## Abstract

Let H be the Jacobi operator Hu(n) = u(n - 1) + u(n + 1) + v(n)u(n), u(0) = 0 acting on l^{2}(Z^{+}) where the potential v is real and v(n] → 0 as n → ∞. Let P be the orthogonal projection onto a closed linear subspace ℒ ⊂ l^{2}(Z^{+}), In a recent paper E.B. Davies defines the second order spectrum Spec_{2} (H, ℒ) of H relative to ℒ as the set of z ∈ C such that the restriction to ℒ of the operator P(H - z] ^{2}P is not invertible within the space ℒ. The present paper is devoted to study properties of Spec_{2} (H, ℒ} when ℒ is large but finite dimensional. Our particular interest is the connection between this set and the spectrum of H. In our main result we provide sharp bounds in terms of the potential v for the asymptotic behaviour of Spec_{2}(H, ℒ) as ℒ increases towards l^{2}(Z ^{+}).

Original language | Spanish |
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Pages (from-to) | 111-113 |

Number of pages | 3 |

Journal | Revista Mexicana de Fisica |

Volume | 49 |

Issue number | Suppl. 3 |

Publication status | Published - Nov 2003 |

## Keywords

- Estimation of the spectrum
- Non-self-adjoint projection methods
- Spectral theory of Jacobi operators

## ASJC Scopus subject areas

- General Physics and Astronomy