Lectures related to Spectral theorem

Quantum Mechanics: Postulates and Observables

Explains the postulates of quantum mechanics and the representation of observables by operators.

Spectral Expansion: Eisenstein Series

Covers the spectral expansion of Eisenstein series and the properties of hyperbolic surfaces.

Spectral Theorem: Second

Covers the spectral theorem, focusing on the second part and orthonormal sequences in a separable Hilbert space.

Diagonalization of Symmetric Matrices

Covers the diagonalization of symmetric matrices, the spectral theorem, and the use of spectral decomposition.

Quantum Measurement Postulate

Explores the verification of PVM properties and the spectral theorem for unbounded operators in quantum mechanics.

Dynamical Approaches to Spectral Theory of Operators

Covers dynamical approaches to the spectral theory of operators.

Spectral Theorem Recap

Revisits the spectral theorem for symmetric matrices, emphasizing orthogonally diagonalizable properties and its equivalence with symmetric bilinear forms.

Low Rank Approximations

Explores low rank approximations, spectral theorems, and orthogonality in matrices.

Spectral Analysis: Hyperbolic Surfaces

Explores the spectral analysis of hyperbolic surfaces through the trace formula and its applications in understanding geometric and spectral properties.

Spectral Decomposition of Unbounded Operators

Explores the spectral decomposition of non-bounded operators and presents the spectral theorem for self-adjoint non-bounded operators.

Decomposition Spectral: Symmetric Matrices

Covers the decomposition of symmetric matrices into eigenvalues and eigenvectors.

Linear Algebra: Quantum Mechanics

Covers the application of linear algebra concepts to Quantum Mechanics, including spectral theorem and Brillouin zone.

Spectral Decomposition

Explores spectral and singular value decompositions of matrices.

Linear Response and Complex Diffusivity

Explores martingale-based linear response, complex diffusivity, and Nyquist relation in stochastic systems with time-dependent perturbation.

Linear Operators: Basis Transformation and Eigenvalues

Explores basis transformation, eigenvalues, and linear operators in inner product spaces, emphasizing their significance in Quantum Mechanics.

Eisenstein Series and Spectral Decomposition

Covers Eisenstein series properties and spectral decomposition of functions.

Linear Algebra: Decomposition Theorems

Covers the proof of primary decomposition theorem for endomorphisms of finite-dimensional vector spaces.

Spectral Theory: Friedrichs Extension Theorem

Covers the proof of Friedrichs extension theorem and the properties of the spectrum and resolvent map.

Spectral Decomposition of Bounded Self-Adjoint Operators

Explores the spectral decomposition of self-adjoint operators on Hilbert spaces.

Spectral Decomposition of Symmetric Matrices

Explores the spectral decomposition of symmetric matrices, including diagonalization and orthogonal basis change matrices.