10 research outputs found
Using Spectral Method as an Approximation for Solving Hyperbolic PDEs
We demonstrate an application of the spectral method as a numerical
approximation for solving Hyperbolic PDEs. In this method a finite basis is
used for approximating the solutions. In particular, we demonstrate a set of
such solutions for cases which would be otherwise almost impossible to solve by
the more routine methods such as the Finite Difference Method. Eigenvalue
problems are included in the class of PDEs that are solvable by this method.
Although any complete orthonormal basis can be used, we discuss two
particularly interesting bases: the Fourier basis and the quantum oscillator
eigenfunction basis. We compare and discuss the relative advantages of each of
these two bases.Comment: 19 pages, 14 figures. to appear in Computer Physics Communicatio
A method for solving systems of non-linear differential equations with moving singularities
We present a method for solving a class of initial valued, coupled,
non-linear differential equations with `moving singularities' subject to some
subsidiary conditions. We show that this type of singularities can be
adequately treated by establishing certain `moving' jump conditions across
them. We show how a first integral of the differential equations, if available,
can also be used for checking the accuracy of the numerical solution.Comment: 9 pages, 7 eps figures, to appear in Comput. Phys. Co
Cosmological dynamics in tomographic probability representation
The probability representation for quantum states of the universe in which
the states are described by a fair probability distribution instead of wave
function (or density matrix) is developed to consider cosmological dynamics.
The evolution of the universe state is described by standard positive
transition probability (tomographic transition probability) instead of the
complex transition probability amplitude (Feynman path integral) of the
standard approach. The latter one is expressed in terms of the tomographic
transition probability. Examples of minisuperspaces in the framework of the
suggested approach are presented. Possibility of observational applications of
the universe tomographs are discussed.Comment: 16 page
Stephani-Schutz quantum cosmology
We study the Stephani quantum cosmological model in the presence of a
cosmological constant in radiation dominated Universe. In the present work the
Schutz's variational formalism which recovers the notion of time is applied.
This gives rise to Wheeler-DeWitt equations which can be cast in the form of
Schr\"odinger equations for the scale factor. We find their eigenvalues and
eigenfunctions by using the Spectral Method. Then we use the eigenfunctions in
order to construct wave packets and evaluate the time-dependent expectation
value of the scale factor, which is found to oscillate between non-zero finite
maximum and minimum values. Since the expectation value of the scale factor
never tends to the singular point, we have an initial indication that this
model may not have singularities at the quantum level.Comment: 6 pages, 4 figures, 1 table, to appear in PL
Accurate energy spectrum for double-well potential: periodic basis
We present a variational study of employing the trigonometric basis functions
satisfying periodic boundary condition for the accurate calculation of
eigenvalues and eigenfunctions of quartic double-well oscillators. Contrary to
usual Dirichlet boundary condition, imposing periodic boundary condition on the
basis functions results in the existence of an inflection point with vanishing
curvature in the graph of the energy versus the domain of the variable. We show
that this boundary condition results in a higher accuracy in comparison to
Dirichlet boundary condition. This is due to the fact that the periodic basis
functions are not necessarily forced to vanish at the boundaries and can
properly fit themselves to the exact solutions.Comment: 15 pages, 5 figures, to appear in Molecular Physic
Multi-dimensional classical and quantum cosmology: Exact solutions, signature transition and stabilization
We study the classical and quantum cosmology of a -dimensional
spacetime minimally coupled to a scalar field and present exact solutions for
the resulting field equations for the case where the universe is spatially
flat. These solutions exhibit signature transition from a Euclidean to a
Lorentzian domain and lead to stabilization of the internal space, in contrast
to the solutions which do not undergo signature transition. The corresponding
quantum cosmology is described by the Wheeler-DeWitt equation which has exact
solutions in the mini-superspace, resulting in wavefunctions peaking around the
classical paths. Such solutions admit parametrizations corresponding to metric
solutions of the field equations that admit signature transition.Comment: 15 pages, two figures, to appear in JHE
Radiative Correction to the Dirichlet Casimir Energy for Theory in Two Spatial Dimensions
In this paper, we calculate the next to the leading order Casimir energy for
real massive and massless scalar fields within theory,
confined between two parallel plates with the Dirichlet boundary condition in
two spatial dimensions. Our results are finite in both cases, in sharp contrast
to the infinite result reported previously for the massless case. In this paper
we use a renormalization procedure introduced earlier, which naturally
incorporates the boundary conditions. As a result our radiative correction term
is different from the previously calculated value. We further use a
regularization procedure which help us to obtain the finite results without
resorting to any analytic continuation techniques.Comment: 8 pages, 3 figure
Tomographic entropy and cosmology
The probability representation of quantum mechanics including propagators and
tomograms of quantum states of the universe and its application to quantum
gravity and cosmology are reviewed. The minisuperspaces modeled by oscillator,
free pointlike particle and repulsive oscillator are considered. The notion of
tomographic entropy and its properties are used to find some inequalities for
the tomographic probability determining the quantum state of the universe. The
sense of the inequality as a lower bound for the entropy is clarified.Comment: 19 page