Letters in Mathematical Physics

, 91:45

Convergence Radii for Eigenvalues of Tri-Diagonal Matrices


  • James Adduci
    • Department of MathematicsThe Ohio State University
    • Sabanci University
  • Boris Mityagin
    • Department of MathematicsThe Ohio State University

DOI: 10.1007/s11005-009-0366-8

Cite this article as:
Adduci, J., Djakov, P. & Mityagin, B. Lett Math Phys (2010) 91: 45. doi:10.1007/s11005-009-0366-8


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 47B36Secondary 47A10


tri-diagonal matrixoperator familyeigenvalues

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© Springer 2009