Asymptotic Behaviors of the Eigenvalues of Schrödinger Operator with Critical Potential

Abstract

We study the asymptotic behaviors of the discrete eigenvalue of Schrödinger operator with We obtain the leading terms of discrete eigenvalues of when the eigenvalues tend to 0. In particular, we obtain the asymptotic behaviors of eigenvalues when has singularity at .

1. Introduction

This paper is devoted to the study of the asymptotic behaviors of the eigenvalues of a class of Schrödinger operators. This problem is related to low-energy spectral analysis for Schrödinger operators, which has been studied in many works (see [1–7]). References [1–3] are concerned with perturbation of a constant elliptic differential operator by a term decaying like , , as . The spectral analysis of Schrödinger operators with potentials of critical decay (decaying like , as ) is studied in [4–7]. The complex potentials are considered in [7].

The asymptotic behaviors of the eigenvalues of Schrödinger operator have been studied by many authors. In [8], Klaus and Simon got the leading term of the eigenvalue of Schrödinger operator with fast decaying potential. The asymptotic behavior of the eigenvalue of Schrödinger operator with periodic potential has been studied by Fassari and Klaus [9]. We have studied the asymptotic behavior of the smallest eigenvalue of Schrödinger operator with potential of critical decay [6]. In this paper, we will consider the asymptotic behavior of all other eigenvalues. The main tool we used in this paper is Birman-Schwinger kernel which was originated in the seventies. But Birman-Schwinger kernel is still a very important tool for the spectrum problem of Schrödinger operators (see [10–14]).

In this paper, we want to study the asymptotic behavior of discrete eigenvalue of Schrödinger operator , . Here, . is the polar coordinate on , and is a real continuous function.

The assumptions used in this paper are as follows. Suppose that is a nonzero real continuous function and satisfies Here, . The assumption on is that Here, denotes Laplace operator on the sphere . If (3) holds, then in (see [5]).

In Section 2 [6], it is shown that, under the assumption on , has discrete eigenvalues when is large enough, and each discrete eigenvalue tends to zero at some . Set If , then the Birman-Schwinger kernel is a bounded operator for (see [6]). Since there is a one-to-one correspondence between the eigenvalues of and the eigenvalues of , the asymptotic expansion of the smallest eigenvalue of has been obtained through the asymptotic expansion of the eigenvalue of in [6], if (2) holds with . In this paper, we will get the asymptotic behaviors of all discrete eigenvalues of in the case of . The eigenfunction corresponding to the smallest eigenvalue is a positive function; we can obtain the leading term of the smallest eigenvalue easily. The eigenfunctions corresponding to all other eigenvalues may not be positive. Therefore, it is much more difficult to obtain the leading term of the eigenvalue.

If , has singularity at (see [5]). Thus, the Birman-Schwinger kernel also has singularity at . The other goal of this paper is to obtain the asymptotic behavior of discrete eigenvalue when . The main difficulty of this situation is the singularity of . In Section 4, the discrete eigenvalue of will be studied through the operator for some fixed . Finally, we obtain the leading term of discrete eigenvalue of , when the assumption (2) and (3)with .

The plan of this work is as follows. In Section 2, we recall some known results for , especially the asymptotic expansion of for near 0. We obtain the asymptotic behavior of discrete eigenvalue of for the case of in Section 3. Section 4 concentrates on the asymptotic expansion of discrete eigenvalue in the case of .

Let us introduce some notations first.

Notation 1. The scalar product on and is denoted by and that on by . ,,, denotes the weighted Sobolev space of order with volume element . The duality between and is identified with product. Denote . Notation stands for the space of continuous linear operators from to . The complex plans is slit along positive real axis so that and with are holomorphic there.

2. Some Results for

Consider the operator on ,. Assume that is the polar coordinate on . Then, the condition implies that , in (see [5]). First, we recall some results on the resolvent of Schrödinger operator . Define For , let denote the multiplicity of as the eigenvalue of . Let ,, denote an orthogonal basis of consisting of eigenfunctions of : Let denote the orthogonal projection in onto the subspace spanned by the eigenfunctions of associated with the eigenvalue : Defined by , Here, and is the largest integer which is not larger than . For , let be the largest integer strictly less than . When , set . Define by , if , , otherwise. One has .

