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SUMMARYIn this paper we prove a L p -decomposition where one of the components is the kernel of a first-order differential operator that factorizes the non-stationary Schrödinger operator − −i* t .

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator ∆ (α,β,γ) + := D 1+α x + 0 + D 1+β y + 0 + D 1+γ z + 0 , where (α, β, γ) ∈ ]0, 1] 3 , and the fractional derivatives D 1+α x + 0 , D 1+β y + 0 , D 1+γ z + 0 are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator ∆ (α,β,γ) + in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

The aim of this work is to study the numerical solution of the nonlinear Schrödinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the nonstationary Schrödinger operator and we show the convergence of the numerical scheme. Numerical examples are given at the end of the article.

In this paper, we study the solutions to the Schrödinger equation on some conformally flat cylinders and on the n-torus. First, we apply an appropriate regularization procedure. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the regularized parabolic-type Dirac operator. We study their fundamental properties, give representation formulas of all these solutions in terms of multiperiodic generalizations of the elliptic functions in the context of the regularized parabolic-type Dirac operator. Furthermore, we also develop some integral representation formulas. In particular, we set up a Green type integral formula for the solutions to the homogeneous regularized Schrödinger equation on cylinders and n-tori. Then, we treat the inhomogeneous Schrödinger equation with prescribed boundary conditions in Lipschitz domains on these manifolds. We present an L p -decomposition where one of the components is the kernel of the first-order differential operator that factorizes the cylindrical (resp. toroidal) Schrödinger operator. Finally, we study the behavior of our results in the limit case where the regularization parameter tends to zero.

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In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann–Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers are constructed. Moreover, we establish the fractional Cauchy–Kovalevskaya extension (FCK extension) theorem for fractional monogenic functions defined on ℝd. Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We study the connection between the FCK extension of functions of the form xPl and the classical Gegenbauer polynomials. Finally, we present an example of an FCK extension.

In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation with time-fractional derivatives of orders α ∈]0, 1] and β ∈]1, 2] in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS expressed in terms of a multivariate MittagLeffler function in the Fourier domain. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension n and of the fractional parameters α and β.