Spectral theory of unbounded operators in Hilbert spaces with application to the foundations of quantum mechanics

• Grant number: 25/17609-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: October 01, 2025
• End date: September 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Walter Alberto de Siqueira Pedra
• Grantee: Brian Reis
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

Quantum mechanics (QM), one of the pillars of modern physics, is formulated on a sophisticated mathematical basis, centered on Hilbert spaces and linear operators. However, undergraduate courses often simplify the formalism, treating all operators as if they were finite-dimensional matrices or bounded operators, which is not the case for fundamental observables such as position, momentum and energy (the Hamiltonian). These are, in fact, unbounded operators whose mathematical treatment requires tools from functional analysis, in particular Spectral Theory. This project proposes a systematic study of the spectral theory of unbounded self-adjoint operators, with the aim of solidifying the mathematical basis for understanding QM. We will investigate the subtleties of the definition of domain, the crucial distinction between symmetric and self-adjoint operators, and the Spectral Theorem. Finally, we will apply this rigorous formalism to canonical quantum systems, such as the free particle and the harmonic oscillator.particle and the quantum harmonic oscillator, contrasting the results with the traditional heuristic approach and highlighting how mathematical rigor prevents paradoxes and guarantees the consistency of the theory.