Rarefied sets at infinity associated with the Schrödinger operator

R E S E A R C H

Open Access

Rarefied sets at infinity associated with the

Schrödinger operator

Gaixian Xue

*_{Correspondence:}

[email protected] School of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450046, China

Abstract

This paper gives some criteria fora-rarefied sets at infinity associated with the Schrödinger operator in a cone. Our proofs are based on estimating Greena-potential with a positive measure by connecting with a kind of density of the modified

measure. Meanwhile, the geometrical property of thisa-rarefied sets at infinity is also considered. By giving an example, we show that the reverse of this property is not true.

Keywords: rarefied set; Schrödinger operator; Greena-potential

1 Introduction and results

LetRandR+ be the set of all real numbers and the set of all positive real numbers,

re-spectively. We denote byRn_{(n≥) then-dimensional Euclidean space. A point inR}n_is

denoted byP= (X,xn),X= (x,x, . . . ,xn–). The Euclidean distance between two pointsP

andQinRn_{is denoted by|P–Q|. Also|P–O|with the originOofR}n_{is simply denoted}

by|P|. The boundary and the closure of a setSinRn_{are denoted by∂SandS, respectively.}

We introduce a system of spherical coordinates (r,),= (θ,θ, . . . ,θn–), inRnwhich

are related to Cartesian coordinates (x,x, . . . ,xn–,xn) byxn=rcosθ.

LetDbe an arbitrary domain inRn_{and letA}

a denote the class of non-negative radial

potentialsa(P),i.e., ≤a(P) =a(r),P= (r,)∈D, such thata∈Lb

loc(D) with someb>n/

ifn≥ and withb=  ifn=  orn= . Ifa∈Aa, then the Schrödinger operator

Scha= –+a(P)I= ,

whereis the Laplace operator andI is the identical operator, can be extended in the usual way from the spaceC_∞(D) to an essentially self-adjoint operator onL_{(D) (see [,}

Ch. ]). We will denote it bySchaas well. This last one has a Greena-functionGaD(P,Q).

HereGa

D(P,Q) is positive onDand its inner normal derivative∂GDa(P,Q)/∂nQ≥, where ∂/∂nQdenotes the differentiation atQalong the inward normal intoD.

We call a functionu≡–∞that is upper semi-continuous inDa subfunction with re-spect to the Schrödinger operatorSchaif its values belong to the interval [–∞,∞) and at

each pointP∈Dwith  <r<r(P) the generalized mean-value inequality (see [])

u(P)≤

S(P,r)

u(Q)∂G

a

B(P,r)(P,Q)

∂nQ

dσ(Q)

is satisfied, whereG_Ba_(P,r)(P,Q) is the Greena-function ofScha inB(P,r) anddσ(Q) is a

surface measure on the sphereS(P,r) =∂B(P,r).

If –uis a subfunction, then we callua superfunction. If a functionuis both subfunction and superfunction, it is, clearly, continuous and is called ana-harmonic function (with respect to the Schrödinger operatorScha).

The unit sphere and the upper half unit sphere in Rn_{are denoted byS}n–_andSn– + ,

respectively. For simplicity, a point (,) on Sn– and the set {; (,)∈} for a set

,⊂Sn–_{, are often identified withand, respectively. For two sets⊂R} + and

⊂Sn–, the set {(r,)∈Rn;r∈, (,)∈}inRn is simply denoted by×. By

Cn() we denote the setR+×inRnwith the domainonSn–. We call it a cone. We

denote the setI×with an interval onRbyCn(;I).

We shall say that a setH⊂Cn() has a covering{rj,Rj}if there exists a sequence of balls {Bj}with centers inCn() such thatH⊂j∞=Bj, whererjis the radius ofBjandRjis the

distance from the origin to the center ofBj.

From now on, we always assume D=Cn(). For the sake of brevity, we shall write Ga

(P,Q) instead ofGaCn()(P,Q). Throughout this paper, letcdenote various positive

con-stants, because we do not need to specify them. Moreover, appearing in the expression in the following sections will be a sufficiently small positive number.

