Coupling constant limits of Schrödinger operators with critical potentials

  • Xiaoyao Jia1Email author and

    Affiliated with

    • Yan Zhao2

      Affiliated with

      Boundary Value Problems20132013:62

      DOI: 10.1186/1687-2770-2013-62

      Received: 8 December 2012

      Accepted: 4 March 2013

      Published: 27 March 2013

      Abstract

      A family of Schrödinger operators, P ( λ ) = P 0 + λ V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq1_HTML.gif, is studied in this paper. Here P 0 = Δ + f ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq2_HTML.gif with f ( x ) 1 | x | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq3_HTML.gif when | x | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq4_HTML.gif is large enough and V ( x ) = O ( | x | 2 ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq5_HTML.gif for some ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq6_HTML.gif. We show that each discrete eigenvalue of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif tends to 0 when λ tends to some λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq8_HTML.gif. We get asymptotic behavior of the smallest discrete eigenvalue when λ tends to λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq8_HTML.gif.

      Keywords

      Schrödinger operator critical potential asymptotic expansion

      1 Introduction

      In this paper, we consider a family of Schrödinger operators P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif which are the perturbation of P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq9_HTML.gif in the form
      P ( λ ) = P 0 + λ V for  λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equa_HTML.gif
      on L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif, d 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq11_HTML.gif. Here P 0 = Δ + q ( θ ) r 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq12_HTML.gif. ( r , θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq13_HTML.gif are the polar coordinates on R d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq14_HTML.gif, and q ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq15_HTML.gif is a real continuous function. V 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq16_HTML.gif is a non-zero continuous function satisfying
      | V ( x ) | C x ρ 0 for some  ρ 0 > 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equ1_HTML.gif
      (1)
      Here x = ( 1 + | x | 2 ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq17_HTML.gif. Let Δ s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq18_HTML.gif denote the Laplace operator on the sphere S d 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq19_HTML.gif. Assume that
      Δ s + q ( θ ) > 1 4 ( d 2 ) 2 on  L 2 ( S d 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equ2_HTML.gif
      (2)

      If (2) holds, then P 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq20_HTML.gif in L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif (see [1]).

      Under the assumption on V, we know that P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif has discrete eigenvalues when λ is large enough, and each discrete eigenvalue tends to zero when λ tends to some λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq8_HTML.gif (see Section 2). We study the asymptotic behaviors of the discrete eigenvalues of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif in this paper. The asymptotic behaviors for Schrödinger operators with fast decaying potentials were studied by Klaus and Simon [2]. In [2], they studied the convergence rate of discrete eigenvalues of H ( λ ) = Δ + λ V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq21_HTML.gif when λ λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq22_HTML.gif. λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq8_HTML.gif is the value at which some discrete eigenvalue e i ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq23_HTML.gif tends to zero. The main method they used in their paper is the Birman-Schwinger technique.

      In order to use the Birman-Schwinger technique to P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif, we need to get the asymptotic expansion of ( P 0 α ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq24_HTML.gif for α near zero, α < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq25_HTML.gif, which was studied by Wang [1]. In this paper, we first show that there exists some λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq8_HTML.gif such that when λ > λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq26_HTML.gif, P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif has discrete eigenvalues. Then, we define the Birman-Schwinger kernel K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif for P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif and find that there is one-to-one correspondence between the discrete eigenvalues of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif and the discrete eigenvalues of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif. Hence, the asymptotic expansion of the discrete eigenvalue of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif can be got through the asymptotic expansion of the discrete eigenvalue of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif. In our main results, we need to use that K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif is a bounded operator from L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq28_HTML.gif to L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq28_HTML.gif. To get that, we add a strong condition on V (i.e., ρ 0 > 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq29_HTML.gif in (1)). We show that K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif is a family of compact operators converging to K ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq30_HTML.gif and obtain the asymptotic expansions of the discrete eigenvalues of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif by functional calculus. After that, the convergence rate of the smallest discrete eigenvalue of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif is obtained.

      Here is the plan of our work. In Section 2, we recall some results of P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq9_HTML.gif and define the Birman-Schwinger kernel K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif for P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif. The relationship between the eigenvalues of these two kinds of operators is studied. In Section 3, we first study the asymptotic behavior of the discrete eigenvalues of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif. Then the convergence rate of the smallest discrete eigenvalue of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif is obtained. We get the leading term and the estimate of the remainder term of the smallest discrete eigenvalue.

      Let us introduce some notations first.

      Notation The scalar product on L 2 ( R + ; r d 1 d r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq31_HTML.gif and L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif is denoted by , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq32_HTML.gif and that on L 2 ( S d 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq33_HTML.gif by ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq34_HTML.gif. H r , s ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq35_HTML.gif, r Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq36_HTML.gif, s R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq37_HTML.gif, denotes the weighted Sobolev space of order r with volume element x 2 s d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq38_HTML.gif. The duality between H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq39_HTML.gif and H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq40_HTML.gif is identified with the L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq28_HTML.gif product. Denote H 0 , s = L 2 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq41_HTML.gif. Notation L ( H r , s , H r , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq42_HTML.gif stands for the space of continuous linear operators from H r , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq43_HTML.gif to H r , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq44_HTML.gif. The complex plane ℂ is slit along positive real axis so that z ν = e ν ln z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq45_HTML.gif and ln z = ln | z | + i arg z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq46_HTML.gif with 0 < arg z < 2 π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq47_HTML.gif are holomorphic there.

      2 Some results for P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq9_HTML.gif

      Assume that ( r , θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq13_HTML.gif are the polar coordinates on R d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq14_HTML.gif. Then the condition
      Δ s + q ( θ ) > 1 4 ( d 2 ) 2 on  L 2 ( S d 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equb_HTML.gif
      implies
      Δ + q ( θ ) r 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equ3_HTML.gif
      (3)

      in L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif (see [1]).

