Asymptotic Behaviors of the Eigenvalues of Schrödinger Operator with Critical Potential

and Applied Analysis 3

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][11][12][13][14]).
The assumptions used in this paper are as follows.Suppose that  ≤ 0 is a nonzero real continuous function and satisfies | ()| ≤ ⟨⟩ − 0 , for some  0 > 2. (2) Here, ⟨⟩ = (1 + || 2 ) 1/2 .The assumption on () is that Here, Δ  denotes Laplace operator on the sphere S −1 .If (3) holds, then  0 ≥ 0 in  2 (R  ) (see [5]).In Section 2 [6], it is shown that, under the assumption on , () has discrete eigenvalues when  is large enough, 2 Abstract and Applied Analysis and each discrete eigenvalue tends to zero at some  0 .Set If 0 ∉  ∞ , then the Birman-Schwinger kernel || 1/2 ( 0 − ) −1 || 1/2 is a bounded operator for  < 0 (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 holds with  0 > 6.In this paper, we will get the asymptotic behaviors of all discrete eigenvalues of () in the case of 0 ∉  ∞ .
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.
The plan of this work is as follows.In Section 2, we recall some known results for  0 , especially the asymptotic expansion of ( 0 −) −1 for  near 0. We obtain the asymptotic behavior of discrete eigenvalue of () for the case of 0 ∉  ∞ in Section 3. Section 4 concentrates on the asymptotic expansion of discrete eigenvalue in the case of 0 ∈  ∞ .
Let us introduce some notations first.
This shows that ( 1 ) < ( 2 ).Hence, using min-max principle again, one has that the eigenvalue of () is monotone increasing about .Note that, for any  < 0, ( 0 − ) −1 is a bounded operator in  2 (R  ).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.
, one says that 0 is the resonance of ().A nonzero function  ∈ N() \  2 is called a resonant state of () at 0.

The Case
is the eigenvalue of (0), and the multiplicity of  −1  is .
We can obtain Propositions 8, 10, and 11, in the same way that Lemmas 7.1 and 7.2 [8] were obtained.

( b )
The multiplicity of  as the eigenvalue of () is exactly the multiplicity of  −1 as the eigenvalue of ().Proposition 3. (a) Let  > 0. The negative eigenvalue of the () is monotone decreasing and continuous about .(b) Let  < 0. The eigenvalue of the () is monotone increasing and continuous about .