Letbe a domain onSn–_{with smooth boundary. Consider the Dirichlet problem}

(n+λ)ϕ=  on,

ϕ=  on∂,

wherenis the spherical part of the Laplace operatorn

n= n– 

r

∂ ∂r+

∂ ∂r +

n r .

We denote the least positive eigenvalue of this boundary value problem byλand the nor-malized positive eigenfunction corresponding toλbyϕ(). In order to ensure the exis-tence ofλand a smoothϕ(), we put a rather strong assumption on: ifn≥, then

is aC,α_{-domain ( <α< ) onS}n–_{surrounded by a finite number of mutually disjoint}

closed hypersurfaces (e.g., see [, pp.-] for the definition ofC,α_-domain). For any (,)∈, we have (see [, pp.-])

c–rϕ()≤δ(P)≤crϕ(), ()

whereP= (r,)∈Cn() andδ(P) =dist(P,∂Cn()).

Solutions of an ordinary differential equation

–Q (r) –n– 

r Q(r) +

λ

r+a(r)

Q(r) = ,  <r<∞. ()

It is known (see, for example, []) that if the potential a∈Aa, then equation () has a

We will also consider the classBa, consisting of the potentialsa∈Aa, such that there

exists the finite limitlim_r→∞r_{a(r) =k∈[,∞) and, moreover,r}–_|r_{a(r) –k| ∈L(,∞). If}

a∈Ba, then the (sub)superfunctions are continuous (see []).

In the rest of paper, we assume thata∈Ba and we shall suppress this assumption for

simplicity. Denote

ι±_k = –n±

(n– )_{+ (k+λ)}

 ,

then the solutions to equation () have the asymptotic (see [])

c–rι+k≤_V(r)≤_crι+k_{, c}–_rι–k≤_W(r)≤_crι–k_{, asr}→ ∞_{. ()}

Letνbe any positive measure onCn() such that the Greena-potential

Gaν(P) =

Cn()

Ga(P,Q)dν(Q)≡+∞

for anyP∈Cn(). Then the positive measurem(ν) onRnis defined by

dm(ν)(Q) =

W(t)ϕ()dν(Q), Q= (t,)∈Cn(; (, +∞)),

, Q∈Rn_–C

n(; (, +∞)).

Remark  We remark that the total massm(ν) is finite (see [, Lemma ]).

For eachP= (r,)∈Rn–{O}, the maximal functionM(P;λ,β) is defined by

M(P;λ,β) = sup

<ρ<r_

λ(B(P,ρ))

ρβ ,

whereβ≥ andλis a positive measure onRn_{. The set}

P= (r,)∈Rn–{O};M(P;λ,β)rβ>

is denoted byE( ;λ,β).

It is known that the Martin boundary ofCn() is the set∂Cn()∪ {∞}, each of which

is a minimal Martin boundary point. ForP∈Cn() andQ∈∂Cn()∪ {∞}, the Martin

kernel can be defined byMa(P,Q). If the reference pointPis chosen suitably, then we have

Ma(P,∞) =V(r)ϕ() and Ma(P,O) =cW(r)ϕ() ()

for anyP= (r,)∈Cn().

In [], Longet al.introduced the notations ofa-thin (with respect to the Schrödinger operatorScha) at a point,a-polar set (with respect to the Schrödinger operatorScha) and a-rarefied sets at infinity (with respect to the Schrödinger operatorScha), which

to bea-thin at a pointQif there is a fine neighborhoodEofQwhich does not intersect

H\{Q}. OtherwiseHis said to be nota-thin atQonCn(). A setHinRnis called a

po-lar set if there is a superfunctionuon some open setEsuch thatH⊂ {P∈E;u(P) =∞}. A subsetHofCn() is said to bea-rarefied at infinity onCn() if there exists a positive

superfunctionv(P) onCn() such that

inf

P∈Cn()

v(P)

Ma

(P,∞)

≡

and

H⊂P= (r,)∈Cn();v(P)≥V(r) .

