Eigenproblem for Jacobi matrices: hypergeometric series solution

Abstract

We study the perturbative power-series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The(small) expansion parameters are being the entries of the two diagonals of length d-1 sandwiching the principal diagonal, which gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series of 3d-5 variables in the generic case, or 2d-3 variables for the eigenvalue growing from a corner matrix element. To derive the result, we first rewrite the spectral problem for a Jacobi matrix as an equivalent system of cubic equations, which are then resolved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.Comment: Latex, 20 pages; v2: corrected typos, added section with example