      Now, we recall some results on the resolvent and the Schrödinger group for the unperturbed operator P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq9_HTML.gif. Let
      σ = { ν ; ν = λ + ( d 2 ) 2 4 , λ σ ( Δ s + q ( θ ) ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equc_HTML.gif
      Denote
      σ k = σ [ 0 , k ] , k N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equd_HTML.gif
      For ν σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq48_HTML.gif, let n ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq49_HTML.gif denote the multiplicity of λ ν = ν 2 ( d 2 ) 2 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq50_HTML.gif as the eigenvalue of Δ s + q ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq51_HTML.gif. Let φ ν ( j ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq52_HTML.gif, ν σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq48_HTML.gif, 1 j n ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq53_HTML.gif denote an orthogonal basis of L 2 ( S d 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq54_HTML.gif consisting of eigenfunctions of Δ s + q ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq51_HTML.gif:
      ( Δ s + q ( θ ) ) φ ν ( j ) = λ ν φ ν ( j ) , ( φ ν ( i ) , φ ν ( j ) ) = δ i j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Eque_HTML.gif
      Let π ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq55_HTML.gif denote the orthogonal projection in L 2 ( S d 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq54_HTML.gif onto the subspace spanned by the eigenfunctions of Δ s + q ( θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq51_HTML.gif associated with the eigenvalue λ ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq56_HTML.gif, and let π ν ( i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq57_HTML.gif denote the orthogonal projection in L 2 ( S d 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq54_HTML.gif onto the eigenfunction φ ν ( i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq58_HTML.gif:
      π ν f = j = 1 n ν ( f , φ ν ( j ) ) φ ν ( j ) , f L 2 ( S n 1 ) , π ν ( i ) f = ( f , φ ν ( i ) ) φ ν ( i ) , f L 2 ( S d 1 ) , 1 i n ν . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equf_HTML.gif
      Denote for ν σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq48_HTML.gif
      z ν = { z ν if  ν N , z ln z if  ν N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equg_HTML.gif

      Here ν = ν [ ν ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq59_HTML.gif, and [ ν ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq60_HTML.gif is the largest integer which is not larger than ν. For ν > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq61_HTML.gif, let [ ν ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq62_HTML.gif be the largest integer strictly less than ν. When ν = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq63_HTML.gif, set [ ν ] = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq64_HTML.gif. Define δ ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq65_HTML.gif by δ ν = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq66_HTML.gif, if ν σ N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq67_HTML.gif, δ ν = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq68_HTML.gif, otherwise. One has [ ν ] = [ ν ] + δ ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq69_HTML.gif.

      The following is the asymptotic expansion for the resolvent R 0 ( z ) = ( P 0 z ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq70_HTML.gif.

      Theorem 2.1 (Theorem 2.2 [1])

      The following asymptotic expansion holds for z near zero with z > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq71_HTML.gif:
      R 0 ( z ) = δ 0 ln z G 0 , 0 π 0 + j = 0 N z j F j + ν σ N z ν j = [ ν ] N 1 z j G ν , j + δ ν π ν + R 0 ( N ) ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equh_HTML.gif
      in L ( 1 , s ; 1 , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq72_HTML.gif, s > 2 N + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq73_HTML.gif. Here
      G ν , j ( r , τ ) = { b ν , j ( r τ ) j + ν f j [ ν ] ( r , τ ; ν ) , ν N , ( i r τ ) j j ! f j [ ν ] ( r , τ ; 0 ) , ν N , F j L ( 1 , s ; 1 , s ) , s > 2 j + 1 , R 0 ( N ) ( z ) = O ( | z | N + ϵ ) L ( 1 , s ; 1 , s ) , s > 2 N + 1 , ϵ > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equi_HTML.gif
      Here
      b ν , j = i j e i ν π / 2 Γ ( 1 ν ) ν ( ν + 1 ) ( ν + j ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equj_HTML.gif
      for 0 ν < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq74_HTML.gif, and
      f j ( r , τ , ν ) = ( r τ ) 1 2 ( n 2 ) P j , ν ( ρ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equk_HTML.gif
      with P j , ν ( ρ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq75_HTML.gif a polynomial in ρ of degree j:
      P j , ν ( ρ ) = i j a ν j ! 1 1 ( ρ + 1 2 θ ) j ( 1 θ 2 ) ν 1 2 d θ , a ν = e i π ν / 2 2 2 ν + 1 π 1 / 2 Γ ( ν + 1 / 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equl_HTML.gif

      First, we show that P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif has discrete eigenvalues when λ is large enough. In fact, we need only to show that there exists a function ψ L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq76_HTML.gif such that ψ , P ( λ ) ψ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq77_HTML.gif.

      From the assumption on V, we know that there exists a point x 0 R d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq78_HTML.gif such that V ( x 0 ) = inf x R d V ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq79_HTML.gif. Choose δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq80_HTML.gif small enough such that for all x B ( x 0 , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq81_HTML.gif, V ( x ) < 1 2 V ( x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq82_HTML.gif. For ψ C 0 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq83_HTML.gif, ψ ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq84_HTML.gif, supp ψ B ( x 0 , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq85_HTML.gif, one has
      ψ , P ( λ ) ψ = ψ , P 0 ψ + λ ψ , V ψ < ψ , P 0 ψ + λ 2 V ( x 0 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equm_HTML.gif

      when λ is large enough, one has ψ , P ( λ ) ψ < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq86_HTML.gif. This means that P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif has discrete eigenvalues when λ is large enough.

      P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif has a continuous spectrum [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq87_HTML.gif for λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq88_HTML.gif because lim | x | V ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq89_HTML.gif exists and equals zero (see [3]). We know that σ ( P ( 0 ) ) = σ ( P 0 ) = [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq90_HTML.gif. Hence, from the continuity of a discrete spectrum of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif, we know that there exists some λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq8_HTML.gif such that when λ > λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq26_HTML.gif, P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif has eigenvalues less than zero, and when λ λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq91_HTML.gif, σ ( P ( λ ) ) = [ 0 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq92_HTML.gif. So, P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif has an eigenvalue e 1 ( λ ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq93_HTML.gif at the bottom of its spectrum for λ > λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq94_HTML.gif. In Section 3 (Proposition 3.1), we prove that e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif is simple and the corresponding eigenfunction can be chosen to be positive everywhere. (There are many results about the simplicity of the smallest eigenvalue of the Schrödinger operator without singularity, but there is no result which can be used directly, because the potential we use in this paper has singularity at zero. Theorem XIII.48 [4] can treat the Schrödinger operator with the potential which has singularity at zero, but the positivity of potential is needed. Hence, we give this result.) From the discussion above and the continuity of a discrete spectrum, one has that e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif tends to zero at some λ. The asymptotic behavior of e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif is studied in this paper.

