The aim of this paper is to give a new criterion for a-minimally thin sets at infinity with respect to the Schrödinger operator in a cone, which supplement the results obtained by T. Zhao.
Keywords:minimally thin set; Schrödinger operator; Green a-potential
1 Introduction and results
Let R and be the set of all real numbers and the set of all positive real numbers, respectively. We denote by ( ) the n-dimensional Euclidean space. A point in is denoted by , . The Euclidean distance between two points P and Q in is denoted by . Also with the origin O of is simply denoted by . The boundary and the closure of a set S in are denoted by ∂S and , respectively. Further, intS, diamS, and stand for the interior of S, the diameter of S, and the distance between and , respectively.
We introduce a system of spherical coordinates , , in which are related to cartesian coordinates by .
If , then the stationary Schrödinger operator
where Δ is the Laplace operator and I is the identical operator, can be extended in the usual way from the space to an essentially self-adjoint operator on (see [, Ch. 11]). We will denote it as well. This last one has a Green a-function . Here is positive on D and its inner normal derivative , where denotes the differentiation at Q along the inward normal into D.
We call a function that is upper semi-continuous in D a subfunction with respect to the Schrödinger operator if its values belong to the interval and at each point with we have the generalized mean-value inequality (see [, Ch. 11])
satisfied, where is the Green a-function of in and is a surface measure on the sphere . If −u is a subfunction, then we call u a superfunction (with respect to the Schrödinger operator ).
The unit sphere and the upper half unit sphere in are denoted by and , respectively. For simplicity, a point on and the set for a set Ω, , are often identified with Θ and Ω, respectively. For two sets and , the set in is simply denoted by . By , we denote the set in with the domain Ω on . We call it a cone. We denote the set with an interval on R by .
From now on, we always assume . For the sake of brevity, we shall write instead of . We shall also write for two positive functions and , if and only if there exists a positive constant c such that .
Let Ω be a domain on with smooth boundary. Consider the Dirichlet problem
where is the spherical part of the Laplace operata
We denote the least positive eigenvalue of this boundary value problem by λ and the normalized positive eigenfunction corresponding to λ by . In order to ensure the existence of λ and a smooth , we put a rather strong assumption on Ω: if , then Ω is a -domain ( ) on surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [, pp.88-89] for the definition of -domain).
For any , we have (see [, pp.7-8])
where and .
Solutions of an ordinary differential equation (see [, p.217])
It is well known (see, for example, ) that if the potential , then equation (2) has a fundamental system of positive solutions such that V and W are increasing and decreasing, respectively.
We will also consider the class , consisting of the potentials such that there exists the finite limit , and, moreover, . If , then the (sub)superfunctions are continuous (see ). In the rest of paper, we assume that and we shall suppress this assumption for simplicity.
then the solutions to equation (2) have the asymptotic (see )
It is well known that the Martin boundary of is the set , each of which is a minimal Martin boundary point. For and , the Martin kernel can be defined by . If the reference point P is chosen suitably, then we have
for any .
In [, p.67], Zhao introduce the notations of a-thin (with respect to the Schrödinger operator ) at a point, a-polar set (with respect to the Schrödinger operator ) and a-minimal thin sets at infinity (with respect to the Schrödinger operator ). A set H in is said to be a-thin at a point Q if there is a fine neighborhood E of Q which does not intersect . Otherwise H is said to be not a-thin at Q on . A set H in is called a polar set if there is a superfunction u on some open set E such that . A subset H of is said to be a-minimal thin at on , if there exists a point such that
where is the regularized reduced function of relative to H (with respect to the Schrödinger operator ).
Let H be a bounded subset of . Then is bounded on and hence the greatest a-harmonic minorant of is zero. When by we denote the Green a-potential with a positive measure μ on , we see from the Riesz decomposition theorem that there exists a unique positive measure on such that
for any and is concentrated on , where
The Green a-energy (with respect to the Schrödinger operator ) of is defined by
Also, we can define a measure on
In [, Theorem 5.4.3], Long gave a criterion that characterizes a-minimally thin sets at infinity in a cone.
Theorem AA subsetHof is a-minimally thin at infinity on if and only if
where and .
In recent work, Zhao (see [, Theorems 1 and 2]) proved the following results. For similar results in the half space with respect to the Schrödinger operator, we refer the reader to the papers by Ren and Su (see [9,10]).
Theorem BThe following statements are equivalent.
(I) A subsetHof is a-minimally thin at infinity on .
(II) There exists a positive superfunction on such that
(III) There exists a positive superfunction on such that even if for any , there exists satisfying .
Theorem CIf a subsetHof is a-minimally thin at infinity on , then we have
Remark From equation (3), we immediately know that equation (6) is equivalent to
This paper aims to show that the sharpness of the characterization of an a-minimally thin set in Theorem C. In order to do this, we introduce the Whitney cubes in a cone.
A cube is the form
where j, are integers. The Whitney cubes of are a family of cubes having the following properties:
(II) ( ).
Theorem 1IfHis a union of cubes from the Whitney cubes of , then equation (7) is also sufficient forHto be a-minimally thin at infinity with respect to .
From the Remark and Theorem 1, we have the following.
Corollary 1Let be a positive superfunction on such that equation (5) holds. Then we have
Corollary 2LetHbe a Borel measurable subset of satisfying
If is a non-negative superfunction on andcis a positive number such that for all , then for all .
To prove our results, we need some lemmas.
Lemma 1Let be a cube from the Whitney cubes of . Then there exists a constant cindependent ofksuch that
Proof If we apply a result of Long (see [, Theorem 6.1.3]) for compact set , we obtain a measure μ on , , such that
for any . Also there exists a positive measure on such that
for any .
Let , , be the center of , the diameter of , the distance between and , respectively. Then we have and . Then from equation (1) we have
for any . We can also prove that
for any and any . Hence we obtain
from equations (8), (9), (10), and (11). Since
from equations (3), (9), and (10), we have from (12)
we obtain from equation (13)
On the other hand, we have from equation (1)
which, together with equation (14), gives the conclusion of Lemma 1. □
3 Proof of Theorem 1
Let be a family of cubes from the Whitney cubes of such that . Let be a subfamily of such that , where .
Since is a countably subadditive set function (see [, p.49]), we have
for . Hence for we see from Lemma 1
which, together with equation (1), gives
for . Thus equations (15), (16), and (17) give
for . Finally we obtain from equation (1)
which shows with Theorem A that H is a-minimally thin at infinity with respect to .
The author declares that they have no competing interests.
This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102.
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