Theorem 1 ([5] Theorem 2.2). The following asymptotic expansion holds for near 0: with , in . Here, Here, for , and with being a polynomial in of degree :

Let denote the negative eigenvalues of (counting multiplicity). Suppose that , for convenience, if there are at most negative eigenvalues. Using the same discussion as in [6], we know that each negative eigenvalue tends to 0, as tends to some value. Suppose that , at . Then, from the definition of , one has . In [6], we have studied the asymptotic behavior of the smallest eigenvalue of , namely, , as . In this paper, we investigate the asymptotic behaviors of all eigenvalues of . Suppose that there are exactly eigenvalues of tending to 0 at . Without loss of generality, suppose that .

As in [6], we define a family of Birman-Schwinger kernel operators, which are used to study the eigenvalues of . For , set

The following result shows the relation between the eigenvalues of and the eigenvalues of .

Proposition 2 ([6] Proposition 2.2). Let . Then,(a) let  Then, is injective from to , and is injective from to .(b) The multiplicity of as the eigenvalue of is exactly the multiplicity of as the eigenvalue of .

Proposition 3. (a) Let . The negative eigenvalue of the is monotone decreasing and continuous about .
(b) Let . The eigenvalue of the is monotone increasing and continuous about .

Proof. (a) Suppose , are two arbitrary positive numbers. By Lemma 3.4 [6], one has This means that the negative eigenvalue of is continuous about .
By min-max principle (Theorem XIII.1 [15]), the negative eigenvalue of has the following form: Since and , it is easy to see that . Hence, is monotone decreasing about . It follows that the eigenvalue of is monotone decreasing about .
(b) For , This shows that . Hence, using min-max principle again, one has that the eigenvalue of is monotone increasing about .
Note that, for any , is a bounded operator in . By (20) and Lemma 3.4 [6], we can get that, for any ,, with some large enough. Here, is the eigenvalue of . It implies that the eigenvalue of is continuous about .

We give the definition of resonance which will be used to investigate the asymptotic behavior of the eigenvalue of later.

Definition 4. Set . If , one says that 0 is the resonance of . A nonzero function is called a resonant state of at 0.

3. The Case

Proposition 5. Assume that . Then, is the eigenvalue of , and the multiplicity of is .

Proof. is the eigenvalue of ; thus, is the eigenvalue of by Proposition 2. Notice that , as , and one gets that is the eigenvalue of , by Lemma 3.5 [6].
Suppose that the multiplicity of is . We will prove that in the following. First, using Lemma 3.5 [6] again, we know that there are eigenvalues (counting multiplicity) of tending to . Suppose that these eigenvalues are . Since the eigenvalue of is continuous and monotone increasing, for a fixed near with , there exists a unique such that . It follows that is the eigenvalue of . This fact shows that , since there are eigenvalues of that tend to 0 at . On the other hand, note that, for any , and is continuous and monotone decreasing. Therefore, for a fixed , there exists a unique such that . It follows that is the eigenvalue of . One has that and , as . It means that there are at least eigenvalues (counting multiplicity) of tending to . Thus, . Therefore, . This ends the proof.

Since is a compact operator, then we can suppose that all of the eigenvalues of can be denoted by (counting multiplicity), and the corresponding eigenvectors are , respectively. Set Then, By Lemma 3.5 [6] and Proposition 5, we know that there are eigenvalues (counting multiplicity) of that tend to , as . Without loss of generality, we can suppose that and the eigenvalues tend to , as . Moreover, we can choose a set of eigenvectors, , of such that , , and , as . Then, by Lemma 3.5 [6], is the eigenvector of corresponding to . Thus, . It follows that by (22). In the second step of the last equality, we use Proposition 3.2 [6]. Thus, . By Theorem 4.1 [4], where . will be called a -resonant state of , if for some .

Proposition 6. (I) Assume is a -resonant state of . Then (a) with be a nonzero constant depending only on ;(b),  , with , .
(II) Assume is an eigenvector of . Then, , , .