      To study the eigenvalues of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif, we first define a family of Birman-Schwinger kernel operators. Let
      K ( z ) = | V | 1 / 2 ( P 0 z ) 1 | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equn_HTML.gif
      for z σ ( P 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq96_HTML.gif, and
      K ( 0 ) = | V | 1 / 2 F 0 | V | 1 / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equo_HTML.gif

      Then we have the following result.

      Proposition 2.2 Let α < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq25_HTML.gif. Then
      1. (a)
        Let
        A = { ψ L 2 ( R d ) ; ( P ( λ ) α ) ψ = 0 } , B = { ϕ L 2 ( R d ) ; K ( α ) ϕ = λ 1 ϕ } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equp_HTML.gif
         
      Then | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq97_HTML.gif is injective from A to B, and ( P 0 α ) 1 | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq98_HTML.gif is injective from B to A.
      1. (b)

        The multiplicity of α as the eigenvalue of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif is exactly the multiplicity of λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq99_HTML.gif as the eigenvalue of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif.

         
      Proof (a) First, we prove that | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq97_HTML.gif is injective from A to B. Note that if ψ A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq100_HTML.gif, then
      K ( α ) ϕ = λ 1 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equq_HTML.gif
      with ϕ = | V | 1 / 2 ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq101_HTML.gif. And if ϕ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq102_HTML.gif, then
      ψ = λ ( P 0 α ) 1 V ψ = λ ( P 0 α ) 1 | V | 1 / 2 ϕ = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equr_HTML.gif

      It follows that | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq97_HTML.gif is injective from A to B.

      Next, we show that ( P 0 α ) 1 | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq98_HTML.gif is injective from B to A. If ϕ B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq103_HTML.gif, then
      ( P ( λ ) α ) ψ = 0 , with  ψ = ( P 0 α ) 1 | V | 1 / 2 ϕ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equs_HTML.gif
      And if ψ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq104_HTML.gif, then
      0 = | V | 1 / 2 ψ = K ( α ) ϕ = λ 1 ϕ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equt_HTML.gif

      It follows that ( P 0 α ) 1 | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq98_HTML.gif is injective from B to A.

      (b) From (a), one has dim A = dim B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq105_HTML.gif. This means that the multiplicity of α as the eigenvalue of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif is exactly the multiplicity of λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq99_HTML.gif as the eigenvalue of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif. □

      From the last proposition, we know that there exists one-to-one correspondence between the discrete eigenvalues of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif and the discrete eigenvalues of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif. Hence, we can study the eigenvalues of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif first.

      3 Asymptotic expansion of the eigenvalues

      If P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq9_HTML.gif and V are defined as above, we show that if P 0 + V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq106_HTML.gif has the eigenvalue less than zero, then the smallest eigenvalue of P 0 + V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq106_HTML.gif is simple. We use Theorems XIII.44, XIII.45 [4] to prove it.

      Proposition 3.1 Suppose P 0 + V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq106_HTML.gif has an eigenvalue at the bottom of its spectrum. Then this eigenvalue is simple and the corresponding eigenfunction can be chosen to be a positive function.

      Proof Let 0 χ ( t ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq107_HTML.gif be a smooth nonincreasing function such that χ ( t ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq108_HTML.gif if | t | < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq109_HTML.gif and χ ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq110_HTML.gif if t > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq111_HTML.gif. Let χ n ( t ) = χ ( t / n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq112_HTML.gif. Set V n = χ n ( r ) q ( θ ) r 2 + V x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq113_HTML.gif, H 0 = Δ + x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq114_HTML.gif, H n = H 0 + V n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq115_HTML.gif, H = P = P 0 + V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq116_HTML.gif. From the proof of Theorem XIII.47 [4], we know that e t H 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq117_HTML.gif is positivity preserving and { e t H 0 } L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq118_HTML.gif acts irreducibly on L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq28_HTML.gif. Hence, by Theorem XIII.45 [4], if H n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq119_HTML.gif converges to H and H V n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq120_HTML.gif converges to H 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq121_HTML.gif in the strong resolvent sense, then e t H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq122_HTML.gif is positivity preserving and { e t H } L http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq123_HTML.gif acts irreducibly on L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq28_HTML.gif. By Theorems XIII.43 and XIII.44 [4], we can get the result. Since C 0 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq124_HTML.gif is the core for all P n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq125_HTML.gif and P, and for any ψ C 0 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq126_HTML.gif, V n ψ ( q ( θ ) r 2 + V ) ψ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq127_HTML.gif in L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq28_HTML.gif, then we have the necessary strong resolvent convergence by Theorem VIII.25(a) [5]. This ends the proof. □

      Proposition 3.2 Assume that 0 σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq128_HTML.gif. P 0 F 0 u = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq129_HTML.gif in H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq130_HTML.gif for any u H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq131_HTML.gif, s > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq132_HTML.gif.

      Proof If u H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq131_HTML.gif, then F 0 u H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq133_HTML.gif. For any test function ϕ C 0 ( R n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq134_HTML.gif, we have P 0 F 0 u , ϕ = u , F 0 P 0 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq135_HTML.gif. If 0 σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq136_HTML.gif, then we have lim z 0 ( P 0 z ) 1 = F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq137_HTML.gif in H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq138_HTML.gif for z > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq139_HTML.gif. It follows u , F 0 P 0 ϕ = lim z 0 u , ( P 0 z ) 1 P 0 ϕ = lim z 0 u , ϕ z ( P 0 z ) 1 ϕ = u , ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq140_HTML.gif because ϕ and P 0 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq141_HTML.gif belong to H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq130_HTML.gif. Hence, P 0 F 0 u = u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq129_HTML.gif in H 1 , s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq130_HTML.gif. □

      Proposition 3.3 Assume that 0 σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq128_HTML.gif. K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif is a compact operator for α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq142_HTML.gif. And K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq27_HTML.gif converges to K ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq30_HTML.gif in operator norm sense.

