Mathematics and Statistics School, Henan University of Science and Technology, No. 263, Luo-Long District, Kai-Yuan Road, Luoyang City, Henan Province, 471023, China

College of Civil Engineering and Architecture, Zhejiang University, B505 Anzhong Building, 866 Yuhangtang Road, Hangzhou, Zhejiang Province, 310058, China

Abstract

A family of Schrödinger operators,

1 Introduction

In this paper, we consider a family of Schrödinger operators

on

Here

If (2) holds, then

Under the assumption on

In order to use the Birman-Schwinger technique to

Here is the plan of our work. In Section 2, we recall some results of

Let us introduce some notations first.

**Notation** The scalar product on

2 Some results for

Assume that

implies

in

Now, we recall some results on the resolvent and the Schrödinger group for the unperturbed operator

Denote

For

Let

Denote for

Here

The following is the asymptotic expansion for the resolvent

**Theorem 2.1** (Theorem 2.2

First, we show that

From the assumption on

when

To study the eigenvalues of

for

Then we have the following result.

**Proposition 2.2**

(a)

(b)

with

It follows that

Next, we show that

And if

It follows that

(b) From (a), one has

From the last proposition, we know that there exists one-to-one correspondence between the discrete eigenvalues of

3 Asymptotic expansion of the eigenvalues

If

**Proposition 3.1**

**Proposition 3.2**

**Proposition 3.3**

and if

**Lemma 3.4**

This ends the proof. □

**Lemma 3.5**

(a)

(b)

It follows that

(b) Because

Then

Let

and

In the last equality, we use the fact

and

It follows that

Let

Here,

Then, by the min-max principle,

Then we have the following.

**Lemma 3.6**

Then

Since

then

Then

In particular,

In the last step, we use that

Similarly, we can get

and

□

First, we study the asymptotic expansion of the smallest eigenvalue

**Lemma 3.7**

In the last equality, we use the fact that

with

**Theorem 3.8**

(a)

(b)

(c)

(a) By Theorem 2.1, one has

Then if

Here

In the last equality, we use the fact

Hence,

It follows

So,

Since

(b) If

By Lemma 3.7, one has

Then we have

with

one has

with

(c) If

By Lemma 3.7, we know that

with

with

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

This research is supported by the Natural Science Foundation of China (11101127,11271110) and the Natural Science Foundation of Educational Department of Henan Province (2011B110014).