We give some criteria for

This paper is concerned with some properties for the generalized subharmonic functions associated with the stationary Schrödinger operator. More precisely the minimally thin sets and rarefied sets about these generalized subharmonic functions will be studied. The research on minimal thinness has been exploited a little and attracted many mathematicians. In 1949 Lelong-Ferrand [

To state our results, we will need some notations and preliminary results. As usual, denote by

For

Relative to the system of spherical coordinates, the Laplace operator

Let

For the identical operator

Suppose that a function

Denote by

For simplicity, the point

Let

Let

An important role will be played by the solutions of the ordinary differential equation

Denote

If

We recall that

The remainder of the paper is organized as follows: in Section

In this section, we will state our main results. Before passing to our main results, we need some definitions.

Martin introduced the so-called Martin functions associated with the Laplace operator (see Brelot [

It is well known that the Martin boundary

Let

If

Now we can state our main theorems.

Let

If

Let

there exists a positive superfunction

there is an

A set

Let

Let

there is an

there is an

Let

The generalized Green energy

A subset

A subset

A subset

When

Set

Let

From the definition of

Let

Let

A cone

Let

In our arguments we need the following results.

Let

If

If

If

Since

Consider

Let

Let

Let

Let

By the Riesz decomposition theorem, we have a unique measure

Since

Let

Since

To see (

To prove (

If

Following the same method of Armitage and Gardiner [

First we assume that (b) holds, and let

Next we assume (a) holds, and let

Finally we assume (c) holds, then there is a Martin topology neighborhood

Obviously we see that (c) implies (b). If (b) holds, then there exist

Finally we assume (a) holds. By Lemma

Clearly (c) implies (b). To prove that (b) implies (a), we suppose that (b) holds and choose

Next we suppose that (a) holds. By Lemma

Since (

When

By applying the Riesz decomposition theorem to the superfunction

Since the measure

Next we will prove the sufficiency. Since

Let a subset

We see from (

Suppose that a subset

By Lemma

Since

The authors wish to express their appreciation to the referee for her or his careful reading and some useful suggestions which led to an improvement of their original paper. The work is supported by SRFDP (No. 20100003110004) and NSF of China (No. 10671022 and No. 11101039).