Non-Archimedean mathematics meets quantum mechanics

Non-Archimedean mathematics, and in particular nonstandard analysis, is based on fields that contain infinitesimal and infinite elements. Within this framework, we construct a space of generalized functions known as ultrafunctions, which provides a richer setting for the description of physical systems in quantum mechanics. In this talk, I will introduce the basic ideas of nonstandard analysis and present the construction and key properties of ultrafunctions. I will then discuss their advantages in applications to quantum mechanics, with particular emphasis on the Schrödinger equation for Hamiltonians with singular potentials..