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 $L_{2}$-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

Poles of regular quaternionic functions

This paper studies the singularities of Cullen-regular functions of one quaternionic variable. The quaternionic Laurent series prove to be Cullen-regular. The singularities of Cullen-regular functions are thus classified as removable, essential or poles. The quaternionic analogues of meromorphic complex functions, called semiregular functions, turn out to be quotients of Cullen-regular functions with respect to an appropriate division operation. This allows a detailed study of the poles and their distribution.Comment: 14 page

Some recent developments in the transmutation operator approach

This is a brief overviewof some recent developments in the transmutation operator approach to practical solution of mathematical physics problems. It introduces basic notions and results of transmutation theory, and gives a brief historical survey with some important references. Mainly applications to linear ordinary and partial differential equations and to related boundary value and spectral problems are discusse

One-Dimensional and Multi-Dimensional Integral Transforms of Buschman–Erdélyi Type with Legendre Functions in Kernels

This paper consists of two parts. In the first part we give a brief survey of results on Buschman–Erdélyi operators, which are transmutations for the Bessel singular operator. Main properties and applications of Buschman–Erdélyi operators are outlined. In the second part of the paper we consider multi-dimensional integral transforms of Buschman–Erdélyi type with Legendre functions in kernels. Complete proofs are given in this part, main tools are based on Mellin transform properties and usage of Fox H-functions

Complex-Distance Potential Theory and Hyperbolic Equations

An extension of potential theory in R^n is obtained by continuing the Euclidean distance function holomorphically to C^n. The resulting Newtonian potential is generated by an extended source distribution D(z) in C^n whose restriction to R^n is the delta function. This provides a natural model for extended particles in physics. In C^n, interpreted as complex spacetime, D(z) acts as a propagator generating solutions of the wave equation from their initial values. This gives a new connection between elliptic and hyperbolic equations that does not assume analyticity of the Cauchy data. Generalized to Clifford analysis, it induces a similar connection between solutions of elliptic and hyperbolic Dirac equations. There is a natural application to the time-dependent, inhomogeneous Dirac and Maxwell equations, and the `electromagnetic wavelets' introduced previously are an example.Comment: 25 pages, submited to Proceedings of 5th Intern. Conf. on Clifford Algebras, Ixtapa, June 24 - July 4, 199

Observation of Electron-Hole Puddles in Graphene Using a Scanning Single Electron Transistor

The electronic density of states of graphene is equivalent to that of relativistic electrons. In the absence of disorder or external doping the Fermi energy lies at the Dirac point where the density of states vanishes. Although transport measurements at high carrier densities indicate rather high mobilities, many questions pertaining to disorder remain unanswered. In particular, it has been argued theoretically, that when the average carrier density is zero, the inescapable presence of disorder will lead to electron and hole puddles with equal probability. In this work, we use a scanning single electron transistor to image the carrier density landscape of graphene in the vicinity of the neutrality point. Our results clearly show the electron-hole puddles expected theoretically. In addition, our measurement technique enables to determine locally the density of states in graphene. In contrast to previously studied massive two dimensional electron systems, the kinetic contribution to the density of states accounts quantitatively for the measured signal. Our results suggests that exchange and correlation effects are either weak or have canceling contributions.Comment: 13 pages, 5 figure

The Static Maxwell System in Three Dimensional Axially Symmetric Inhomogeneous Media and Axially Symmetric Generalization of the Cauchy–Riemann System

In this paper we discuss different generalizations of the Cauchy–Riemann system and their connection with the static Maxwell system. In particular, this allows us to present relations between slice-monogenic functions and hypermonogenic functions, as well as to provide a physical interpretation of slice-monogenic functions. Furthermore, we present an explicit and complete set of basic solutions of a new class of axial-hypermonogenic functions in R^3. In the end we determine the symmetry operators for the class of axial-hypermonogenic functions

Necessary condition for the existence of an intertwining operator and classification of transmutations on its basis

The authors study second-order ordinary differential operators with functional coefficients for all derivatives and the Volterra integral operator with a definite kernel. Results of the paper establish a hyperbolic equation and additional conditions that allow one to construct a kernel according to the OD

Prognostic Value of the PRAME Gene Expression in T-Cell Lymphoproliferative Disorders

Background. T-cell lymphomas (T-CL) represent a heterogeneous group of malignant lymphoproliferative disorders characterized by unfavorable prognosis. The cancer-testis PRAME gene is notable for its spontaneous expression in transformed cells as observed in solid tumors, B-cell lymphoproliferative and chronic myeloproliferative diseases. Activity and clinical significance of PRAME in T-CL was not studied before, which determines the relevance and provides ground for the present trial. Aim. To assess the clinical significance of the PRAME gene expression in T-CL. Materials & Methods. PRAME gene expression level was measured in samples of lymph nodes, blood, and bone marrow from 35 T-CL patients. Among them 3 patients received allogeneic hematopoietic stem cell transplantation, and 6 patients received autologous hematopoietic stem cell transplantation. A correlation was established between the PRAME expression in bone marrow and peripheral blood with morphological markers of disseminated disease with bone marrow lesions and leukemic blood. PRAME expression level was correlated with survival parameters and tumor proliferative activity (Ki-67). Results. PRAME activity was observed in 21 (60 %) patients. PRAME hyperexpression is associated with advanced stages of disease (p = 0.0734), bone marrow lesions (p = 0.0289), leukemic blood (p = 0.0187), worsening of the overall survival (OS) (p = 0.0787) and event-free survival (EFS) (p = 0.7185), also after hematopoietic stem cell transplantation (p = 0.2661 for OS and p = 0.0452 for EFS), and with a high Ki-67 expression level (p = 0.0155). Conclusion. PRAME expression in T-CL is often observed and related with unfavorable clinical prognosis