Introduction to spectral theory in Hilbert spaces and applications

• Grant number: 23/05997-5
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: July 01, 2023
• End date: June 30, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Nataliia Goloshchapova
• Grantee: Lucas Nunes Fernandes Teles
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The objective of this project is to introduce the student to operator theory and spectral theory, culminating in the study of the application of these results to differential equations and operators that appear in quantum mechanics.The planned duration is 12 months. During this period the student will study 5 major topics.The first part will be devoted to the review of compact operators in normed spaces and their spectrum.In the second step, we will introduce bounded and unbounded operators in Hilbert spaces.The third part will be the main one, that is, we will study Spectral Theorem for bounded and unbounded operators in Hilbert spaces.In the penultimate part, the student will learn uniparametric groups and their applications in solving differential equations in Hilbert spaces.The last part will be linked with other applications of the Spectral Theorem:  Quantum interpretation of spectral measurements, spectral mapping, Min-Max theorem.The analysis of the results will be done from weekly meetings. In addition, the student will write two reports: the first will serve to report the main results explored and the other expository will introduce the content studied in more detail.