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RETRACTED ARTICLE: Minimally thin sets associated with the stationary Schrödinger operator
Journal of Inequalities and Applications volume 2014, Article number: 67 (2014)
This paper gives some new criteria for a-minimally thin sets at infinity with respect to the Schrödinger operator in a cone, which supplement the results obtained by Long-Gao-Deng.
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 O the origin of is simply denoted by . The boundary and the closure of a set S in are denoted by ∂S and , respectively.
We introduce a system of spherical coordinates , , in which are related to Cartesian coordinates by .
Let D be an arbitrary domain in and denote the class of nonnegative radial potentials , i.e. , , such that with some if and with if or .
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 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 it values belong to the interval and at each point with the generalized mean-value inequality (see )
is 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 superfunctions (with respect to the Schrödinger operator ). If a function u is both subfunction and superfunction, it is, clearly, continuous and is called an a-harmonic function (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 . Throughout this paper, let c denote various positive constants, because we do not need to specify them.
Let Ω be a domain on with smooth boundary. Consider the Dirichlet problem
where is the spherical part of the Laplace operator
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 .
We study solutions of an ordinary differential equation,
and W is monotonically decreasing with
We will also consider the class , consisting of the potentials such that the finite limit exists, and moreover, . If , then the (sub)superfunctions are continuous (see ). In the rest of this paper, we assume that and we shall suppress this assumption for simplicity.
It is well known that the Martin boundary of is the set , each point 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 , Long-Gao-Deng 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 ), which generalized earlier notations obtained by Brelot and Miyamoto (see [7, 8]). 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 (see [, Theorem 2]) that there exists a unique positive measure on such that (see [, p.6])
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
Recently, Long-Gao-Deng (see [, Theorem 2.5]) gave a criterion that characterizes a-minimally thin sets at infinity in a cone.
Theorem A A subset H of is a-minimally thin at infinity on if and only if
where and .
In this paper, we shall obtain a series of new criteria for a-minimally thin sets at infinity on , which complemented Theorem A by the way completely different from theirs. Our results are essentially based on Ren and Su (see [9, 10]).
First we have the following.
Theorem 1 The following statements are equivalent.
A subset H of is a-minimally thin at infinity on .
There exists a positive superfunction on such that(5)
There exists a positive superfunction on such that even if for any , there exists satisfying .
Next we shall state Theorem 2, which is the main result in this paper.
Theorem 2 If a subset H of is a-minimally thin at infinity on , then we have
In our discussions, the following estimate for the Green a-potential is fundamental, as follows from .
for any and any satisfying .
Lemma 2 If H is a bounded Borel subset of , then
Proof For any and any positive number , there exists a positive constant such that
from [, p.178], where Cap denotes the Newtonian capacity. Then there exists a positive constant c depending only on and n such that
for every (see [, Theorem 2]).
It is well known that the Green a-energy also can be represented as (see [, p.57])
From equation (1) and Lemma 1 we have
equipped with the norm
and further , where denotes the closure of in .
Since quasi everywhere on H and hence a.e. on H, we have from equation (7)
which gives the conclusion of Lemma 2. □
3 Proof of Theorem 1
We shall show that (II) follows from (I). Since
for any and is concentrated on , we have
for any and hence from Lemma 1
for any and any integer j satisfying .
If we define a measure μ on by
From equation (9), (I), and Theorem A, we know that is a finite superfunction on and
for any () and from Lemma 1
for any and
If we set , where
and , then
and is equal to H except a polar set . If we define a positive measure η on such that is identically +∞ on and define a measure ν on by , then
If we put , then this shows that is the function required in (II).
Now we shall show that (III) follows from (II). Let be the function in (II). It follows that for any . On the other hand, from equation (5) we can find a point such that . Therefore satisfies (III) with .
Finally, we shall prove that (I) follows from (III). Let be the function in (III). If we put
then we have
where is a positive constant depending only on ∞ and v. Since there exists satisfying , we note that . Now we obtain for any . Hence by a result of [, p.69], H is a-minimally thin at infinity on with respect to the Schrödinger operator, which is the statement of (I). Thus we complete the proof of Theorem 1.
4 Proof of Theorem 2
First of all, we remark that
where is the set in equation (10) and is the n-dimensional Lebesgue measure of .
By using Lemma 2, we obtain
If H is a-minimally thin at infinity on , then from Theorem A, equations (3), (11), and (12), we have
which is the conclusion of Theorem 2.
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This work was supported by the National Natural Science Foundation of China under Grants Nos. 11301140 and U1304102. The author would like to thank two anonymous referees for numerous insightful comments and suggestions, which have greatly improved the paper.
The author declares that there is no conflict of interests regarding the publication of this article.
This article has been retracted. Please see the retraction notice for more detail: https://doi.org/10.1186/s13660-021-02575-1
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Zhao, T. RETRACTED ARTICLE: Minimally thin sets associated with the stationary Schrödinger operator. J Inequal Appl 2014, 67 (2014). https://doi.org/10.1186/1029-242X-2014-67
- minimally thin set
- Schrödinger operator
- Green a-potential