Bargmann空间中无界Gribov-Intissar算子的谱逼近（英文）

On Spectral Approximation of Unbounded Gribov-Intissar Operators in Bargmann Space

在[Adv.Math.(China),2015,44(3):335-353]中,我们研究了经典Bargmann空间Bo中的非自伴算子H_μ:H_μ=S_μ+H_λ,其中S_μ=μz d/(dz),H_λ=iλ(z(d^2)/(dz^2)+z^2 d/(dz)),i^2=-1,参数μ,λ都是实数.我们给出了H_μ的谱分析和H_μ的广义特征向量的渐近分析.设ek(z)=(z^k)/((k!)^(1/2)),k=1,2,…是B0的正交基.算子H_μ可以被一列三对角矩阵逼近,此三对角矩阵的主对角线元素为β_k=μk,次对角线元素α_k=iλk(k+1)^(1/2),1≤k≤n,n∈N.对于μ∈C和λ∈C,本文主要研究上述矩阵的特征值z_(k,n)(μ,λ)的局部化,它是多项式P_(n+1)^(μ,λ)(z)的零点,P_(n+1)^(μ,λ)(z)满足三项递推关系:若"∈R和λ∈R,则上述矩阵是复对称的.在这种情况下,我们证明了R上有界变分复值函数∈(z)的存在性,它使得权重为∈(z)的多项式P_n^(μ,λ)(z)是正交的.我们也考虑了H_μ的扰动H_λ'=S_λ'+H_λ,其中S_λ'=λ'z^2(d^2)/(dz^2)+S_μ,λ'∈R,H_λ可以被矩阵(h_(jk)~λ)_(j,k=1)~∞表示.证明了可以通过S_λ'的特征值和有限矩阵(h_(jk)~λ)_(j,k=1)~n的特征值的组合来逼近H_λ'的特征值.

In [Adv. Math. （China）, 2015, 44（3）： 335-353], we have considered on classical Bargmann space B0 the non-selfadjoint Jacobi-Gribov operator Hu defined by Hu = Su ＋ Hλ, where Su=uz d/dz,and Hλ=iλ（z d2/dz2＋z2 d/dz）,i2=-1 and u,λ are real parameters. We have given spectral analysis of Hu and asymptotic analysis of its generalized eigenvectors. Let ek（z）=zk/√ki,k=1,2,… be the normalized basis of B0. This operator can be approx- imated by a sequence of tridiagonal matrices with diagonal βk = uk and off-diagonal entries αk=iλk√k＋1,1≤k≤n,n∈N. For u ∈ C and A ∈ C, the main motive of this paper is the localization of eigenvalues zk,n（μ,λ） of the above matrices which are the zeros of the polynomials P u,λ n＋1（z） satisfying a three-term recurrence：{P u,λ（z）=0; P u,λ 1 （z）=1; α n-1 Pu,λ n-1（z）＋βn Pu,λ n（z）＋α n Pu,λ n＋1（z）=z Pμ,λ n（z）, n≥1. If u∈R and λ∈R, then the above matrices are complex symmetric. In this case we show the existence of complex-valued function λ（z） of bounded variation on R such that the polynomials Pu, λ n（z） are orthogonal with this weight ξ（z）. We consider aiso the foliowing perturbation of Hu defined by Hλ＇=Sλ＇＋Hλ where Sλ＇=λ＇z2 d2/dz2＋Su,with λ＇∈R,and Hλ is represented by the matrix （hλ jk）∞j,k=1 in {ek（z）} k ∞=1.Then we approximate the eigenvalues of Hλ＇ by a combination of the eigenvalues of Sλ＇ and the eigenvalues of finite matrix （hλ jk）n j,k=1.