LetHbe a bounded subset ofCn(). ThenRˆH_Ma

(·,∞)is bounded onCn() and the greatest

a-harmonic minorant ofRˆH

Ma(·,∞)is zero. We see from the Riesz decomposition theorem

(see [, Theorem ]) that there exists a unique positive measureλa_H onCn() such that

(see [, p.])

ˆ RH_Ma

(·,∞)(P) =G

a

λ

a

H(P) ()

for anyP∈Cn() andλaHis concentrated onIH, where

IH=

P∈Cn();His nota-thin atP .

We denote the total massλa_H(Cn()) ofλaHbyλa(H).

By using this positive measureλa_H (with respect to the Schrödinger operatorScha), we

can further define another measureηa

HonCn() by

dηa_H(P) =Ma(P,∞)dλa_H(P)

for anyP∈Cn(). It is easy to see thatηaH(Cn()) < +∞.

Recently, Longet al.(see [, Theorem .]) gave a criterion for a subsetHofCn() to

bea-rarefied set at infinity.

Theorem A A subset H of Cn()is a-rarefied at infinity on Cn()if and only if

∞

j=

λa(Hj)W

j<∞,

where Hj=H∩Cn(; [j, j+))and j= , , , . . . .

In this paper, we shall obtain a series of new criteria fora-rarefied sets at infinity on

Cn(), which complement Theorem A. Our results are essentially based on Qiao and

Deng, Ren and Zhao, Xue (see [, –]). In order to avoid complexity of our proofs, we shall assumen≥. But our results in this paper are also true forn= .

Theorem  A subset H of Cn()is a-rarefied at infinity on Cn()if and only if there exists a positive measureξ_Ha on Cn()such that

GaξHa(P)≡+∞ ()

for any P∈Cn()and

H⊂P= (r,)∈Cn();GaξHa(P)≥V(r) . ()

Next we give the geometrical property ofa-rarefied sets at infinity.

Theorem  If a subset H of Cn()is a-rarefied at infinity on Cn(),then H has a covering

{rj,Rj}(j= , , , . . .)satisfying

∞

j=

rj Rj

V

Rj rj

W

Rj rj

<∞. ()

Finally, by an example we show that the reverse of Theorem  is not true.

Example Put

rj= ·j–·j



–n _{and Rj= }·_j– _{(j= , , , . . .).}

A covering{rj,Rj}satisfies

∞

j=

rj Rj

V

Rj rj

W

Rj rj

≤c ∞

j=

rj Rj

n–

=c ∞

j=

jn––n_{< +}∞

from equation ().

LetCn() be a subset ofCn(),i.e., ⊂. Suppose that this covering is located as

follows: there is an integerjsuch thatBj⊂Cn() andRj> rj forj≥j. Then the set

H=∞_j=jBjis nota-rarefied at infinity onCn(). This fact will be proved in Section . 2 Lemmas

Lemma (see [, Ch. ] and [, Lemma ])

Ga(P,Q)≤cV(t)W(r)ϕ()ϕ()

resp. Ga(P,Q)≤cV(r)W(t)ϕ()ϕ()

for any P= (r,)∈Cn()and any Q= (t,)∈Cn()satisfying r≥t(resp.t≥r).

Lemma  (see [, Lemma ]) Letν be a positive measure on Cn()such that there is

a sequence of points Pi= (ri,i)∈Cn(), ri→+∞(i→+∞)satisfying Gaν(Pi) < +∞

(i= , , . . . ;Q∈Cn()).Then,for a positive number L,

Cn(;(L,+∞))

and

lim

R→+∞ W(R)

V(R)

Cn(;(,R))

V(t)ϕ()dν(Q) = .

