2,204 research outputs found
An effective spectral collocation method for the direct solution of high-order ODEs
This paper reports a new Chebyshev spectral collocation method for directly solving high-order ordinary differential equations (ODEs). The construction of the Chebyshev approximations is based on integration rather than conventional differentiation. This use of integration allows the multiple boundary conditions to be incorporated more efficiently. Numerical results show that the
proposed formulation significantly improves the conditioning of the system and yields more accurate results and faster convergence rates than conventional formulations
Transmutations and spectral parameter power series in eigenvalue problems
We give an overview of recent developments in Sturm-Liouville theory
concerning operators of transmutation (transformation) and spectral parameter
power series (SPPS). The possibility to write down the dispersion
(characteristic) equations corresponding to a variety of spectral problems
related to Sturm-Liouville equations in an analytic form is an attractive
feature of the SPPS method. It is based on a computation of certain systems of
recursive integrals. Considered as families of functions these systems are
complete in the -space and result to be the images of the nonnegative
integer powers of the independent variable under the action of a corresponding
transmutation operator. This recently revealed property of the Delsarte
transmutations opens the way to apply the transmutation operator even when its
integral kernel is unknown and gives the possibility to obtain further
interesting properties concerning the Darboux transformed Schr\"{o}dinger
operators.
We introduce the systems of recursive integrals and the SPPS approach,
explain some of its applications to spectral problems with numerical
illustrations, give the definition and basic properties of transmutation
operators, introduce a parametrized family of transmutation operators, study
their mapping properties and construct the transmutation operators for Darboux
transformed Schr\"{o}dinger operators.Comment: 30 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1111.444
Dirac Hamiltonian with superstrong Coulomb field
We consider the quantum-mechanical problem of a relativistic Dirac particle
moving in the Coulomb field of a point charge . In the literature, it is
often declared that a quantum-mechanical description of such a system does not
exist for charge values exceeding the so-called critical charge with based on the fact that the standard expression for the
lower bound state energy yields complex values at overcritical charges. We show
that from the mathematical standpoint, there is no problem in defining a
self-adjoint Hamiltonian for any value of charge. What is more, the transition
through the critical charge does not lead to any qualitative changes in the
mathematical description of the system. A specific feature of overcritical
charges is a non uniqueness of the self-adjoint Hamiltonian, but this non
uniqueness is also characteristic for charge values less than the critical one
(and larger than the subcritical charge with ). We present the spectra and (generalized) eigenfunctions for all
self-adjoint Hamiltonians. The methods used are the methods of the theory of
self-adjoint extensions of symmetric operators and the Krein method of guiding
functionals. The relation of the constructed one-particle quantum mechanics to
the real physics of electrons in superstrong Coulomb fields where multiparticle
effects may be of crucial importance is an open question.Comment: 44 pages, LaTex file, to be published in Teor.Mat.Fiz.
(Theor.Math.Phys.
A frequency-independent solver for systems of first order linear ordinary differential equations
When a system of first order linear ordinary differential equations has
eigenvalues of large magnitude, its solutions exhibit complicated behaviour,
such as high-frequency oscillations, rapid growth or rapid decay. The cost of
representing such solutions using standard techniques typically grows with the
magnitudes of the eigenvalues. As a consequence, the running times of standard
solvers for ordinary differential equations also grow with the size of these
eigenvalues. The solutions of scalar equations with slowly-varying
coefficients, however, can be efficiently represented via slowly-varying phase
functions, regardless of the magnitudes of the eigenvalues of the corresponding
coefficient matrix. Here, we couple an existing solver for scalar equations
which exploits this observation with a well-known technique for transforming a
system of linear ordinary differential equations into scalar form. The result
is a method for solving a large class of systems of linear ordinary
differential equations in time independent of the magnitudes of the eigenvalues
of their coefficient matrices. We discuss the results of numerical experiments
demonstrating the properties of our algorithm.Comment: arXiv admin note: text overlap with arXiv:2308.0328
Pseudospectral Calculation of Helium Wave Functions, Expectation Values, and Oscillator Strength
The pseudospectral method is a powerful tool for finding highly precise
solutions of Schr\"{o}dinger's equation for few-electron problems. We extend
the method's scope to wave functions with non-zero angular momentum and test it
on several challenging problems. One group of tests involves the determination
of the nonrelativistic electric dipole oscillator strength for the helium
S P transition. The result achieved, , is
comparable to the best in the literature.
Another group of test applications is comprised of well-studied leading order
finite nuclear mass and relativistic corrections for the helium ground state. A
straightforward computation reaches near state-of-the-art accuracy without
requiring the implementation of any special-purpose numerics.
All the relevant quantities tested in this paper -- energy eigenvalues,
S-state expectation values and bound-bound dipole transitions for S and P
states -- converge exponentially with increasing resolution and do so at
roughly the same rate. Each individual calculation samples and weights the
configuration space wave function uniquely but all behave in a qualitatively
similar manner. Quantum mechanical matrix elements are directly and reliably
calculable with pseudospectral methods.
The technical discussion includes a prescription for choosing coordinates and
subdomains to achieve exponential convergence when two-particle Coulomb
singularities are present. The prescription does not account for the wave
function's non-analytic behavior near the three-particle coalescence which
should eventually hinder the rate of the convergence. Nonetheless the effect is
small in the sense that ignoring the higher-order coalescence does not appear
to affect adversely the accuracy of any of the quantities reported nor the rate
at which errors diminish.Comment: 24 pages, 12 figures, 6 tables. To be submitted to Physical Review A.
LANL identifier 'LA-UR-11-10986
Characterization of the functionally graded shear modulus of a half-space
In this article, a method is proposed for determining parameters of the exponentialy varying shear modulus of a functionally graded half-space. The method is based on the analytical solution of the problem of pure shear of an elastic functionally graded half-space by a strip punch. The half-space has the depth-wise exponential variation of its shear modulus, whose parameters are to be determined. The problem is reduced to an integral equation that is then solved by asymptotic methods. The analytical relations for contact stress under the punch, displacement of the free surface outside the contact area and other characteristics of the problem are studied with respect to the shear modulus parameters. The parameters of the functionally graded half-space shear modulus are determined (a) from the coincidence of theoretical and experimental values of contact stresses under the punch and from the coincidence of forces acting on the punch, or (b) from the coincidence of theoretical and experimental values of displacement of the free surface of the half-space outside the contact and coincidence of forces acting on the punch, or (c) from other conditions. The transcendental equations for determination of the shear modulus parameters in cases (a) and (b) are given. By adjusting the parameters of the shear modulus variation, the regions of "approximate-homogeneous" state in the functionally graded half-space are developed
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