Proof. (I) By the definition of and , , , Here, is a constant depending only on . If is a resonant state of , then, by (24), where . If is a -resonant state of , then the leading part of is . It follows that, for with , and there exists at least one with , such that Thus, (26) shows that (II) By (24) and , it is easy to see that for . Thus, for ,

Theorem 7. Assume . , , and are defined as above. If , one of three situations holds.(a) is a nonzero constant independent of , , and .(b).   is a nonzero constant independent of .(c) is a nonzero constant independent of .

Proof. Applying Theorem 1 to , we obtain the asymptotic expansion of for near 0, with , in , , . It follows that, if , in . is the eigenvalue of with multiplicity ; then, there are eigenvalues of (counting multiplicity) tending to , as . Suppose that the eigenvalues of are . We study the asymptotic expansion of first.
Set and . Then, by (22), By Lemma 3.6 [6], the eigenvalue of , , has the following form: Here,
If is -resonant state of with , then by Proposition 6, Hence, Similarly, for . It follows that Thus, . By Proposition 2, for , with such that . Thus, . Since we can get that the leading term of is , with being a constant independent of . This is in case (a).
If is 1-resonant state of , computing similarly, we obtain that . Using again and (41), we can get that the leading part of is , with being a constant independent of . This is in case (b).
If is the eigenfunction of corresponding to eigenvalue , then, for , In the second equation, we use (22) and the definition of . Proposition 6 shows that for , . Thus, by (42), Computing similarly as before, we obtain that . Thus, the leading part of is , with being a constant independent of . This is in case (c). This completes the proof.

4. The Case

Assume that in this section. By Theorem 1, in . Here, with . is the polar coordinate on . Hence, ban will be written as Here, is rank 1 projection with , , , and .

Note that as , . It follows that has singularity at . Hence, it is more complicated than the case . We first study the operator as . Set .

We can obtain Propositions 8, 10, and 11, in the same way that Lemmas 7.1 and 7.2 [8] were obtained.

Proposition 8. (a) For small enough with , and for a fixed with , converges to in norm, as .
(b) For small enough, and for fixed with , Here, .

Remark 9. Proposition 8 shows that, for fixed , with ,

Proposition 10. For fixed , with , is the eigenvalue of .

This proposition follows that is the eigenvalue of .

Proposition 11. Suppose that the multiplicity of as the eigenvalue of is . Set . Then, with or , so that for . And for , .

Suppose that all of the eigenvalues of can be denoted by (counting multiplicity), and the corresponding eigenvectors are , respectively. By Lemma 3.5 [6] and Proposition 5, we know that there are eigenvalues (counting multiplicity) of tending to , as . Without loss of generality, we can suppose that and the eigenvalues tend to , as . Moreover, we can choose a set of eigenvectors, , of such that , , and , as . Then, is the eigenvector of corresponding to , by Lemma 3.5 [6]. Thus, .

Theorem 12. Assume . If , one of the four situations holds.(a) and are two nonzero constants independent of .(b) is a nonzero constant independent of , and .(c) is a nonzero constant independent of .(d) is a nonzero constant independent of .

Proof. By Proposition 10, with Lemma 3.6 [6] shows that the eigenvalue of , has the form As in Theorem 7, we should compute ,.
If , then It follows that . Note that . It is easy to get that It follows that with some . Analyzing similarly as in Theorem 7, we can get the leading term of . We are in case (a).
If , then by Proposition 11 and the definition of , . Thus, In the second and third steps, we use Proposition 11 and the fact that . Then, If ; , such that , set . If , then, by (57), Computing similarly as before, we also can obtain that By (53), one has Computing similarly as in Theorem 7, we can get that the leading term of is . We are in case (b).
If , computing similarly as in Theorem 7, we can get the leading term of , and we are in case (c).
If , then for , It follows that Thus, Computing as before, we obtain By (53), one has Computing similarly as in Theorem 7, we can get the leading term of . We are in case (d). This completes the proof.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors contributed equally to the paper and read and approved the final paper.

Acknowledgment

This research is supported by the Natural Science Foundation of China (11301153, 11101127, and 11271110).