      Proof For α < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq25_HTML.gif, K ( α ) = | V | 1 / 2 ( P 0 α ) 1 | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq143_HTML.gif. Since ( P 0 α ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq144_HTML.gif is a bounded operator from L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif to H 1 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq145_HTML.gif, and V is a compact operator from H 1 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq145_HTML.gif to L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif, then V ( P 0 α ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq146_HTML.gif is a compact operator on L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif. Using a similar method to that in Proposition 2.2, we can show that V ( P 0 α ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq146_HTML.gif and K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq147_HTML.gif have the same non-zero eigenvalues, and for the same eigenvalue e ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq148_HTML.gif, the multiplicity of e ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq148_HTML.gif as the eigenvalue of V ( P 0 α ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq149_HTML.gif and the multiplicity of e ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq148_HTML.gif as the eigenvalue of K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq147_HTML.gif are the same. Hence, K ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq147_HTML.gif is a compact operator. Because
      K ( α ) K ( 0 ) = | V | 1 / 2 [ ( P 0 α ) 1 F 0 ] | V | 1 / 2 = | V | 1 / 2 R 0 ( 0 ) | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equu_HTML.gif

      and if ρ 0 > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq150_HTML.gif, then | V | 1 / 2 R 0 ( 0 ) | V | 1 / 2 = o ( | α | ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq151_HTML.gif in L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif. Hence, K ( α ) K ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq152_HTML.gif in operator norm sense as α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq153_HTML.gif. This means that K ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq30_HTML.gif is a compact operator. □

      Lemma 3.4 Suppose A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq154_HTML.gif, A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq155_HTML.gif are two bounded self-adjoint operators on a Hilbert space H. Set
      μ n ( A i ) = inf ϕ 1 , , ϕ n sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ( ψ , A i ψ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equv_HTML.gif

      then | μ n ( A 1 ) μ n ( A 2 ) | A 1 A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq156_HTML.gif.

      Proof By the definition of μ n ( A i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq157_HTML.gif, one has
      | μ n ( A 1 ) μ n ( A 2 ) | = | inf ϕ 1 , , ϕ n sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ψ , A 1 ψ inf ϕ 1 , , ϕ n sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ψ , A 2 ψ | = | sup ϕ 1 , , ϕ n ( sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ψ , A 1 ψ ) + sup ϕ 1 , , ϕ n ( sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ψ , A 2 ψ ) | | sup ϕ 1 , , ϕ n [ sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ψ , A 1 ψ + sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ψ , A 2 ψ ] | = sup ϕ 1 , , ϕ n | sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ψ , A 1 ψ sup ψ = 1 , ψ [ ϕ 1 , , ϕ n ] ψ , A 2 ψ | sup ϕ 1 , , ϕ n sup ψ = 1 , [ ϕ 1 , , ϕ n ] | ψ , A 1 ψ ψ , A 2 ψ | A 1 A 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equw_HTML.gif

      This ends the proof. □

      Lemma 3.5 Suppose T ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq158_HTML.gif is a family of compact self-adjoint operators on a separable Hilbert space H, and T ( α ) = T 0 + o ( | α | ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq159_HTML.gif for α near zero. Set
      μ k ( α ) = inf ϕ 1 , , ϕ k sup ψ = 1 , ψ [ ϕ 1 , , ϕ k ] ψ , T ( α ) ψ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equx_HTML.gif
      Then:
      1. (a)

        μ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq160_HTML.gif is an eigenvalue of T ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq158_HTML.gif, and μ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq160_HTML.gif converges when α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq161_HTML.gif. Moreover, if μ k ( α ) μ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq162_HTML.gif, then μ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq163_HTML.gif is an eigenvalue of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif.

         
      2. (b)

        Suppose that E 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq165_HTML.gif is an eigenvalue of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif of the multiplicity of m. Then there are m eigenvalues (counting multiplicity), E k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq166_HTML.gif ( 1 k m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq167_HTML.gif), of T ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq158_HTML.gif near E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq168_HTML.gif. Moreover, we can choose { ϕ k ( α ) ; 1 k m } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq169_HTML.gif such that ( ϕ k ( α ) , ϕ j ( α ) ) = δ k j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq170_HTML.gif ( 1 k , j m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq171_HTML.gif), ϕ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq172_HTML.gif is the eigenvector of T ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq158_HTML.gif corresponding to E k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq173_HTML.gif ( E k ( α ) E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq174_HTML.gif), and ϕ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq172_HTML.gif converges as α 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq161_HTML.gif. If ϕ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq172_HTML.gif converges to ϕ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq175_HTML.gif, then ϕ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq175_HTML.gif is the eigenvector of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif corresponding to E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq168_HTML.gif.

         
      Proof (a) By the min-max principle, we know that μ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq160_HTML.gif is an eigenvalue of T ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq158_HTML.gif. By Lemma 3.4, one has
      | μ k ( α ) μ k ( 0 ) | T ( α ) T 0 = O ( | α | ϵ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equy_HTML.gif

      It follows that μ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq160_HTML.gif converges to the eigenvalue of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif.

      (b) Because T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif is a compact operator and E 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq165_HTML.gif is an eigenvalue of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif, then E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq168_HTML.gif is a discrete spectrum of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif. Then there exists a constant δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq176_HTML.gif small enough such that T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif has only one eigenvalue E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq168_HTML.gif in B ( E 0 , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq177_HTML.gif ( = { z C ; | z E 0 | < δ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq178_HTML.gif). For α small enough, T ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq158_HTML.gif has exactly m eigenvalues (counting multiplicity) in B ( E 0 , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq177_HTML.gif because the eigenvalues of T ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq158_HTML.gif converge to the eigenvalues of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif by part (a) of lemma. Suppose the m eigenvalues, near E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq168_HTML.gif, of T ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq158_HTML.gif are E 1 ( α ) , E 2 ( α ) , , E m ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq179_HTML.gif, and the corresponding eigenvectors are ψ 1 ( α ) , ψ 2 ( α ) , , ψ m ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq180_HTML.gif such that ψ k ( α ) , ψ j ( α ) = δ k j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq181_HTML.gif. Let
      P α = 1 2 π i | E E 0 | = δ ( T ( α ) E ) 1 d E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equz_HTML.gif
      Then P α = k = 1 m , ψ k ( α ) ψ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq182_HTML.gif. Let P α ( k ) = , ψ k ( α ) ψ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq183_HTML.gif, then P α = k = 1 m P α ( k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq184_HTML.gif. For α near zero, one has
      P α P 0 = 1 2 π i | E E 0 | = δ ( T ( α ) E ) 1 ( T 0 E ) 1 d E = 1 2 π i | E E 0 | = δ ( T ( α ) E ) 1 ( T 0 T α ) ( T 0 E ) 1 d E = O ( | α | ϵ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equaa_HTML.gif
      Let A = { ϕ ; ϕ = 1 , ϕ Ran P 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq185_HTML.gif. Let ϕ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq175_HTML.gif be an element in A such that ϕ ψ k ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq186_HTML.gif acquires the minimum value. Then we have
      P α ( k ) ϕ k ϕ k P α ( k ) ϕ k ψ k ( α ) + ψ k ( α ) ϕ k = P α ( k ) ϕ k P α ( k ) ψ k ( α ) + ψ k ( α ) ϕ k 2 ψ k ( α ) ϕ k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equab_HTML.gif
      and
      ψ k ( α ) ϕ k P 0 ψ k ( α ) P 0 ψ k ( α ) ψ k ( α ) P 0 ψ k ( α ) P 0 ψ k ( α ) P 0 ψ k ( α ) + P 0 ψ k ( α ) P α ψ k ( α ) = O ( | α | ϵ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equac_HTML.gif
      In the last equality, we use the fact
      P 0 ψ k ( α ) = ( P 0 ψ k ( α ) ψ k ( α ) ) + ψ k ( α ) = 1 + O ( | α | ϵ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equad_HTML.gif
      and
      P 0 ψ k ( α ) P 0 ψ k ( α ) P 0 ψ k ( α ) = P 0 ψ k ( α ) ( 1 P 0 ψ k ( α ) 1 ) = O ( | α | ϵ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equae_HTML.gif

