Boundary behaviors of modified Green’s function with respect to the stationary Schrödinger operator and its applications

In this paper, we construct a modified Green’s function with respect to the stationary Schrödinger operator on cones. As applications, we not only obtain the boundary behaviors of generalized harmonic functions but also characterize the geometrical properties of the exceptional sets with respect to the Schrödinger operator.


Introduction and results
Let R and R + be the set of all real numbers and the set of all positive real numbers, respectively. We denote by R n (n ≥ ) the n-dimensional Euclidean space. A point in R n is denoted by P = (X, x n ), X = (x  , x  , . . . , x n- ). The Euclidean distance between two points P and Q in R n is denoted by |P -Q|. Also |P -O| with the origin O of R n is simply denoted by |P|. The boundary and the closure of a set S in R n are denoted by ∂S and S, respectively. We introduce a system of spherical coordinates (r, ), = (θ  , θ  , . . . , θ n- ), in R n which are related to Cartesian coordinates (x  , x  , . . . , x n- , x n ) by x n = r cos θ  .
The unit sphere and the upper half unit sphere in R n are denoted by S n- and S n- + , respectively. For simplicity, a point (, ) on S n- and the set { ; (, ) ∈ } for a set , ⊂ S n- , are often identified with and , respectively. For two sets ⊂ R + and ⊂ S n- , the set {(r, ) ∈ R n ; r ∈ , (, ) ∈ } in R n is simply denoted by × . In particular, the half space R + × S n- + = {(X, x n ) ∈ R n ; x n > } will be denoted by T n . For P ∈ R n and r > , let B(P, r) denote the open ball with center at P and radius r in R n . S r = ∂B(O, r). By C n ( ), we denote the set R + × in R n with the domain on S n- . We call it a cone. Then T n is a special cone obtained by putting = S n- + . We denote the sets I × and I × ∂ with an interval on R by C n ( ; I) and S n ( ; I). By S n ( ; r) we denote C n ( ) ∩ S r . By S n ( ) we denote S n ( ; (, +∞)), which is ∂C n ( ) -{O}.
We shall say that a set E ⊂ C n ( ) has a covering {r j , R j } if there exists a sequence of balls {B j } with centers in C n ( ) such that E ⊂ ∞ j= 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 non-negative radial potentials a(P), i.e.  ≤ a(P) = a(r), P = (r, ) ∈ C n ( ), such that a ∈ L b loc (C n ( )) with some b > n/ if n ≥  and with b =  if n =  or n = .

R E T R
This article is devoted to the stationary Schrödinger equation Sch a u(P) =u(P) + a(P)u(P) =  for P ∈ C n ( ), where is the Laplace operator and a ∈ A a . These solutions are called generalized harmonic functions (associated with the operator Sch a ). Note that they are (classical) harmonic functions in the case a = . Under these assumptions the operator Sch a can be extended in the usual way from the space C ∞  (C n ( )) to an essentially self-adjoint operator on L  (C n ( )) (see []). We will denote it Sch a as well. The latter has a Green-Sch function G( ; a)(P, Q). Here G( ; a)(P, Q) is positive on C n ( ) and its inner normal derivative ∂G( ; a)(P, Q)/∂n Q ≥ . We denote this derivative by PI( ; a)(P, Q), which is called the Poisson kernel with respect to the stationary Schrödinger operator. We remark that G( ; )(P, Q) and PI( ; )(P, Q) are the Green's function and Poisson kernel of the Laplacian in C n ( ), respectively.
Hence the well-known estimates (see, e.g., [], p.) imply the following inequality: where the symbol M(n) denotes a constant depending only on n.
Let V j (r) (j = , , , . . .) and W j (r) (j = , , , . . .) stand, respectively, for the increasing and non-increasing, as r → +∞, solutions of the equation We shall also consider the class B a , consisting of the potentials a ∈ A a such that there exists a finite limit lim r→∞ r  a(r) = k ∈ [, ∞), moreover, r - |r  a(r) -k| ∈ L(, ∞). If a ∈ B a , then the g.h.f.s. are continuous (see []).
In the rest of this paper, we assume that a ∈ B a and we shall suppress this assumption for simplicity. Further, we use the standard notations u + = max(u, ), u -= -min(u, ),

