Dirichlet Problem for the Schrödinger Operator in a Half Space

and Applied Analysis 3 Denote ιj,k 2 − n ± √ n − 2 2 4k λj ) 2 ( j 0, 1, 2, 3 . . . ) . 1.6 Remark 1.1. ι j,0 j j 0, 1, 2, 3, . . . in the case a 0. It is known see 7 that in the case under consideration the solutions to 1.5 have the asymptotics Vj r ∼ d1r j,k , Wj r ∼ d2r − j,k , as r −→ ∞, 1.7 where d1 and d2 are some positive constants. If a ∈ Aa, it is known that the following expansion for the Green functionGa x, y see 8, Chapter 11 , 1, 9 Ga ( x, y ) ∞ ∑ j 0 1 χ′ 1 Vj ( min |x|, ∣y∣Wj ( max |x|, ∣y∣ ( vj ∑ v 1 φjv Θ φjv Φ ) , 1.8 where |x|/ |y| and χ′ 1 w W1 r , V1 r |r 1 is itsWronskian. The series converges uniformly if either |x| ≤ s|y| or |y| ≤ s|x| 0 < s < 1 . For a nonnegative integer m and two points x, y ∈ H, we put K a,m ( x, y ) { 0 if ∣y ∣∣ < 1, K̃ a,m ( x, y ) if 1 ≤ ∣y∣ < ∞, 1.9


Introduction and Results
Let R and R be the sets of all real numbers and of all positive real numbers, respectively.Let R n n ≥ 2 denote the n-dimensional Euclidean space with points x x , x n , where x x 1 , x 2 , . . ., x n−1 ∈ R n−1 and x n ∈ R. The unit sphere and the upper half unit sphere in R n are denoted by S n−1 and S n−1 , respectively.The boundary and closure of an open set D of R n are denoted by ∂D and D, respectively.The upper half space is the set H { x , x n ∈ R n : x n > 0}, whose boundary is ∂H.
For a set E, E ⊂ R ∪ {0}, we denote {x ∈ H; |x| ∈ E} and {x ∈ ∂H; |x| ∈ E} by HE and ∂HE, respectively.We identify R n with R n−1 × R and R n−1 with R n−1 × {0}, writing typical points x, y ∈ R n as x x , x n , y y , y n , where y y 1 , y 2 , . . ., y n−1 ∈ R n−1 , and putting For x ∈ R n and r > 0, let B x, r denote the open ball with center at x and radius r > 0 in R n .We will say that a set E ⊂ H has a covering {r j , R j } if there exists a sequence of balls {B j } with centers in H such that E ⊂ ∞ j 1 B j , where r j is the radius of B j and R j is the distance between the origin and the center of B j .
Let A a denote the class of nonnegative radial potentials a x , that is, 0 ≤ a x a |x| , x ∈ H, such that a ∈ L b loc H with some b > n/2 if n ≥ 4 and with b 2 if n 2 or n 3.This paper is devoted to the stationary Schr ödinger equation where x ∈ H, Δ is the Laplace operator and a ∈ A a .These solutions are called a-harmonic functions or generalized harmonic functions associated with the operator SSE.Note that they are classical harmonic functions in the case a 0. Under these assumptions the operator SSE can be extended in the usual way from the space C ∞ 0 H to an essentially self-adjoint operator on L 2 H see 1-3 .We will denote it by SSE as well.This last one has a Green function G a x, y .Here, G a x, y is positive on H and its inner normal derivative ∂G a x, y /∂n y ≥ 0. We denote this derivative by P a x, y , which is called the Poisson a-kernel with respect to H. We remark that G x, y and P x, y are the Green function and Poisson kernel of the Laplacian in H, respectively.
Let Δ * be a Laplace-Beltrami operator spherical part of the Laplace on the unit sphere.It is known see, e.g., 4, page 41 that the eigenvalue problem has the eigenvalues λ j j j n − 2 j 0, 1, 2 . . . .Corresponding eigenfunctions are denoted by ϕ jv 1 ≤ v ≤ v j , where v j is the multiplicity of λ j .We norm the eigenfunctions in L 2 S n−1 and ϕ 1 ϕ 11 > 0. Hence, well-known estimates see, e.g., 5, page 14 imply the following inequality: where the symbol M n denotes a constant depending only on n.Let V j r and W j r stand, respectively, for the increasing and nonincreasing, as r → ∞, solutions of the equation normalized under the condition V j 1 W j 1 1.We will also consider the class B a , consisting of the potentials a ∈ A a such that there exists a finite limit lim r → ∞ r 2 a r k ∈ 0, ∞ .Moreover, r −1 |r 2 a r − k| ∈ L 1, ∞ .If a ∈ B a , then solutions of 1.2 are continuous see 6 .
In the rest of paper, we assume that a ∈ B a , and we will suppress this assumption for simplicity.Further, we use the standard notations u max{u, 0}, u − − min{u, 0}, d is the integer part of d and d d {d}, where d is a positive real number.
It is known see 7 that in the case under consideration the solutions to 1.5 have the asymptotics where d 1 and d 2 are some positive constants.
If a ∈ A a , it is known that the following expansion for the Green function G a x, y see 8, Chapter 11 , 1, 9 where |x| / |y| and χ 1 For a nonnegative integer m and two points x, y ∈ H, we put where We introduce another function of

