Time‐dependent operators on some non‐orientable projective orbifolds

Abstract

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.