Lemma (see [, Theorem ]) Letνbe any positive measure on Cn()such that Gaν(P)≡

+∞for any P∈Cn().Then,for a sufficiently large L,

P= (r,)∈Cn

; (L, +∞);Gaν(P)≥V(r)ϕ() ⊂E ;m(ν),n– .

Lemma (see [, Lemma ]) Letλbe any positive measure onRnhaving finite total mass.

Then E( ;λ,n– )has a covering{rj,Rj}(j= , , . . .)satisfying

∞

j=

rj Rj

V

Rj rj

W

Rj rj

<∞. ()

3 Proof of Theorem 1

Suppose that

H⊂ξ_Ha=P= (r,)∈Cn();Gaξ

a

H(P)≥V(r) ()

for a positive measureξ_Ha onCn() satisfying equation ().

We write

Gaν(P) =Ga(,j)(P) +Ga(,j)(P) +Ga(,j)(P),

where

Ga(,j)(P) =

Cn(;(,j–))

Ga(P,Q)dν(Q),

Ga(,j)(P) =

Cn(;[j–_,j+₎₎G

a

(P,Q)dν(Q)

and

Ga(,j)(P) =

Cn(;[j+_,∞))G

a

(P,Q)dν(Q).

Now we shall show the existence of an integerNsuch that for any integerj(≥N), we have

ξ_Ha(j)⊂P= (r,)∈Cn

;j, j+; Ga(,j)(P)≥V(r) ()

for any integerj(≥N).

For anyP= (r,)∈Cn(; [j, j+)), we have

Ga(,j)(P)≤cW(r)ϕ()

and

Ga(,j)(P)≤cV(r)ϕ()

Cn(;[j+_,∞))dm(

ν)(Q)

from Lemma .

By applying Lemma , we can take an integerNsuch that for anyj(≥N),

WjV–j

Cn(;(,j–))

V(t)ϕ()dν(Q)≤  c

and

Cn(;[j+_,∞))dm(

ν)(Q)≤  c.

Thus we obtain

Ga(,j)(P)≤V(r)ϕ() ()

and

Ga(,j)(P)≤V(r)ϕ() ()

for anyP= (r,)∈Cn(; [j, j+)), wherej≥N.

Thus, ifP= (r,)∈(ν)(j) (j≥N), then we obtain

Ga(,j)(P)≥V(r)ϕ()

from equations () and (), which gives equation (). From equations (), () and (), we have

Ga(,j)(P) =

Cn()

Ga(P,Q)dτja(Q)≥Ma(P,∞),

whereP∈Ij(j≥N) and

dτ_ja(Q) =

–jdξ_Ha(Q), Q∈Cn(; [j–, j+)),

, Q∈Cn(; (, j–))∪Cn(; [j+,∞)).

And then we obtain

ηa_HjCn()

≤

Cn()

V(t)ϕ()dτ_ja(Q) =

Cn(;[j–_,j+₎₎V(t)

ϕ()dξ_Ha(Q)

forj≥N. Then we have

∞

j=N

λa(Hj)W

j=

∞

j=N

η_Hja Cn()

Wj≤c

Cn(;[N–_,∞))dm

in which the last integral is finite by Remark . And henceH isa-rarefied set at infinity from Theorem A.

Suppose that

∞

j=

λa(Hj)W

j<∞.

Consider a functionfa

H(P) onCn() defined by

f_Ha(P) =

∞

j=–

ˆ RHj_Ma

(·,∞)(P)

for anyP∈Cn(), whereH–=H∩Cn(; (, )).

If we putμa H()(P) =

_∞

j=–λaHj(P), then from equation () we have that f_Ha(P) =

Cn()

Ga(P,Q)dμaH()(Q)

for anyP∈Cn().

Next we shall show thatf_Ha(P) is always finite onCn(). Take any pointP= (r,)∈Cn()

and a positive integerj(P) satisfyingr≤j(P)+. We write

f_Ha(P) =f_Ha()(P) +f_Ha()(P),

where

f_Ha()(P) =

j(P)+

j=–

Cn()

Ga(P,Q)dλaHj(Q) and

f_Ha()(P) =

∞

j=j(P)+

Cn()

Ga(P,Q)dλaHj(Q).

