)≤ω(x3)γBγ

0B0R2

2γ 1 + ∞

X

k=1 B0R2

2k1

k!+1

√2πk !γ!,

and the claim follows again by applying Theorem 6.1 with γ=σ+1

2.

Acknowledgements

The research was supported by the Czech Science Foundation (GAˇ

CR) within

the project 14-06818S. D.B. acknowledges the support of the University of Os-

trava and the project “Support of Research in the Moravian-Silesian Region

2013”. H. K. was supported by the Gruppo Nazionale per Analisi Matemat-

ica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale

di Alta Matematica (INdAM). The support of MIUR-PRIN2010-11 grant for

the project “Calcolo delle variazioni” (H. K.) is also gratefully acknowledged.

T.W. was in part supported by the DFG project WE 1964/4-1 and the DFG

GRK 1838.

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