      It follows that P α ( k ) ϕ k ϕ k = O ( | α | ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq187_HTML.gif. Let ϕ k ( α ) = P α ( k ) ϕ k P α ( k ) ϕ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq188_HTML.gif. Then ϕ k ( α ) , ϕ j ( α ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq189_HTML.gif for k j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq190_HTML.gif because P α ( k ) P α ( j ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq191_HTML.gif if k j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq192_HTML.gif, and ϕ k ( α ) ϕ k P α ( k ) ϕ k ϕ k + ( 1 1 P α ( k ) ϕ k ) P α ( k ) ϕ k = O ( | α | ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq193_HTML.gif. This ends the proof. □

      Let 0 < α 1 < α 2 < < α i < < α m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq194_HTML.gif and
      T ( β ) = T 0 + i = 1 m β α i ( ln β ) δ i T i + T r ( β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equaf_HTML.gif
      Here, δ i = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq195_HTML.gif or 1, T 0 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq196_HTML.gif, T i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq197_HTML.gif ( 1 i m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq198_HTML.gif), T r ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq199_HTML.gif are compact operators, and T r ( β ) = O ( | β | α m + ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq200_HTML.gif for β near zero. Set
      e s = inf ϕ 1 , , ϕ s sup ψ = 1 , ψ [ ϕ 1 , , ϕ s ] ψ , T 0 ψ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equag_HTML.gif
      Then, by the min-max principle, e s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq201_HTML.gif is an eigenvalue of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif. Moreover, if e s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq202_HTML.gif, then e s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq201_HTML.gif is a discrete eigenvalue of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif because T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif is a compact operator. If e s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq202_HTML.gif is an eigenvalue of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif of multiplicity m, without loss, we can suppose that e s = e s + 1 = = e s + m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq203_HTML.gif. Then there exist exactly m eigenvalues (counting multiplicity), e s ( β ) , e s + 1 ( β ) , , e s + m 1 ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq204_HTML.gif, of T ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq205_HTML.gif near e s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq201_HTML.gif. By Lemma 3.5, we know that there exists a family of normalized eigenvectors { ϕ j ( β ) ; j = s , s + 1 , , s + m 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq206_HTML.gif of T ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq205_HTML.gif such that T ( β ) ϕ j ( β ) = e j ( β ) ϕ j ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq207_HTML.gif, ϕ j ( β ) , ϕ k ( β ) = δ j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq208_HTML.gif ( j , k = s , s + 1 , , s + m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq209_HTML.gif), and ϕ j ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq210_HTML.gif ( j = s , s + 1 , , s + m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq211_HTML.gif) converge as β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq212_HTML.gif. Suppose that ϕ j ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq210_HTML.gif converge to ϕ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq213_HTML.gif for all j such that e j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq214_HTML.gif. Then ϕ s , ϕ j = δ s j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq215_HTML.gif. { ϕ s } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq216_HTML.gif can be extended to a standard orthogonal basis. Set
      T 1 ( β ) = s = 1 m β α s ( ln β ) δ i T s + T r ( β ) , T s j ( β ) = ϕ s , T 1 ( β ) ϕ j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equah_HTML.gif

      Then we have the following.

