Convergence Radii for Eigenvalues of Tri-Diagonal Matrices

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Abstract

Consider a family of infinite tri-diagonal matrices of the form L + zB, where the matrix L is diagonal with entries Lkk = k2, and the matrix B is off-diagonal, with nonzero entries Bk,k+1 = Bk+1,k = kα, 0 ≤ α < 2. The spectrum of L + zB is discrete. For small |z| the nth eigenvalue En (z), En (0) = n2, is a well-defined analytic function. Let Rn be the convergence radius of its Taylor’s series about z = 0. It is proved that
$$R_n \leq C(\alpha) n^{2-\alpha}\quad \text{if}\enspace 0 \leq \alpha <11 /6$$
.

Mathematics Subject Classification (2000)

Primary 47B36 Secondary 47A10 

Keywords

tri-diagonal matrix operator family eigenvalues 

Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Sabanci UniversityTuzlaTurkey

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