Article

Letters in Mathematical Physics

, 91:45

First online:

Convergence Radii for Eigenvalues of Tri-Diagonal Matrices

  • James AdduciAffiliated withDepartment of Mathematics, The Ohio State University
  • , Plamen DjakovAffiliated withSabanci University Email author 
  • , Boris MityaginAffiliated withDepartment of Mathematics, The Ohio State University

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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 L kk  = k 2, and the matrix B is off-diagonal, with nonzero entries B k,k+1 = B k+1,k  = k α , 0 ≤ α < 2. The spectrum of L + zB is discrete. For small |z| the nth eigenvalue E n (z), E n (0) = n 2, is a well-defined analytic function. Let R n 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