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Computation of extremal eigenvalues of large Hermitian and Hamiltonian
matrices in quantum and classical physics
E.V. Tsiper
School of Computational Sciences, George Mason University, Fairfax, VA 22030-44\
44
Center for Computational Materials Science, Naval Research Laboratory, Washingt\
on, DC 20375
Large Hermitian eigenvalue problems appear frequently in electronic
structure theory. The underlying quantum many-body problem is often
referred to as "incomputable" since it involves Hilbert spaces whose
dimensionality grows exponentially with the number of interacting
particles. This complexity is, in fact, the source of the idea of
quantum computation, since a moderately large quantum many-body system
possesses an amount of information greater than any conventional
computer can handle. Yet, exact diagonalization of large matrices
using conventional computers is an invaluable tool that helps us
understand many physical phenomena.
In this presentation I will review the Lanczos recursion and related
Krylov-subspace methods that allow us to compute extremal eigenvalues
of ultra-large Hermitian matrices. I will also discuss a
complementary classical problem that leads to Hamiltonian
(non-Hermitian) eigenvalue equation, and an intricate relation that
exists between its eigenvalues and the eigenvalue differences, or
excitation energies, of the quantum system. I will present a minimum
principle and TDHF-Lanczos recursion that generalize the Rayleigh
minimum principle and the Hermitian Lanczos recursion to the
Hamiltonian case. The presentation will be accompanied by several
examples from solid state physics and quantum chemistry.
Colloquium, George Mason University, October 27, 2005
presentation:
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