Witten deformed exterior derivative and Bessel functions

Abstract

In a recent paper we investigated the internal space of Bessel functions associated with their orders. We found a formula (new) unifying Bessel functions of integer and of real orders. In this paper we study the deformed exterior derivative system $H=d_{\lambda}$ on the puctured plane as a tentative to understand the origin of the formula and find that indeed similar formula occurs. This is no coincidence as we will demonstrate that generating functions of integer order Bessel functions and of real orders are respectively eigenstates of the usual exterior derivative and its deformation. As a direct consequence we rediscover the unifying formula and learn that the system linear in $d_{\lambda}$ is related to Bessel theory much as the system quadratic in ($d_{\lambda}+d_{\lambda}^{*}$) is related to Morse theory.