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Eigenproblem for Jacobi matrices: hypergeometric series solution

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Journal Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
DatePublished - 28 Mar 2008
Issue number1867
Volume366
Number of pages26
Pages (from-to)1089-1114
Original languageEnglish

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 the entries of the two diagonals of length d-1 sandwiching the principal diagonal that gives the unperturbed spectrum.

The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series in 3d-5 variables in the generic case. To derive the result, we first rewrite the spectral problem for the Jacobi matrix as an equivalent system of algebraic equations, which are then solved 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.

    Research areas

  • Jacobi matrix, tridiagonal matrix, Lagrange inversion formula, spectral problem, multivariate hypergeometric series

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