1.15
If these conditions all hold, we write For γ and u, we define the positive measure μ resp., ν on R n by 1.17 We remark that the total mass of μ and ν is finite.Let > 0 and ξ ≥ 0, and let μ be any positive measure on R n having finite mass.For each x x , x n ∈ R n , the maximal function is defined by

1.20
Our first aim is to give the growth properties at infinity for U a, m; u x .
, n and u is a measurable function on ∂H satisfying 1.16 , then there exists a covering where γ is a real number, for this γ and u, we define Obviously, the total mass of μ is also finite.
If we take a 0 in Theorem 1.2, then we immediately have the following growth property based on 1.5 and Remark 1.1.
If u is defined as previously, then the function U 0, m; u x is a harmonic function on H and there exists a covering Remark 1.4.In the case ζ n, 1.26 is a finite sum, and the set E ; μ , 0 is a bounded set and 1.27 holds in H.
Next we are concerned with solutions of the Dirichlet problem for the Schr ödinger operator on H.For related results, we refer the readers to the paper by Kheyfits 1 .
If we take ι γ ,k {γ} ι m 1,k n − 1, then we immediately have the following corollary, which is just Theorem A in the case a 0.

Lemmas
Throughout this paper, let M denote various constants independent of the variables in questions, which may be different from line to line.
Lemma 2.3.Let > 0 and ξ ≥ 0, and let μ be any positive measure on R n having finite total mass.Then, E ; μ, ξ has a covering {r j , R j } j 1, 2, . . .satisfying Proof.Set If x ∈ E j ; μ, ξ , then there exists a positive number ρ x such that Here, E j ; μ, ξ can be covered by the union of a family of balls {B x j,i , ρ j,i : x j,i ∈ E j ; μ, ξ } ρ j,i ρ x j,i .By the Vitali lemma see 13 , there exists Λ j ⊂ E j ; μ, ξ , which is at most countable, such that {B x j,i , ρ j,i : x j,i ∈ Λ j } are disjoint and E j ; μ, ξ ⊂ On the other hand, note that x j,i ∈Λ j B x j,i , ρ j,i ⊂ {x : 2 j−1 ≤ |x| < 2 j 2 }, so that

2.10
Hence, we obtain

Proof of Theorem 1.2
We only prove the case γ ≥ 0, the remaining case γ < 0 can be proved similarly.
For any > 0, there exists The relation G a x, y ≤ G x, y implies this inequality see 14 P a x, y ≤ P x, y .