Sinceλa_Hj is concentrated onIHj⊂Hj∩Cn(), we have that

Cn()

Ga(P,Q)dλa_Hj(Q)≤cV(r)ϕ()

Cn()

W(t)ϕ()dλa_Hj(t,)

≤cV(r)ϕ()WjV–j

Cn()

V(t)ϕ()dλa_Hj(t,)

forj≥j(P) + . Hence we have

f_Ha()(P)≤cV(r)ϕ()

∞

j=j(P)+

η_Hja Cn()

WjV–j, ()

which, together with Theorem A, shows thatf_Ha()(P) is finite and hencef_Ha(P) is also finite for anyP∈Cn().

Since

ˆ RHj_Ma

(·,∞)(P) =M

a

holds onIHjandIHj⊂Hj∩Cn(), we see that for anyP= (r,)∈IHj (j= –, , , , , . . .)

f_Ha(P)≥cRˆHj_Ma

(·,∞)(P)≥V(r)ϕ(). () And hence equation () also holds for anyP= (r,)∈H =∞_j=–IHj. SinceH is equal

toHexcept a polar setH_{, we can take another positive superfunctionf}a

H()(P) onCn()

such thatfa

H()(P) =GaμaH()(P) with a positive measureμHa()(P) onCn() andfHa()(P)

is identically +∞onH_.

Finally, we can define a positive superfunctiongonCn() byg(P) =fHa(P) +fHa()(P) = GaξHa(P) for anyP∈Cn() withξHa =μaH() +μaH(). Also we see from equation () that

equations () and () hold.

Thus we complete the proof of Theorem .

4 Proof of Theorem 2

From Theorem  and Lemma , we have a positive numberLsuch that

H∩Cn

; (L, +∞)⊂E ;mξ_Ha,n– .

Hence by Remark  and Lemma ,E( ;m(ξ_Ha),n– ) has a covering{rj,Rj}(j= , , , . . .)

satisfying equation () and henceH has also a covering{rj,Rj}(j= , , , , . . .) with an

additional finiteBcoveringCn(; (,L]), satisfying equation (), which is the conclusion

of Theorem .

5 Proof of an example

Sinceϕ()≥cfor any∈, we haveMa

(P,∞)≥cV(Rj) for anyP∈Bj, wherej≥j.

Hence we have

ˆ RB_Mja

(·,∞)(P)≥cV(Rj) ()

for anyP∈Bj, wherej≥j.

Take a measureδonCn(),suppδ⊂Bj,δ(Bj) =  such that

Cn()|

P–Q|–ndδ(P) =Cap(Bj)

–

()

for anyQ∈Bj, whereCapdenotes the Newton capacity. Since

Ga(P,Q)≤ |P–Q|–n

for anyP∈Cn() andQ∈Cn() (see [], the casen=  is implicitly contained in []),

Cap(Bj) –λaBj

Cn()

= |P–Q|–ndδ(P)

dλa_Bj(Q)

≥ Ga(P,Q)dλaBj(Q)

dδ(P)

=

ˆ RBj_Ma

(·,∞)d

δ(P)

from equations () and (). Hence we have

λa_BjCn()

≥cCap(Bj)V(Rj)≥crjn–V(Rj). ()

If we observeλa_Hj(Cn()) =λaBj(Cn()), then we have by equation ()

∞

j=j

Wjλa_HjCn()

≥c ∞

j=j

rj Rj

n–

=c ∞

j=j 

j = +∞,

from which it follows by Theorem A thatHis nota-rarefied at infinity onCn().

Competing interests

The author declares that they have no competing interests.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102.

Received: 26 March 2014 Accepted: 30 May 2014 Published: 18 July 2014 References

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doi:10.1186/1029-242X-2014-247