On Generalized Functions in Adelic Quantum Mechanics

TL;DR

This paper explores adelic generalized functions and their significance in adelic quantum mechanics, deriving key formulas and linking them to fundamental mathematical functions like the Riemann zeta function.

Contribution

It introduces the role of adelic generalized functions in quantum mechanics and establishes connections between adelic analysis and number theory.

Findings

Derivation of adelic product formula for Gauss integrals

Connection between Riemann zeta functional relation and quantum states

Demonstration of adelic generalized functions' importance in quantum mechanics

Abstract

Some aspects of adelic generalized functions, as linear continuous functionals on the space of Schwartz-Bruhat functions, are considered. The importance of adelic generalized functions in adelic quantum mechanics is demonstrated. In particular, adelic product formula for Gauss integrals is derived, and the connection between the functional relation for the Riemann zeta function and quantum states of the harmonic oscillator is stated.