3.2
For any fixed point

3.5
Next, we will estimate U a,4 x .Take a sufficiently small positive number d 3 such that I 4 ⊂ B x, 1/2 |x| for any x ∈ Π d 3 , where and divide H into two sets Π d 3 and

3.7
We will consider the case x ∈ Π d 3 .Now put x − y n dy , 3.9 where i x is a positive integer satisfying 2 i x −1 δ x ≤ |x|/2 < 2 i x δ x .

3.13
We only consider U a,6 x in the case m ≥ 1, since U a,6 x ≡ 0 for m 0. By the definition of K a, m , 1.4 , and 2.2 , we see that where W j y u y y dy .

3.19
Analogous to the estimate of q j |x| , we have Thus, we can conclude that Combining 3.5 -3.22 , we obtain that if R is sufficiently large and is sufficiently small, then U a, m; u Finally, there exists an additional finite ball B 0 covering H 0, R , which together with Lemma 2.3 gives the conclusion of Theorem 1.2.

Proof of Theorem 1.5
For any fixed x ∈ H, take a number satisfying R > max{1, 2|x|}.By ι m 1,k ≥ ι γ ,k {γ} − n 1, 1.5 , 1.16 , and 2.2 , we have for any y ∈ ∂H from the arbitrariness of l.Finally, 1.29 and 1.30 follow from 1.22 and 1.23 , respectively, in the case ζ n.Thus, we complete the proof of Theorem 1.5.

Proof of Theorem 1.7
From Corollary 1.6, we have the solution U a, m; u x of the Dirichlet problem on H with u satisfying 1.31 .Consider the function h x − U a, m; u x .Then, it follows that this is a solution of 1.

Corollary 1 . 6 .Theorem 1 . 7 . 1 d
If u is a continuous function on ∂H satisfying an application of Corollary 1.6, we can give a solution of the Dirichlet problem for any continuous function on ∂H.If u is a continuous function on ∂H satisfying 1.31 and h x is a solution of the Dirichlet problem for the Schrödinger operator on H with u satisfying lim |x| → ∞,x∈H |x| −ι m 1,k h x jv ϕ jv Θ V j |x| , 1.34 where x ∈ H and d jv are constants.
then there exists a positive d 3 such that |x−y | ≥ d 3 |x| for any y ∈ ∂H, and hence

4 . 1
Then, U a, m; u x is absolutely convergent and finite for any x ∈ H. Thus U a, m; u x is a solution of 1.2 on H. Now we study the boundary behavior of U a, m; u x .Let y ∈ ∂H be any fixed point and l any positive number satisfying l > max{|y | 1, 1/2 R}.Set χ S l as the characteristic function of S l {y ∈ ∂H, |y | ≤ l}, and writeU a, m; u x U x − U x U x , Notice that U x is the Poisson a-integral of u y χ S 2l , We have lim x → y , x∈H U x u y .Since lim Θ → Φ ϕ jv Θ 0 j 1, 2, 3 . . .; 1 ≤ v ≤ v j as x → y ∈ ∂H, we have lim x → y , x∈H U x0 from the definition of the kernel function K a, m x, y .U x O |x| ι γ ,k {γ}−n 1 ϕ 1 Θ and therefore tends to zero.So the function U a, m; u x can be continuously extended to H such that lim

5 . 2 from 1 .
2 in H and vanishes continuously on ∂H.Since 0 ≤ h − U Ω, a, m; u x ≤ h x U a, m; u − x 5.1 for any x ∈ H, we have lim |x| → ∞,x∈H |x| −ι m 1,k h − U Ω, a, m; u x 0 32 and 1.33 .Then, the conclusion of Theorem 1.7 follows immediately from Lemma 2.2.
We remark that the kernel function P 0, m x, y coincides with ones in Finkelstein and Scheinberg 10 and Siegel and Talvila 11 see 8, Chapter 11 .
y .1.11 The generalized Poisson kernel P a, m x, y with respect to H is defined by ∂H P a, m x, y u y dy , 1.14 where u y is a continuous function on ∂H.If γ is a real number and γ denoted by E ; μ, ξ .About classical solutions of the Dirichlet problem for the Laplacian, Siegel and Talvila cf.11, Corollary 2.1 proved the following result.