We provide a very efficient procedure for obtaining the excited states of a quantum operator H, in any arbitrary chosen energy range, independently from the knowledge of the states at lower energies. Our procedure consists in determining, within the Lanczos algorithm, the ground state of the auxiliary operator A=(H-Et)2, and hence the eigenvalue of H nearest in energy to Et, where Et is any chosen trial energy in the energy range of interest. We show that a variational method exploiting diagonalization of 2×2 Lanczos matrices, combined with a two-pass Lanczos procedure of relatively small number of iterations, produces eigenvalues of H within any desired accuracy, as well as the corresponding eigenfunctions. We discuss in particular the T⊗τ Jahn-Teller model, as a realistic prototype of a quantum system with a very large number of degrees of freedom.

LANCZOS-TYPE ALGORITHM FOR EXCITED-STATES OF VERY-LARGE-SCALE QUANTUM-SYSTEMS

GROSSO, GIUSEPPE;MARTINELLI, LIANA;
1995

Abstract

We provide a very efficient procedure for obtaining the excited states of a quantum operator H, in any arbitrary chosen energy range, independently from the knowledge of the states at lower energies. Our procedure consists in determining, within the Lanczos algorithm, the ground state of the auxiliary operator A=(H-Et)2, and hence the eigenvalue of H nearest in energy to Et, where Et is any chosen trial energy in the energy range of interest. We show that a variational method exploiting diagonalization of 2×2 Lanczos matrices, combined with a two-pass Lanczos procedure of relatively small number of iterations, produces eigenvalues of H within any desired accuracy, as well as the corresponding eigenfunctions. We discuss in particular the T⊗τ Jahn-Teller model, as a realistic prototype of a quantum system with a very large number of degrees of freedom.
Grosso, Giuseppe; Martinelli, Liana; Pastori Parravicini, G.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/202896
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