Regularization of the non‐stationary Schrödinger operator

2010

J. Evol. Equ.

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.

Section: Regularization Of the Schrödinger Operatormentioning

confidence: 98%

“…is the fundamental solution to the Schrödinger operator (see [2]). It has a singularity in all the points of the hyperplane t=0.…”

Section: Regularization Of the Schrödinger Operatormentioning

confidence: 99%

“…For more details about this regularization procedure, we refer the reader to the preceding papers [2] and [15].…”

Section: Regularization Of the Schrödinger Operatormentioning

confidence: 99%

“…For more details about the application of this regularization procedure to the Schrödinger operator in the context of Clifford analysis, see for instance [2,15].…”

Section: Regularization Of the Schrödinger Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Schrödinger equation on cylinders and the n-torus

Kraußhar

2010

J. Evol. Equ.

Time‐dependent operators on some non‐orientable projective orbifolds

Kraußhar

Rodrigues

Math Methods in App Sciences

2015

In this paper, we present an explicit construction for the fundamental solution of the heat operator, the Schrödinger operator, and related first-order parabolic Dirac operators on a class of some conformally flat non-orientable orbifolds. More concretely, we treat a class of projective cylinders and tori where we can study parabolic monogenic sections with values in different pin bundles. We present integral representation formulas together with some elementary tools of harmonic analysis that enable us to solve boundary value problems on these orbifolds. 5305-5319 5305 R. S. KRAUßHAR, M. M. RODRIGUES AND N. VIEIRA example, 'upper' and 'lower' parts with each other. This is translated into an additional symmetry structure, which actually leads to a singularity-in general, we are now dealing with orbifolds in this context. We explain how we can construct associated Cauchy kernels for the parabolic Dirac operator, as well as heat and Schrödinger kernels for this class of orbifolds. Similarly to the oriented case, we can consider distinct pin bundles in the non-oriented case. We show how arbitrary pin sections can be represented by these kernels. We also set up integral representation formulas for the sections in the kernel of the parabolic Dirac (resp. heat and Schrödinger) operator on these orbifolds.This paper also provides a continuation of [11] where explicit formulas for the Cauchy and Green's kernels for the elliptic Dirac and Laplace operators have been developed.The structure of the paper is as follows. In Section 2, we recall some basic facts about Clifford algebras and about factorization and regularization of time-dependent operators. In Section 3, we define the spin and the pin structures that are necessary for the development of our work. In the following section, we study the non-orientable projective counterparts of cylinders and tori. We start by presenting the geometric context. Then we construct the Cauchy and heat kernels, as well as the corresponding integral formulas. Section 5 is dedicated to the study of the Schrödinger operator. Here, we start by studying the inhomogeneous Schrödinger equation in orbifolds in terms of a proper Teodorescu and Cauchy operators. In this last section, we finally present a Hodge-type decomposition for the regularized case, and we will study the behavior of our result when the regularization parameter tends to zero.

Fundamental solution of the time‐fractional telegraph Dirac operator

Ferreira

Rodrigues

Math Methods in App Sciences

2017

In this work, we obtain the fundamental solution (FS) of the multidimensional time-fractional telegraph Dirac operator where the 2 time-fractional derivatives of orders ∈]0, 1] and ∈]1, 2] are in the Caputo sense. Explicit integral and series representation of the FS are obtained for any dimension. We present and discuss some plots of the FS for some particular values of the dimension and of the fractional parameters and . Finally, using the FS, we study some Poisson and Cauchy problems.