In this article, a solution of the Dirichlet problem for the Schrödinger operator on a cone is constructed by the generalized Poisson integral with a slowly growing continuous boundary function. A solution of the Poisson integral for any continuous boundary function is also given explicitly by the Poisson integral with the generalized Poisson kernel depending on this boundary function.
MSC: 31B05, 31B10.
Keywords:Dirichlet problem; stationary Schrödinger equation; cone
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 the n-dimensional Euclidean space by (). A point in is denoted by , where . 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.
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 .
For and , let denote an open ball with a center at P and radius r in . . By , we denote the set in with the domain Ω on . We call it a cone. We denote the sets and with an interval on R by and . By we denote . By we denote which is . We denote the -dimensional volume elements induced by the Euclidean metric on by .
This article is devoted to the stationary Schrödinger equation
where , Δ is the Laplace operator and . These solutions called a-harmonic functions or generalized harmonic functions are associated with the operator . Note that they are (classical) harmonic functions in the case . Under these assumptions, the operator can be extended in the usual way from the space to an essentially self-adjoint operator on (see [1-3]). We will denote it as well. This last one has a Green’s function . Here is positive on and its inner normal derivative . We denote this derivative by , which is called the Poisson a-kernel with respect to . We remark that and are the Green’s function and Poisson kernel of the Laplacian in respectively.
Let be a Laplace-Beltrami operator (the spherical part of the Laplace) on and () be the eigenvalues of the eigenvalue problem for on Ω (see, e.g., , p. 41])
for (see Courant and Hilbert ), where .
In order to ensure the existences of (). 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). Then () and on ∂Ω (here and below, denotes differentiation along the interior normal).
Hence well-known estimates (see, e.g., , p. 14]) imply the following inequality:
We shall also consider the class , consisting of the potentials such that there exists a finite limit ; moreover, . If , then the solutions of Equation (1.1) are continuous (see ).
In the rest of the article, we assume that and we shall suppress this assumption for simplicity. Further, we use the standard notations , , is the integer part of d and , where d is a positive real number.
It is known (see ) that in the case under consideration the solutions to Equation (1.4) have the asymptotics
With regard to classical solutions of the Dirichlet problem for the Laplacian, Yoshida and Miyamoto , Theorem 1] proved the following result.is a classical solution of the Dirichlet problem onwithgand satisfies
Throughout this article, let M denote various constants independent of the variables in questions, which may be different from line to line.
Lemma 2 (see )
For a nonnegative integerm, we have
Lemma 3 (see , Theorem 1])
Lemma 4Obviously, the conclusion of Lemma 3 holds true if (2.5) is replaced by
from (1.5) and
(2.6) gives that (2.5) holds, from which the conclusion immediately follows. □
3 Proof of Theorem 1
The relation implies this inequality (see )
By (2.3) and (3.2), we consider the inequality
We first have
Since we see from (1.2)
Thus we can conclude that
4 Proof of Theorem 2
5 Proof of Theorem 3
From Corollary, we have the solution of the Dirichlet problem on with u satisfying (1.9). Consider the function . Then it follows that this is the solution of Equation (1.1) in and vanishes continuously on .
from (1.10) and (1.11). Then the conclusions of Theorem 3 follow immediately from Lemma 4.
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
The authors would like to thank anonymous reviewers for their valuable comments and suggestions about improving the quality of the manuscript. This work is supported by The National Natural Science Foundation of China under Grant 11071020 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20100003110004.
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