Dirichlet problem for the Schrödinger operator on a cone

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.

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.

We introduce a system of spherical coordinates , , in which are related to Cartesian coordinates by .

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 .

Let denote the class of nonnegative radial potentials , i.e., , , such that with some if and with if or .

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.

Given a domain and a continuous function u on , we say that h is a solution of the Dirichlet problem for the Schrödinger operator on D with u if in D and

for every . Note that h is a solution of the classical Dirichlet problem for the Laplacian in the case .

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., [4], p. 41])

Corresponding eigenfunctions are denoted by (), where is the multiplicity of . We set , norm the eigenfunctions in and . Then there exist two positive constants and such that

for (see Courant and Hilbert [5]), 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 [6], 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., [7], p. 14]) imply the following inequality:

where the symbol denotes a constant depending only on n.

Let and stand, respectively, for the increasing and nonincreasing, as , solutions of the equation

normalized under the condition .

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 [8]).

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.

Denote

It is known (see [9]) that in the case under consideration the solutions to Equation (1.4) have the asymptotics

where and are some positive constants.

If , it is known that the following expansion for the Green function (see [10], Ch. 11], [1,11])

where and , is their Wronskian. The series converges uniformly if either or ().

For a nonnegative integer m and two points , , we put

where

We introduce another function of and

The generalized Poisson kernel (, ) with respect to is defined by

In fact,

We remark that the kernel function coincides with the one in Yoshida and Miyamoto [12] (see [10], Ch. 11]).

Put

where is a continuous function on and is a surface area element on .

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 onsatisfying

thenis a classical solution of the Dirichlet problem onwithgand satisfies

Our first aim is to give growth properties at infinity for .

Theorem 1Let (resp. ), (resp. ) and

Ifuis a measurable function onsatisfying

then

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

Theorem 2Letγandbe as in Theorem 1. Ifuis a continuous function onsatisfying (1.6), thenis a solution of the Dirichlet problem for the Schrödinger operator onwithuand (1.7) (resp. (1.8)) holds.

If we take , then we immediately have the following corollary, which is just Theorem A in the case .

CorollaryIfuis a continuous function onsatisfying

thenis a solution of the Dirichlet problem for the Schrödinger operator onwithuand satisfies

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

Theorem 3Ifuis a continuous function onsatisfying (1.9) andis a solution of the Dirichlet problem for the Schrödinger operator onwithusatisfying

then

whereandare constants.

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 anyand anysatisfying (resp. );

for anyand 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 anyandsatisfying (), whereis a constant dependent ofn, mands.

Lemma 3 (see [2], Theorem 1])

Ifis a solution of Equation (1.1) onsatisfying

then

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. □

We only prove the case , the remaining case can be proved similarly.

For any , there exists such that

The relation implies this inequality (see [14])

For and any fixed point satisfying , let , , , , , and , we write

where

By , (1.6), (2.1) and (3.1), we have the following growth estimates

We obtain by , (2.2) and (3.1)

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 . Take a sufficiently small positive number such that for any , where

and divide into two sets and .

If , then there exists a positive such that for any , and hence

which is similar to the estimate of .

We shall consider the case . Now put

Since , we have

where is a positive integer satisfying .

Since we see from (1.2)

for . Similar to the estimate of , we obtain

for .

So

We only consider in the case , since for . By the definition of , (1.3) and Lemma 2, we see

where

To estimate , we write

where

Notice that

Thus, by , (1.5) and (1.6) we conclude

Analogous to the estimate of , we have

Thus we can conclude that

which yields

By , (1.5), (2.4) and (3.1) we have

Combining (3.3)–(3.11), we obtain that if is sufficiently large and ϵ is sufficiently small, then as , where . Then we complete the proof of Theorem 1.

For any fixed , take a number satisfying (). By , (1.4), (1.6) and (2.4), we have

Thus is finite for any . Since is a generalized harmonic function of for any fixed , is also a generalized harmonic function of . That is to say, is a solution of Equation (1.1) on .

Now we study the boundary behavior of . Let be any fixed point and l be any positive number satisfying .

Set is a characteristic function of and write

where

Notice that is the Poisson a-integral of , we have . Since (; ) as , we have from the definition of the kernel function . , and therefore tends to zero.

So the function can be continuously extended to such that

for any from the arbitrariness of l. Thus we complete the proof of Theorem 2 from Theorem 1.

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 .

Since

for any , we have

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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