Selçuk Ş. Bayin

Consistency problem of the solutions of the space fractional Schrödinger equation

Recently, consistency of the infinite square well solution of the space fractional Schrodinger equation has been the subject of some controversy. Hawkins and Schwarz [J. Math. Phys.54, 014101 (Year: 2013)]10.1063/1.4772533 objected to the way certain integrals are evaluated to show the consistency of the infinite square well solutions of the space fractional Schrodinger equation[S. S. Bayin, J. Math. Phys.53, 042105 (Year: 2012)10.1063/1.4705268; S. S. Bayin, J. Math. Phys.53, 084101 (Year: 2012)]10.1063/1.4739758. Here, we show for general n that as far as the integral representation of the solution in the momentum space is concerned, there is no inconsistency. To pinpoint the source of a possible inconsistency, we also scrutinize the different representations of the Riesz derivative that plays a central role in this controversy and show that they all have the same Fourier transform, when evaluated with consistent assumptions.

The Casimir energy of the twisted string loop: Uniform and two segment loops

We calculate the Casimir energy of a two segment loop of string with one normal boundary point and one twisted boundary point. The energy is renormalized relative to the twisted uniform loop. The use of the twisted loop in simplifying untwisted loop calculations is discussed.

Fluid sources for Bianchi I and III space-times

Four analytic solutions to the Einstein field equations are presented. The solutions are parametrized to have either Bianchi I or Bianchi III symmetry. The associated fluid parameters are given and some of them are discussed in detail.

Comment on “On the consistency of the solutions of the space fractional Schrödinger equation” [J. Math. Phys. 53, 042105 (2012)]

Recently we have reanalyzed the consistency of the solutions of the space fractional Schrodinger equation found in a piecewise manner, and showed that an exact and a proper treatment of the relevant integrals prove that they are consistent. In this comment, for clarity, we present additional information about the critical integrals and describe how their analytic continuation is accomplished.

Fractional boundaries for fluid spheres

A single Israel layer can be created when two metrics adjoin with no continuous metric derivative across the boundary. The properties of the layer depend only on the two metrics it separates. By using a fractional derivative match, a family of Israel layers can be created between the same two metrics. The family is indexed by the order of the fractional derivative. The method is applied to Tolman IV and V interiors and a Schwarzschild vacuum exterior. The method creates new ranges of modeling parameters for fluid spheres. A thin shell analysis clarifies pressure/tension in the family of boundary layers.

AN OPEN SINGULARITY-FREE COSMOLOGICAL MODEL WITH INFLATION

In the light of recent observations which point to an open universe (Ω0 < 1), we construct an open singularity-free cosmological model by reconsidering a model originally constructed for a closed universe. Our model starts from a nonsingular state called prematter, governed by an inflationary equation of state P = (γp - 1)ρ where γp (≃ 10-3) is a small positive parameter representing the initial vacuum dominance of the universe. Unlike the closed models universe cannot be initially static hence, starts with an initial expansion rate represented by the initial value of the Hubble constant H(0). Therefore, our model is a two-parameter universe model (γp,H(0)). Comparing the predictions of this model for the present properties of the universe with the recent observational results, we argue that the model constructed in this work could be used as a realistic universe model.

Casimir energy of the massless conformal scalar field on S-2 by the point-splitting method

We calculate the Casimir energy of the massless conformal scalar field on the surface (S-2) of a 3 dimensional Riemann sphere by using the point-splitting, mode sum and the ζ-function renormalization methods. We also consider the half space case with both the Dirichlet and the Neumann boundary conditions. This problem is interesting since the Casimir energy could be calculated analytically by various methods, thus allowing us to compare different regularization schemes.

Definition of the Riesz derivative and its application to space fractional quantum mechanics

We investigate and compare different representations of the Riesz derivative, which plays an important role in anomalous diffusion and space fractional quantum mechanics. In particular, we show that a certain representation of the Riesz derivative, R x α , that is generally given as also valid for α = 1, behaves no differently than the other definition given in terms of its Fourier transform. In the light of this, we discuss the α → 1 limit of the space fractional quantum mechanics and its consistency.

Relativistic Stars with Anisotropic Pressure

In this article we present various analytic solutions for anisotropic fluid spheres in general relativity.’ First we consider generalizations of t h e p = a p solution to the case where pressure is anisotropic and study the effects of anisotropy on the structure of neutron stars. Next we study radiating anisotropic fluid spheres and present three classes of analytic solutions. We also study slowly rotating anisotropic fluid spheres and present two analytic solutions corresponding to the nonradiating case. One of these solutions corresponds to uniform rotation, while the other corresponds to differential rotation. We also present differential equations to be solved for slowly rotating and radiating anisotropic fluids. Anisotropic pressure could be introduced by the existence of a solid core, by the presence of type-p superfluid, by the complexity of interactions, or by the existence of an external field. Recently it has been suggested that cooling of neutron stars might be accompanied by a phase transition from one anisotropic superfluid to another with significantly different properties.’ Besides these, an interesting way to generate anisotropic pressure is through the presence of two perfect fluids with the energymomentum tensor given as

AN OPEN INFLATIONARY MODEL FOR DIMENSIONAL REDUCTION AND ITS EFFECTS ON THE OBSERVABLE PARAMETERS OF THE UNIVERSE

Assuming that higher dimensions existed in the early stages of the universe where the evolution was inflationary, we construct an open, singularity-free, spatially homogeneous and isotropic cosmological model to study the effects of dimensional reduction that may have taken place during the early stages of the universe. We consider dimensional reduction to take place in a stepwise manner and interpret each step as a phase transition. By imposing suitable boundary conditions we trace their effects on the present day parameters of the universe.