On the Absence of Spurious Eigenstates in an Iterative Algorithm Proposed By Waxman.pdf

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On the Absence of Spurious Eigenstates in an Iterative Algorithm Proposed By Waxman.pdf

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On the Absence of Spurious Eigenstates in an Iterative Algorithm Proposed By Waxman

a r X i v : q u a n t - p h / 0 6 0 2 1 3 5 v 1 1 6 F e b 2 0 0 6 On the Absence of Spurious Eigenstates in an Iterative Algorithm Proposed By Waxman R. A. Andrew, H. G. Miller?, and A. R. Plastino Department of Physics, University of Pretoria, Pretoria 0002, South Africa Abstract We discuss a remarkable property of an iterative algorithm for eigenvalue problems recently ad- vanced by Waxman that constitutes a clear advantage over other iterative procedures. In quantum mechanics, as well as in other fields, it is often necessary to deal with operators exhibiting both a continuum and a discrete spectrum. For this kind of operators, the problem of identifying spurious eigenpairs which appear in iterative algorithms like the Lanczos algorithm does not occur in the algorithm proposed by Waxman. PACS 03.65.Ge, 02.60.Lj ? E-Mail: hmiller@maple.up.ac.za 1 The Hamiltonian operator which describes a quantum mechanical system generally pos- sesses both a continuum as well as a discrete spectrum. A similar situation also occurs in other fields, such as theoretical population genetics [1]. In many cases one is only inter- ested in a few of the lower-lying bound states of the system. When only bound states are present iterative algorithms such as the Lanczos algorithm [2] yield good approximations to the lower-lying eignstates with good convergence properties [3, 4, 5, 6]. On the other hand, the presence of the continuum leads to complications which can be circumvented but not without introducing spurious eigensolutions that need to be identified and eliminated [7]. Such spurious eigensolutions, however, do not occur in an algorihthm recently proposed by Waxman[8]. We compare the two algorithms and demonstrate this behaviour in a simple numerical example. The presence of the continuum leads to complications in the Lanczos algorithm[2]. Find- ing a suitable start vector is by no means trivial[9] since the Lanczos algorithm can only be applied to states which are normalizable