19,562 research outputs found
Perturbation splitting for more accurate eigenvalues
Let be a symmetric tridiagonal matrix with entries and
eigenvalues of different magnitudes. For some , small entrywise
relative perturbations induce small errors in the eigenvalues,
independently of the size of the entries of the matrix; this is
certainly true when the perturbed matrix can be written as
with small . Even if it is
not possible to express in this way the perturbations in every
entry of , much can be gained by doing so for as many as
possible entries of larger magnitude. We propose a technique which
consists of splitting multiplicative and additive perturbations
to produce new error bounds which, for some matrices, are much
sharper than the usual ones. Such bounds may be useful in the
development of improved software for the tridiagonal eigenvalue
problem, and we describe their role in the context of a mixed
precision bisection-like procedure. Using the very same idea of
splitting perturbations (multiplicative and additive), we show
that when defines well its eigenvalues, the numerical values
of the pivots in the usual decomposition may
be used to compute approximations with high relative precision.Fundação para a Ciência e Tecnologia (FCT) - POCI 201
Comparing periodic-orbit theory to perturbation theory in the asymmetric infinite square well
An infinite square well with a discontinuous step is one of the simplest
systems to exhibit non-Newtonian ray-splitting periodic orbits in the
semiclassical limit. This system is analyzed using both time-independent
perturbation theory (PT) and periodic-orbit theory and the approximate formulas
for the energy eigenvalues derived from these two approaches are compared. The
periodic orbits of the system can be divided into classes according to how many
times they reflect from the potential step. Different classes of orbits
contribute to different orders of PT. The dominant term in the second-order PT
correction is due to non-Newtonian orbits that reflect from the step exactly
once. In the limit in which PT converges the periodic-orbit theory results
agree with those of PT, but outside of this limit the periodic-orbit theory
gives much more accurate results for energies above the potential step.Comment: 22 pages, 2 figures, 2 tables, submitted to Physical Review
Excitation energies from density functional perturbation theory
We consider two perturbative schemes to calculate excitation energies, each
employing the Kohn-Sham Hamiltonian as the unperturbed system. Using accurate
exchange-correlation potentials generated from essentially exact densities and
their exchange components determined by a recently proposed method, we evaluate
energy differences between the ground state and excited states in first-order
perturbation theory for the Helium, ionized Lithium and Beryllium atoms. It was
recently observed that the zeroth-order excitations energies, simply given by
the difference of the Kohn-Sham eigenvalues, almost always lie between the
singlet and triplet experimental excitations energies, corrected for
relativistic and finite nuclear mass effects. The first-order corrections
provide about a factor of two improvement in one of the perturbative schemes
but not in the other. The excitation energies within perturbation theory are
compared to the excitations obtained within SCF and time-dependent
density functional theory. We also calculate the excitation energies in
perturbation theory using approximate functionals such as the local density
approximation and the optimized effective potential method with and without the
Colle-Salvetti correlation contribution
Quantization with operators appropriate to shapes of trajectories and classical perturbation theory
Quantization is discussed for molecular systems having a zeroth order pair of doubly degenerate
normal modes. Algebraic quantization is employed using quantum operators appropriate to the
shape of the classical trajectories or wave functions, together with Birkhoff-Gustavson
perturbation theory and the W eyl correspondence for operators. The results are compared with a
previous algebraic quantization made with operators not appropriate to the trajectory shape.
Analogous results are given for a uniform semiclassical quantization based on Mathieu functions of fractional order. The relative sensitivities of these two methods (AQ and US) to the use of operators and coordinates related to and not related to the trajectory shape is discussed. The
arguments are illustrated using principally a Hamiltonian for which many previous results are available
The double well potential in quantum mechanics: a simple, numerically exact formulation
The double well potential is arguably one of the most important potentials in
quantum mechanics, because the solution contains the notion of a state as a
linear superposition of `classical' states, a concept which has become very
important in quantum information theory. It is therefore desirable to have
solutions to simple double well potentials that are accessible to the
undergraduate student. We describe a method for obtaining the numerically exact
eigenenergies and eigenstates for such a model, along with the energies
obtained through the Wentzel-Kramers-Brillouin (WKB) approximation. The exact
solution is accessible with elementary mathematics, though numerical solutions
are required. We also find that the WKB approximation is remarkably accurate,
not just for the ground state, but for the excited states as well.Comment: 10 pages, 4 figures; suitable for undergraduate courses in quantum
mechanic
Small x Resummation with Quarks: Deep-Inelastic Scattering
We extend our previous results on small-x resummation in the pure Yang--Mills
theory to full QCD with nf quark flavours, with a resummed two-by-two matrix of
resummed quark and gluon splitting functions. We also construct the
corresponding deep-inelastic coefficient functions, and show how these can be
combined with parton densities to give fully resummed deep-inelastic structure
functions F_2 and F_L at the next-to-leading logarithmic level. We discuss how
this resummation can be performed in different factorization schemes, including
the commonly used MSbar scheme. We study the importance of the resummation
effects by comparison with fixed-order perturbative results, and we discuss the
corresponding renormalization and factorization scale variation uncertainties.
We find that for x below 0.01 the resummation effects are comparable in size to
the fixed order NNLO corrections, but differ in shape. We finally discuss the
phenomenological impact of the small-x resummation, specifically in the
extraction of parton distribution from present day experiments and their
extrapolation to the kinematics relevant for future colliders such as the LHCComment: 45 pages, 16 figures, plain TeX with harvma
Quantum Dot Potentials: Symanzik Scaling, Resurgent Expansions and Quantum Dynamics
This article is concerned with a special class of the ``double-well-like''
potentials that occur naturally in the analysis of finite quantum systems.
Special attention is paid, in particular, to the so-called Fokker-Planck
potential, which has a particular property: the perturbation series for the
ground-state energy vanishes to all orders in the coupling parameter, but the
actual ground-state energy is positive and dominated by instanton
configurations of the form exp(-a/g), where a is the instanton action. The
instanton effects are most naturally taken into account within the modified
Bohr-Sommerfeld quantization conditions whose expansion leads to the
generalized perturbative expansions (so-called resurgent expansions) for the
energy values of the Fokker-Planck potential. Until now, these resurgent
expansions have been mainly applied for small values of coupling parameter g,
while much less attention has been paid to the strong-coupling regime. In this
contribution, we compare the energy values, obtained by directly resumming
generalized Bohr-Sommerfeld quantization conditions, to the strong-coupling
expansion, for which we determine the first few expansion coefficients in
powers of g^(-2/3). Detailed calculations are performed for a wide range of
coupling parameters g and indicate a considerable overlap between the regions
of validity of the weak-coupling resurgent series and of the strong-coupling
expansion. Apart from the analysis of the energy spectrum of the Fokker-Planck
Hamiltonian, we also briefly discuss the computation of its eigenfunctions.
These eigenfunctions may be utilized for the numerical integration of the
(single-particle) time-dependent Schroedinger equation and, hence, for studying
the dynamical evolution of the wavepackets in the double-well-like potentials.Comment: 13 pages; RevTe
Tunnel splittings for one dimensional potential wells revisited
The WKB and instanton answers for the tunnel splitting of the ground state in
a symmetric double well potential are both reduced to an expression involving
only the functionals of the potential, without the need for solving any
auxilliary problems. This formula is applied to simple model problems. The
prefactor for the splitting in the text book by Landau and Lifshitz is amended
so as to apply to the ground and low lying excited states.Comment: Revtex; 1 ps figur
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