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.
Keywords:
Dirichlet problem; stationary Schrödinger equation; cone1 Introduction and results
Let R and
We introduce a system of spherical coordinates
The unit sphere and the upper half unit sphere in
For
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 wellknown estimates (see, e.g., [7], p. 14]) imply the following inequality:
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 [9]) that in the case under consideration the solutions to Equation (1.4) have the asymptotics
where
If
where
For a nonnegative integer m and two points
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 [12], Theorem 1] proved the following result.
Theorem AIfuis a continuous function on
Our first aim is to give growth properties at infinity for
Theorem 1Let
Ifuis a measurable function on
Next, we are concerned with solutions of the Dirichlet problem for the Schrödinger
operator on
Theorem 2Letγand
If we take
CorollaryIfuis a continuous function on
By using Corollary, we can give a solution of the Dirichlet problem for any continuous
function on
Theorem 3Ifuis a continuous function on
then
where
2 Lemmas
Throughout this article, let M denote various constants independent of the variables in questions, which may be different from line to line.
Lemma 1
for any
for any
Proof (2.1) and (2.2) are obtained by Kheyfits (see [10], Ch. 11]). (2.3) follows from Azarin (see [13], Lemma 4 and Remark]). □
Lemma 2 (see [1])
For a nonnegative integerm, we have
for any
Lemma 3 (see [2], Theorem 1])
If
Lemma 4Obviously, the conclusion of Lemma 3 holds true if (2.5) is replaced by
Proof Since
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.
References

Kheyfits, A: Dirichlet problem for the Schrödinger operator in a halfspace with boundary data of arbitrary growth at infinity. Diff. Integr. Equ.. 10, 153–164 (1997)

Kheyfits, A: Liouville theorems for generalized harmonic functions. Potential Anal.. 16, 93–101 (2002). Publisher Full Text

Reed, M, Simon, B: Methods of Modern Mathematical Physics, Academic Press, London (1970)

Rosenblum, G, Solomyak, M, Shubin, M: Spectral Theory of Differential Operators, VINITI, Moscow (1989)

Courant, R, Hilbert, D: Methods of Mathematical Physics, Interscience, New York (1953)

Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1977)

Simon, B: Schrödinger semigroups. Bull. Am. Math. Soc.. 7, 447–526 (1982). Publisher Full Text

Hartman, P: Ordinary Differential Equations, Wiley, New York (1964)

Escassut, A, Tutschke, W, Yang, CC: Some Topics on Value Distribution and Differentiability in Complex and Padic Analysis, Science Press, Beijing (2008)

Kheyfits, A: Representation of the analytic functions of infinite order in a halfplane. Izv. Akad. Nauk Armjan SSR Ser. Mat.. 6(6), 472–476 (1971)

Yoshida, H, Miyamoto, I: Solutions of the Dirichlet problem on a cone with continuous data. J. Math. Soc. Jpn.. 50(1), 71–93 (1998). Publisher Full Text

Azarin, VS: Generalization of a theorem of Hayman on subharmonic functions in an mdimensional cone. Trans. Am. Math. Soc.. 80(2), 119–138 (1969)

Ancona, A: First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains. J. Anal. Math.. 72, 45–92 (1997). Publisher Full Text