No abstract

The fractional Schrödinger equation is solved for a free particle and for an infinite square potential well. The fundamental solution of the Cauchy problem for a free particle, the energy levels and the normalized wave functions of a particle in a potential well are obtained. In the barrier penetration problem, the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined. In the quantum scattering problem, according to the fractional Schrödinger equation, the Green’s function of the Lippmann-Schwinger integral equation is given.

The space fractional Schrödinger equation with linear potential, delta-function potential, and Coulomb potential is studied under momentum representation using Fourier transformation. By use of Mellin transform and its inverse transform, we obtain the energy levels and wave functions expressed in H function for a particle in linear potential field. The wave function expressed also by the H function and the unique energy level of the bound state for the particle of even parity state in delta-function potential well, which is proved to have no action on the particle of odd parity state, is also obtained. The integral form of the wave functions for a particle in Coulomb potential field is shown and the corresponding energy levels which have been discussed in Laskin’s paper [Phys. Rev. E 66, 056108 (2002)] are proved to satisfy an equality of infinite limit of the H function. All of these results contain the ones of the standard quantum mechanics as their special cases.

In this paper the generalized fractional Schrödinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schrödinger equation and the ones in standard quantum.

No abstract

Short chain carboxylic acids are well known as the precursors of fatty acid and polyketide biosynthesis. Iso-fatty acids, which are important for the control of membrane fluidity, are formed from branched chain starter units (isovaleryl-CoA and isobutyryl-CoA), which in turn are derived from the degradation of leucine and valine, respectively. Branched chain carboxylic acids are also employed as starter molecules for the biosynthesis of secondary metabolites, e.g. the therapeutically important anthelmintic agent avermectin or the electron transport inhibitor myxothiazol. During our studies on myxothiazol biosynthesis in the myxobacterium, Stigmatella aurantiaca, we detected a novel biosynthetic route to isovaleric acid. After cloning and inactivation of the branched chain keto acid dehydrogenase complex, which is responsible for the degradation of branched chain amino acids, the strain is still able to produce iso-fatty acids and myxothiazol. Incorporation studies employing deuterated leucine show that it can only serve as precursor in the wild type strain but not in the esg mutant. Feeding experiments using 13 C-labeled precursors show that isovalerate is efficiently made from acetate, giving rise to a labeling pattern in myxothiazol that provides evidence for a novel branch of the mevalonate pathway involving the intermediate 3,3-dimethylacryloyl-CoA. 3,3-Dimethylacrylic acid was synthesized in deuterated form and fed to the esg mutant, resulting in strong incorporation into myxothiazol and iso-fatty acids. Similar experiments employing Myxococcus xanthus revealed that the discovered biosynthetic route described is present in other myxobacteria as well.Fatty acids (FA) 1 are important building blocks of cell membranes. The various groups of prokaryotes differ remarkably in the structure and synthesis of fatty acid-derived lipids, which serve as reliable systematic marker molecules in chemotaxonomy (1). Unsaturated and branched chain (or iso-) FA increase the fluidity of the membrane and fulfill the function of thermal adaptation. The higher the content of these FA, the lower is the solid-to-liquid phase transition temperature of lipids.The biosynthesis of FA has been elucidated in detail, revealing that the difference between the biosynthetic pathways to FA and iso-FA lies only in the respective acylated primer acyl carrier proteins and in the condensing enzymes, which prefer modified primer acyl chains for elongation instead of the "normal" acetyl primer (1, 2). The modified primers used for iso-FA biosynthesis are the coenzyme A esters of isovaleric acid (IVA, resulting in iso-odd FA with an uneven number of carbon atoms), isobutyric acid (IBA, resulting in iso-even FA with an even number of carbon atoms), or 2-methyl butyric acid (2MBA, resulting in ante-iso FA, in which the methyl branch is located on the third carbon atom counted from the -end of the chain), which can be generated via the degradation pathway from branched chain amino acids or from intermediates of the biosyntheses of these essential amino...

No abstract

No abstract