Onirban Islam
Postdoctoral Researcher, Institute for Mathematics, University of Potsdam, Germany
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School on Mathematics for Physics 1: Operators on Hilbert Spaces
Instructor
Teaching Assistants
Tanmoy Kumar Sarkar
BRAC University
Atonu Roy Chowdhury
Lecturer, Department of Computer Science and Engineering, BRAC University
Abstract
This short course is intended to offer a gentle introduction to unbounded operators on Hilbert spaces pertinent to quantum mechanics. Our starting point will be to point out the differences between operators on finite-dimensional and infinite-dimensional Hilbert spaces, and to understand why the latter is essential for formulating quantum mechanics in a mathematically rigorous fashion. We will begin with the basics of unbounded operators on Hilbert spaces and rigged Hilbert spaces in order to explain the standard Heisenberg matrix mechanics and Dirac bra-ket formalisms. Our aim is to comprehend the quantum mechanical simple harmonic oscillator and its spectrum from an operator-theoretic viewpoint. A rudimentary background in linear algebra is assumed, although a few topics will be recalled in due course in order to motivate the analysis in an infinite-dimensional Hilbert space setting.
This school introduces operator-theoretic methods that are central to a rigorous treatment of quantum mechanics. It is designed for students who want a careful bridge from finite-dimensional linear algebra to the infinite-dimensional Hilbert space setting used in modern quantum theory.
Course Outline and Schedule
| Session | Topic | Date and time (BST) | Lecturer / Speaker | Lecture Notes | Recording |
|---|---|---|---|---|---|
| L01 | Finite-dimensional Hilbert Spaces | 01/05/2026, 7:00 PM - 9:00 PM | Onirban Islam | Lecture Notes | Recording |
| L02 | Infinite-dimensional Hilbert Spaces | 02/05/2026, 7:00 PM - 9:00 PM | Onirban Islam | Lecture Notes | Recording |
| L03 | Bounded Operators | 03/05/2026, 7:00 PM - 9:00 PM | Onirban Islam | Lecture Notes | Recording |
| T01 | Real Numbers: Density, Completeness, and Equivalent Characterizations | 09/05/2026, 7:00 PM - 9:00 PM | Tanmoy Kumar Sarkar | Lecture Notes | Recording |
| L04 | Unbounded Operators | 15/05/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L05 | Self-adjoint Operators | 16/05/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| T02 | Tutorial Session | 23/05/2026, 7:00 PM - 9:00 PM | To be announced | Pending | Pending |
| L06 | Spectrum | 30/05/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L07 | Spectral Theorem - I | 31/05/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L08 | Spectral Theorem - II | 05/06/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L09 | Spectral Theorem - III | 06/06/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| T03 | Tutorial Session | 07/06/2026, 7:00 PM - 9:00 PM | To be announced | Pending | Pending |
| L10 | Algebraic and Topological Duals | 12/06/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L11 | Gelfand Triplet and Dirac Bra-ket Formalism | 13/06/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L12 | Position and Momentum Operators in Quantum Mechanics | 14/06/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L13 | Canonical Commutation Relations in Quantum Mechanics | 19/06/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| T04 | Tutorial Session | 21/06/2026, 7:00 PM - 9:00 PM | To be announced | Pending | Pending |
| L14 | One-parameter Group of Unitary Operators | 03/07/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L15 | Spectral Theory of a Quantum Harmonic Oscillator | 04/07/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
| L16 | Trace-class Operators and Partition Function of a Quantum Harmonic Oscillator | 05/07/2026, 7:00 PM - 9:00 PM | Onirban Islam | Pending | Pending |