      Lemma 3.6 T ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq205_HTML.gif, e s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq201_HTML.gif are given as before. Then the eigenvalue of T ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq205_HTML.gif, e j ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq217_HTML.gif ( j = s , s + 1 , , s + m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq211_HTML.gif) has the following form:
      e j ( β ) = e s + n = 0 a n ( j ) ( β ) n = 0 b n ( j ) ( β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equai_HTML.gif
      Here
      a 0 ( j ) ( β ) = T j j ( β ) , a 1 ( j ) ( β ) = { k ; e k e s } ( e k e s ) 1 T j k ( β ) T k j ( β ) , a 2 ( j ) ( β ) = k j l ( e k e s ) 1 ( e l e s ) 1 T j k ( β ) T k l ( β ) T l j ( β ) a 2 ( j ) ( β ) = 2 { k ; e k e s } ( e k e s ) 1 T j k ( β ) T k j ( β ) T j j ( β ) , a n ( j ) ( β ) = ( 1 ) n 2 π i | E e s | = δ ( e s E ) 1 i 1 , i 2 , , i n ( e i 1 E ) 1 ( e i n E ) 1 a n ( j ) ( β ) = × T j i 1 T i 1 i 2 T i n j d E for n > 2 , b 0 ( j ) ( β ) = 1 , b 1 ( j ) ( β ) = 0 , b 2 ( j ) ( β ) = { k ; e k e s } ( e s e k ) 2 T j k ( β ) T k j ( β ) , b n ( j ) ( β ) = ( 1 ) n 2 π i | E e s | = δ ( e s E ) 2 i 1 , i 2 , , i n 1 ( e i 1 E ) 1 ( e i n 1 E ) 1 b n ( j ) ( β ) = × T j i 1 T i 1 i 2 T i n 1 j d E for n > 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equaj_HTML.gif
      Proof If e s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq202_HTML.gif, then e s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq201_HTML.gif is the discrete eigenvalue of T 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq164_HTML.gif. Suppose that the multiplicity of e s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq201_HTML.gif is m, and suppose that e s = e s + 1 = = e s + m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq218_HTML.gif as before. Hence, we can choose δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq219_HTML.gif small enough such that there is only one eigenvalue e s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq201_HTML.gif in B ( e s , δ ) = { z C ; | z e s | < δ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq220_HTML.gif. We know that e j ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq217_HTML.gif ( j = s , s + 1 , , s + m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq211_HTML.gif) converge to e s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq201_HTML.gif. It follows that if δ is small enough, there are exactly m eigenvalues (counting multiplicity) of T ( β ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq205_HTML.gif in B ( e s , δ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq221_HTML.gif for β small. Set
      P β 1 2 π i | E e s | = δ ( T ( β ) E ) 1 d E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equak_HTML.gif
      Then
      e j ( β ) = ϕ j , T ( β ) P β ϕ j ϕ j , P β ϕ j = ϕ j , T 0 P β ϕ j ϕ j , P β ϕ j + ϕ j , T 1 ( β ) P β ϕ j ϕ j , P β ϕ j = e s + ϕ j , T 1 ( β ) P β ϕ j ϕ j , P β ϕ j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equal_HTML.gif
      Since
      ( T ( β ) E ) 1 = ( T 0 E ) 1 ( I + T 1 ( β ) ( T 0 E ) 1 ) 1 = ( T 0 E ) 1 n = 0 ( 1 ) n [ T 1 ( β ) ( T 0 E ) 1 ] n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equam_HTML.gif
      then
      ϕ j , P β ϕ j = 1 2 π i | E e s | = δ ϕ j , ( T 0 E ) 1 n = 0 ( 1 ) n [ ( T 1 ( β ) ) ( T 0 E ) 1 ] n ϕ j d E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equan_HTML.gif
      Then
      b n ( j ) ( β ) = ( 1 ) n 2 π i | E e s | = δ ϕ j , ( T 0 E ) 1 [ ( T 1 ( β ) ) ( T 0 E ) 1 ] n ϕ j d E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equao_HTML.gif
      In particular,
      b 0 ( j ) ( β ) = 1 2 π i | E e s | = δ ϕ j , ( T 0 E ) 1 ϕ j d E = 1 , b 1 ( j ) ( β ) = 1 2 π i | E e s | = δ ϕ j , ( T 0 E ) 1 [ ( T 1 ( β ) ) ( T 0 E ) 1 ] ϕ j d E b 1 ( j ) ( β ) = 1 2 π i | E e s | = δ ( e s E ) 2 ϕ j , T 1 ( β ) ϕ j d E = 0 , b 2 ( j ) ( β ) = ( 1 ) 2 2 π i | E e s | = δ ϕ j , ( T 0 E ) 1 [ ( T 1 ( β ) ) ( T 0 E ) 1 ] 2 ϕ j d E b 2 ( j ) ( β ) = 1 2 π i | E e s | = δ ( e s E ) 2 ϕ j , [ ( T 1 ( β ) ) ( T 0 E ) 1 ( T 1 ( β ) ) ] ϕ j d E b 2 ( j ) ( β ) = 1 2 π i | E e s | = δ ( e s E ) 2 k ϕ j , ( T 1 ( β ) ) ϕ k ( e k E ) 1 ϕ j , ( T 1 ( β ) ) ϕ k d E b 2 ( j ) ( β ) = 1 2 π i | E e s | = δ ( e s E ) 2 { k ; e k e s } ϕ j , ( T 1 ( β ) ) ϕ k ( e k E ) 1 ϕ j , ( T 1 ( β ) ) ϕ k d E b 2 ( j ) ( β ) = 1 2 π i | E e s | = δ ( e s E ) 2 { k ; e k = e s } ϕ j , ( T 1 ( β ) ) ϕ k ( e k E ) 1 ϕ j , ( T 1 ( β ) ) ϕ k d E b 2 ( j ) ( β ) = { k ; e k e s } ( e s e k ) 2 T j k ( β ) T k j ( β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equap_HTML.gif
      In the last step, we use that
      { k ; e k e s } | ( e k e j ) 2 T j k ( β ) T k j ( β ) | C { k ; e k e s } ( | T j k ( β ) | 2 + | T k j ( β ) | 2 ) C ( T 1 ( β ) ϕ j 2 + T 1 ( β ) ϕ j 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equaq_HTML.gif
      Similarly, we can get
      b n ( j ) ( β ) = ( 1 ) n 2 π i | E e s | = δ ( e s E ) 2 i 1 , i 2 , , i n 1 ( e i 1 E ) 1 ( e i n 1 E ) 1 × T j i 1 T i 1 i 2 T i n 1 j d E ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equar_HTML.gif
      and
      a 0 ( j ) ( β ) = T j j ( β ) , a 1 ( j ) ( β ) = { k ; e k e s } ( e k e s ) 1 T j k ( β ) T k j ( β ) , a 2 ( j ) ( β ) = { l , k ; e k e j e l } ( e k e s ) 1 ( e l e s ) 1 T j k ( β ) T k l ( β ) T l j ( β ) a 2 ( j ) ( β ) = 2 { k ; e k e s } ( e k e s ) 1 T j k ( β ) T k j ( β ) T j j ( β ) , a n ( j ) ( β ) = ( 1 ) n 2 π i | E e s | = δ ϕ j , [ ( T 1 ( β ) ) ( T 0 E ) 1 ] n + 1 ϕ j d E a n ( j ) ( β ) = ( 1 ) n 2 π i | E e s | = δ ( e s E ) 1 ϕ j , [ ( T 1 ( β ) ) ( T 0 E ) 1 ] n T 1 ( β ) ϕ j d E a n ( j ) ( β ) = ( 1 ) n 2 π i | E e s | = δ ( e s E ) 1 i 1 ϕ j , [ ( T 1 ( β ) ) ( T 0 E ) 1 ] n 1 a n ( j ) ( β ) = × T 1 ( β ) ϕ i 1 ( e s E ) 1 T i 1 j ( β ) d E a n ( j ) ( β ) = a n ( j ) ( β ) = ( 1 ) n 2 π i | E e s | = δ ( e s E ) 1 i 1 , i 2 , , i n ( e i 1 E ) 1 ( e i n E ) 1 a n ( j ) ( β ) = × T j i 1 T i 1 i 2 T i n j d E . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equas_HTML.gif

       □

      First, we study the asymptotic expansion of the smallest eigenvalue e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif. By Proposition 3.1, we know that e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif is a simple eigenvalue of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif, and the corresponding eigenfunction can be chosen to be positive. We suppose u ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq222_HTML.gif is a positive eigenfunction corresponding to e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif. Then u ˜ ( λ ) = | V | 1 / 2 u ( λ ) L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq223_HTML.gif. Without loss of generality, we can suppose that u ˜ ( λ ) L 2 ( R d ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq224_HTML.gif. Then we can get the following result.

