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.
| Original language | English |
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| Pages (from-to) | 1089-1114 |
| Number of pages | 26 |
| Journal | Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences |
| Volume | 366 |
| Issue number | 1867 |
| DOIs | |
| Publication status | Published - 28 Mar 2008 |
Keywords
- Jacobi matrix
- tridiagonal matrix
- Lagrange inversion formula
- spectral problem
- multivariate hypergeometric series