R E T R
It is well known (see []) that in the case under consideration the solutions to equation (.) have the asymptotics where d  and d  are some positive constants.
If a ∈ A a , it is well known that the following expansion holds for the Green's function G( ; a)(P, Q) (see [], Chapter ): The expansion (.) can also be rewritten in terms of the Gegenbauer polynomials. For a non-negative integer m and two points P = (r, ), Q = (t, ) ∈ C n ( ), we put If we modify the Green's function with respect to the stationary Schrödinger operator on cones as follows: for two points P = (r, ), Q = (t, ) ∈ C n ( ), then the modified Poisson kernel with respect to the stationary Schrödinger operator on cones can be defined by We remark that PI( ; a, )(P, Q) = PI( ; a)(P, Q).
In this paper, we shall use the modified Poisson integrals with respect to the stationary Schrödinger operator defined by is a continuous function on ∂C n ( ) and dσ Q is the surface area element on S n ( ).
If γ is a real number and γ ≥  (resp. γ < ), we assume in addition that  ≤ p < ∞, and in the case p = , If these conditions all hold, we write γ ∈ C (k, p, m, n) (resp. γ ∈ D(k, p, m, n)).
We remark that the total masses of μ and ν are finite. Let p > -, > ,  ≤ ζ ≤ np and μ be any positive measure on R n having finite mass. For each P = (r, ) ∈ R n -{O}, the maximal function with respect to the stationary Schrödinger operator is defined by (see []) The set [], Theorem ) gave the asymptotic behavior of PI  (m, u)(P) at infinity on cones.

Theorem A If u is a continuous function on ∂C n ( ) satisfying
Now we have the following.
Theorem  If p > -, γ ∈ C (k, p, m, n) (resp. γ ∈ D(k, p, m, n)) and u is a measurable function on ∂C n ( ) satisfying (.), then there exists a covering {r j , R j } of E( ; μ, ζ ) (resp. E( ; ν, ζ )) (⊂ C n ( )) satisfying such that As an application of modified Green's function with respect to the stationary Schrödinger operator and Theorem , we give the solutions of the Dirichlet problem for the Schrödinger operator on C n ( ).

Theorem  If u is a continuous function on
then the function PI a (m, u)(P) satisfies

A R T I C L E 2 Lemmas
Throughout this paper, let M denote various constants independent of the variables in questions, which may be different from line to line.

A R T I C L E
On the other hand, note that P j,i ∈ j B(P j,i , ρ j,i ) ⊂ (P = (r, ) :  j- ≤ r <  j+ ), so that Hence we obtain where B(P  , ) (P  = (, , . . . , ) ∈ R n ) is the ball which covers {P = (r, ) ∈ R n ; r < }.

Proof of Theorem 1
We only prove the case p > - and γ ≥ , the remaining cases can be proved similarly. For any > , there exists R >  such that We first have U  (P) = Mϕ  ( ) which is similar to the estimate of U  (P). Next, we shall estimate U  (P).
If P = (r, ) ∈ C n ( ) -(c), then there exists a positive c such that |P -Q| ≥ c r for any Q ∈ S n ( ), and hence U  (P) ≤ Mϕ  ( ) which is similar to the estimate of U  (P). We shall consider the case P = (r, ) ∈ (c). Now put where i(P) is a positive integer satisfying  i(P)- δ(P) ≤ r  <  i(P) δ(P). Since rϕ  ( ) ≤ Mδ(P) (P = (r, ) ∈ C n ( )), similar to the estimate of U  (P), we obtain For any fixed P = (r, ) ∈ C n ( ), take a number satisfying R > max(, r s ) ( < s <   ). By (.) and Lemma , we have Then PI a (m, u)(P) is absolutely convergent and finite for any P ∈ C n ( ). Thus PI a (m, u)(P) is a generalized harmonic function on C n ( ). Now we study the boundary behavior of PI a (m, u)(P). Let Q = (t , ) ∈ ∂C n ( ) be any fixed point and l be any positive number satisfying l > max(t + ,   R).