      Lemma 3.7 Assume that 0 σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq128_HTML.gif. Set ν 0 = min { ν ; ν σ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq225_HTML.gif. u ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq226_HTML.gif is defined as above. Then u ˜ ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq227_HTML.gif converges in L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif when λ λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq228_HTML.gif. If ν 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq229_HTML.gif and u ˜ ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq227_HTML.gif converges to ϕ, then ϕ is the eigenfunction of K ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq30_HTML.gif, and ϕ , | V | 1 / 2 G ν 0 , 0 π ν 0 | V | 1 / 2 ϕ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq230_HTML.gif.

      Proof By the assumption of u ˜ ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq227_HTML.gif, one has K ( e 1 ( λ ) ) u ˜ ( λ ) = λ 1 u ˜ ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq231_HTML.gif. One can check that u ˜ ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq227_HTML.gif converges in L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq10_HTML.gif as λ λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq232_HTML.gif by Lemma 3.5. And also, by Lemma 3.5, we know that ϕ is the normalized eigenfunction of K ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq30_HTML.gif corresponding to E 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq168_HTML.gif. ϕ is a positive function since u ˜ ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq227_HTML.gif is a positive function. Let u = F 0 | V | 1 / 2 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq233_HTML.gif, then P ( λ 0 ) u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq234_HTML.gif and u is a positive function because | V | 1 / 2 u = | V | 1 / 2 F 0 | V | 1 / 2 ϕ = K ( 0 ) ϕ = λ 0 1 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq235_HTML.gif. Then
      ϕ , | V | 1 / 2 G ν 0 , 0 π ν 0 | V | 1 / 2 ϕ = λ 0 2 | V | 1 / 2 u , | V | 1 / 2 G ν 0 , 0 π ν 0 | V | 1 / 2 | V | 1 / 2 u = λ 0 2 V u , G ν 0 , 0 π ν 0 V u = λ 0 2 C ν 0 | V u , | y | n 2 2 + ν 0 φ ν 0 | 2 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equat_HTML.gif
      In the last equality, we use the fact that
      G ν 0 , 0 = ( r τ ) ν 0 b ν 0 , 0 f 0 = d ν 0 b ν 0 , 0 ( r τ ) d 2 2 + ν 0 = C ν 0 ( r τ ) d 2 2 + ν 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equau_HTML.gif

      with d ν 0 = e 1 2 i π ν 0 2 2 ν 0 + 1 Γ ( ν 0 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq236_HTML.gif, and C ν 0 = d ν 0 b ν 0 , 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq237_HTML.gif. This ends the proof. □

      Theorem 3.8 Assume that 0 σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq128_HTML.gif. ϕ is defined in Lemma  3.7. If ρ 0 > 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq238_HTML.gif, one of three exclusive situations holds:
      1. (a)
        If σ 1 = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq239_HTML.gif, then
        e 1 ( λ ) = c ( λ λ 0 ) + o ( λ λ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equav_HTML.gif
         
      with c = ( λ 0 F 0 | V | 1 2 ϕ ) 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq240_HTML.gif.
      1. (b)
        If ν 0 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq241_HTML.gif, then
        e 1 ( λ ) = c λ λ 0 ln ( λ λ 0 ) + o ( λ λ 0 ln ( λ λ 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equaw_HTML.gif
         
      with c = λ 0 2 ϕ , | V | 1 / 2 G 1 , 0 π 1 | V | 1 / 2 ϕ 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq242_HTML.gif.
      1. (c)
        If ν 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq243_HTML.gif, then
        e 1 ( λ ) = c ( ( λ λ 0 ) 1 ν 0 ) + o ( ( λ λ 0 ) 1 ν 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equax_HTML.gif
         

      with c = λ 0 2 ϕ , | V | 1 / 2 G ν 0 , 0 π ν 0 | V | 1 / 2 ϕ 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq244_HTML.gif.

      Proof
      1. (a)
        By Theorem 2.1, one has
        R 0 ( α ) = F 0 + α F 1 + R 0 ( 1 ) ( α ) , in  L ( 1 , s ; 1 , s ) , s > 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equay_HTML.gif
         
