Lecture Notes on Methods of Mathematical Physics (PHY503)

Fall 2021, Jacobus Verbaarschot

Below you find the lecture notes of the first few lectures of last year. This year will be mostly the same at least for the beginning of the class.

Lecture 1, August 23
Functionals, Functional Derivatives, Euler-Lagrange equations, Constraints

Lecture 2, August 25, Missed lecture

Lecture 3, August 27
Catenary

Lecture 4, August 30
>Action principle, Noether's theorem, Linear Vector spaces, linear maps

Lecture 5, September 1
Dual space, Orthonormal basis, Scalar product

Lecture 6, September 3
Direct Sum, Qoutient space, Co-kernel

Lecture 7, September 8
Adjoint Operator, Direct Sum, Projector

Lecture 8, September 10
Rank, Determinants, Fredholm Alternative

Lecture 9, September 13
Adjugate Matrix, Characteristic Equation, Caley's Theorem

Lecture 10, September 15
Diagonaliation

Lecture 11, September 17
Jordan Canical Form, Quadratic Forms

Lecture 12, September 20
Symplectic Forms

Lecture 13, September 22
Function Spaces, Convergence, Norm, Cauchy sequence, Complete Spapes

Lecture 14, September 24
Banach Space, Hilbert Space, Cauchy-Schartz Inequality, Triangular Inequality

Lecture 15, September 27
Orthogonal Polynomials, Best Approximation, Parcival's Theorem, Three Step Recursion Relation

Lecture 16, September 29
Weierstrass approximation theorem, Construction of Orthogonal Polnomials, Legendre Polynomials, Hermite Polynomials

Lecture 17, October 1
Tsebychev Polynomials, Distribution, Test Functions

Lecture 18, October 4
Weak Derivative, Principe Value Integral

Lecture 19, October 6
Linear Differential Equations, Wronskian

Lecture 20, October 8
Normal Form, Singular Points

Lecture 21, October 13
This will be a prerecorded Zoom lecture, I sent you the link and password. The lecture notes are the review I recorded. Note that October 11 is a Holiday.

Lecture 22, October 15
Midterm Exam proctored by Prof. Teaney in the usual class room

Lecture 23, October 18
This will be a zoom lecture at the usual time. I will send you the link. Inhomogenouse differential Equations. Singualr Points. Linear Differential Equation, Adjoint Operator

Lecture 24, October 21
Sturm-Liouville Operator

Lecture 25, October 23
Self-Adjointness, Eigenvalues of Self-Adjoint Operators

Lecture 26, October 25
Self-Adjoint Boundary Conditions

Lecture 27, October 28
Heterjunction and Self-Adjointness

Lecture 28, October 30
Spectrum of an Operator, Completeness

Lecture 29, November 1
Operator Methods, Harmonic Oscillator

Lecture 30, November 3
Phase Shifts, Continuous Spectrum

Lecture 31, November 5
Answered questiŧons

Lecture 32, November 8
Completeness, Green's Functions

Lecture 33, November 10
Green's Function of hormaonic oscillator

Lecture 34, November 12
Caldeira-Leggett Model

Lecture 35, November 15
Example of Green's function

Lecture 36, November 17
Symmetry of Green's functions

Lecture 37, November 19
Partial Differential Equations

Lecture 38, November 22
Eigenfunction expansions of Green's functions

Lecture 38, November 29
Causality and Analyticity

Lecture 40, December 1
Plemelj Formulae, Resolvent

Lecture 41, December 3
Resolvent

Lecture 42, December 6
Gelfand-Dikii Equations

Final Exam, December 15, 2.15-5 pm in regular class room