Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou, 450002, P.R. China

School of Mathematical Science, Laboratory of Mathematics and Complex Systems, MOE Beijing Normal University, Beijing, 100875, P.R. China

Abstract

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.

1 Introduction and results

Let **R** and **S** in **S** and

We introduce a system of spherical coordinates

The unit sphere and the upper half unit sphere in

For **R** by

Let

This article is devoted to the stationary Schrödinger equation

where

Given a domain

for every

Let

Corresponding eigenfunctions are denoted by

for

In order to ensure the existences of

Hence well-known estimates (see, e.g.,

where the symbol

Let

normalized under the condition

We shall also consider the class

In the rest of the article, we assume that

Denote

It is known (see

where

If

where

For a nonnegative integer

where

We introduce another function of

The generalized Poisson kernel

In fact,

We remark that the kernel function

Put

where

With regard to classical solutions of the Dirichlet problem for the Laplacian, Yoshida and Miyamoto

**Theorem A**

Our first aim is to give growth properties at infinity for

**Theorem 1**

Next, we are concerned with solutions of the Dirichlet problem for the Schrödinger operator on

**Theorem 2**

If we take

**Corollary**

By using Corollary, we can give a solution of the Dirichlet problem for any continuous function on

**Theorem 3**

2 Lemmas

Throughout this article, let

**Lemma 1**

**Lemma 2** (see

**Lemma 3** (see

**Lemma 4**

from (1.5) and

(2.6) gives that (2.5) holds, from which the conclusion immediately follows. □

3 Proof of Theorem 1

We only prove the case

For any

The relation

For

where

By

We obtain by

By (2.3) and (3.2), we consider the inequality

where

We first have

which is similar to the estimate of

Next, we shall estimate

and divide

If

which is similar to the estimate of

We shall consider the case

Since

where

Since we see from (1.2)

for

for

So

We only consider

where

To estimate

where

Notice that

Thus, by

Analogous to the estimate of

Thus we can conclude that

which yields

By

Combining (3.3)–(3.11), we obtain that if

4 Proof of Theorem 2

For any fixed

Thus

Now we study the boundary behavior of

Set

where

Notice that

So the function

for any

5 Proof of Theorem 3

From Corollary, we have the solution

Since

for any

from (1.10) and (1.11). Then the conclusions of Theorem 3 follow immediately from Lemma 4.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

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.