      Then if ρ 0 > 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq245_HTML.gif, we can get K ( α ) = K ( 0 ) + | V | 1 / 2 ( α F 1 + R 0 ( 1 ) ( α ) ) | V | 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq246_HTML.gif in L ( 0 , 0 ; 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq247_HTML.gif. Because e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif is the simple eigenvalue of P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif, then λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq99_HTML.gif is the simple eigenvalue of K ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq248_HTML.gif. Since V 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq16_HTML.gif, one has that P ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq7_HTML.gif is monotonous with respect to λ and so is the e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif. Hence, K ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq248_HTML.gif and the eigenvalues of K ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq248_HTML.gif are monotonous with respect to λ. Therefore, we have that λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq99_HTML.gif is the biggest eigenvalue of K ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq248_HTML.gif. If not, suppose that a > λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq249_HTML.gif is an eigenvalue of K ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq248_HTML.gif, then by the continuity and monotony of the eigenvalue of K ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq248_HTML.gif with respect to λ, we know that there exists a constant λ < λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq250_HTML.gif such that λ σ ( K ( e 1 ( λ ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq251_HTML.gif. It follows that e 1 ( λ ) < e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq252_HTML.gif is an eigenvalue of P 0 + λ V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq253_HTML.gif. This is contradictory to that e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif is the smallest eigenvalue. By Lemma 3.7, we know the normalized eigenfunction u ˜ ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq254_HTML.gif of K ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq248_HTML.gif converges to ϕ. It follows u ˜ ( λ ) = P λ ϕ P λ ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq255_HTML.gif with P λ = 1 2 π i | E E 0 | = δ ( K ( e 1 ( λ ) ) E ) 1 d E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq256_HTML.gif. Then
      μ ( e 1 ( λ ) ) = u ˜ ( λ ) , K ( e 1 ( λ ) ) u ˜ ( λ ) = ϕ , K ( e 1 ( λ ) ) P λ ϕ ϕ , P λ ϕ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equaz_HTML.gif
      Here μ ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq257_HTML.gif is the eigenvalue of K ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq258_HTML.gif corresponding to the eigenfunction u ˜ ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq227_HTML.gif. By Lemma 3.6, we should compute ϕ , | V | 1 / 2 F 1 | V | 1 / 2 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq259_HTML.gif. Let ψ = F 0 | V | 1 / 2 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq260_HTML.gif. From the definition of ϕ, one has K ( 0 ) ϕ = λ 0 1 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq261_HTML.gif. Hence,
      ( P 0 + λ 0 V ) ψ = P 0 F 0 | V | 1 / 2 ϕ + λ 0 V F 0 | V | 1 / 2 ϕ = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equba_HTML.gif
      In the last equality, we use the fact P 0 F 0 | V | 1 / 2 ϕ = | V | 1 / 2 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq262_HTML.gif, which can be obtained by Proposition 3.2. Since ν 0 > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq263_HTML.gif, we have ψ L 2 ( R d ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq76_HTML.gif by Theorem 3.1 [1]. So, ψ is the ground state of P ( λ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq264_HTML.gif. We also have
      | V | 1 / 2 ψ = K ( 0 ) ϕ = λ 0 1 ϕ , ( P 0 α ) 1 | V | ψ = λ 0 1 ( ψ + α R 0 ( α ) ψ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbb_HTML.gif
      Hence,
      | V | 1 / 2 F 1 | V | 1 / 2 ϕ = λ 0 | V | 1 / 2 F 1 | V | 1 / 2 | V | 1 / 2 ψ = λ 0 α 1 | V | 1 / 2 ( R 0 ( α ) F 0 ) | V | ψ + O ( | α | ϵ ) = λ 0 α 1 | V | 1 / 2 λ 0 1 ( ψ + α R 0 ( α ) ψ ψ ) + O ( | α | ϵ ) = | V | 1 / 2 R 0 ( α ) ψ + O ( | α | ϵ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbc_HTML.gif
      It follows
      ϕ , | V | 1 / 2 F 1 | V | 1 / 2 ϕ = lim α 0 | V | 1 / 2 ϕ , R 0 ( α ) ψ = F 0 | V | 1 / 2 ϕ , ψ = ψ 2 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbd_HTML.gif
      So, μ ( α ) = λ 0 1 + c 1 α + o ( | α | 1 + ϵ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq265_HTML.gif with c 1 = F 0 | V | 1 2 ϕ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq266_HTML.gif. By the Proposition 2.2, one has μ ( e 1 ( λ ) ) = λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq267_HTML.gif. It follows
      λ 1 = λ 0 1 + c 1 e 1 ( λ ) + O ( | e 1 ( λ ) | 1 + ϵ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Eqube_HTML.gif
      Since λ 1 = λ 0 1 λ 0 2 ( λ λ 0 ) + O ( | λ λ 0 | 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq268_HTML.gif, we can get the leading term of e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif is c ( λ λ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq269_HTML.gif with c = ( λ 0 F 0 | V | 1 2 ϕ ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq270_HTML.gif.
      1. (b)
        If ν 0 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq271_HTML.gif, then
        K ( α ) = K ( 0 ) + α ln α | V | 1 / 2 G 1 , 0 π 1 | V | 1 / 2 + O ( α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbf_HTML.gif
         
      By Lemma 3.7, one has
      ϕ , | V | 1 / 2 G 1 , 0 π 1 | V | 1 / 2 ϕ 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbg_HTML.gif
      Then we have
      μ ( α ) = λ 0 1 + c 1 α ln α + o ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbh_HTML.gif
      with c 1 = ϕ , | V | 1 / 2 G 1 , 0 π 1 | V | 1 / 2 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq272_HTML.gif. As in (a), using μ ( e 1 ( λ ) ) = λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq273_HTML.gif and
      λ 1 = λ 0 1 λ 0 2 ( λ λ 0 ) + O ( | λ λ 0 | 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbi_HTML.gif
      one has λ 2 ( λ λ 0 ) + O ( | λ λ 0 | 2 ) = c e 1 ( λ ) ln e 1 ( λ ) + o ( e 1 ( λ ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq274_HTML.gif. To get the leading term of e 1 ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq95_HTML.gif, we can suppose that e 1 ( λ ) = ( λ λ 0 ) f ( λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq275_HTML.gif. Then, by comparing the leading term, we can get f ( λ ) = 1 / ln ( λ λ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq276_HTML.gif. It follows
      e 1 ( λ ) = c λ λ 0 ln ( λ λ 0 ) + o ( λ λ 0 ln ( λ λ 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbj_HTML.gif
      with c = λ 0 2 ϕ , | V | 1 / 2 G 1 , 0 π 1 | V | 1 / 2 ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq277_HTML.gif.
      1. (c)
        If ν 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq278_HTML.gif, one has
        K ( α ) = K ( 0 ) + 0 < ν 1 α ν | V | 1 / 2 G ν , 0 π ν | V | 1 / 2 + O ( | α | ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbk_HTML.gif
         
      By Lemma 3.7, we know that ϕ , | V | 1 / 2 G ν 0 , 0 π ν 0 | V | 1 / 2 ϕ 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq279_HTML.gif. Using the same argument as before, we can conclude
      μ ( α ) = λ 0 1 + c 1 α ν 0 + o ( | α | ν 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbl_HTML.gif
      with c 1 = ϕ , | V | 1 / 2 G ν 0 , 0 π ν 0 | V | 1 / 2 ϕ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq280_HTML.gif. As above, we can get that
      e 1 ( λ ) = c ( λ λ 0 ) 1 ν 0 + o ( ( λ λ 0 ) 1 ν 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_Equbm_HTML.gif

      with c = λ 0 2 ϕ , | V | 1 / 2 G ν 0 , 0 π ν 0 | V | 1 / 2 ϕ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-62/MediaObjects/13661_2012_Article_314_IEq281_HTML.gif. □

      Declarations

      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).

      Authors’ Affiliations

      (1)
      Mathematics and Statistics School, Henan University of Science and Technology
      (2)
      College of Civil Engineering and Architecture, Zhejiang University

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      Copyright

      © Jia and Zhao; licensee Springer 2013

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