Math 224a: Mathematical Methods for the Physical Sciences. Fall 2019.

Instructor: Nikhil Srivastava, email: firstname at

Lectures: TTh 2:10-3:30pm, Hildebrand B56.

Office Hours: T: 3:40-4:40pm, Th 5-6pm (1035 Evans)

Textbooks. Much of the material will be drawn from the following two books. Both are available to Cal students for free on Oskicat or Springerlink.
Robert Richtmyer, Principles of Advanced Mathematical Physics, Volume I
Reed and Simon, Functional Analysis, Vol I
Jonathan Dimock, Quantum mechanics and quantum field theory [electronic resource] : a mathematical primer
Folland, Intro to PDE, 2e
Trefethen, Approximation Theory and Approximation Practice
I will also draw on several other resources and frequently post lecture notes on this webpage.


Syllabus The course will survey methods for solving the fundamental problems of mathematical physics. The overall purpose of the course will be to develop a functional analytic framework for understanding and approximating solutions of differential equations, with an emphasis on physical examples. The content can broadly be divided into three parts:

  1. Functional Analysis. (3 weeks) Lp spaces, Hilbert spaces, distributions, Schwartz functions, Fourier transform, potential theory.
  2. Spectral Theory. (7 weeks) Linear operators, adjoint, spectrum and resolvent, spectral theorem for bounded s.a. operators, Fredholm alternative, Green's functions, Sturm-Liouville theory.
  3. Orthogonal Polynomials. (4 weeks) Classical orthogonal polynomials; approximation and interpolation theory.

Class Schedule

#DateTopics ReadingsNotesRemarks
1 Th 8/29 Lebesgue integral, monotone and dominated convergence, completeness of L1 RS I.3 lec1
2 T 9/3 L2, Hilbert spaces, separability, orthonormal bases. RS II.1,II.3 lec2
3 Th 9/5 Weierstrass thm, separability of L2, projections, dual space, Riesz-Frechet thm RS II.2-II.3 lec3
4 T 9/10 norm, adjoint, positivity, square root RS VI.1-VI.2, VI.4 lec4
5 Th 9/12 range and kernel, polar decomposition, compact operators RS VI.4-5 lec5
6 T 9/17 spectral thm for compact operators RS VI.5 lec6
7 Th 9/19 consequences of spectral thm, trace class operators, Fredholm alternative RS VI.6 lec7
8 T 9/24 basic ODE theory lec8
9 Th 9/26 Green's function, completness of eigenfunctions, regular SL theory lec9
10 T 9/31 group work lec10
Th 10/3 no lecture
11 T 10/8 oscillation theory lec11
Th 10/10 power outage
12 T 10/15 resolvent, spectrum RS VI.3 lec12
13 Th 10/17 uniform boundedness, spectral radius, multiplication operators RS I.4, VI.3, VI.1 lec13
14 T 10/22 cts functional calculus, spectral theorem RS VII.1-2 lec14
15 Th 10/24 group work lec15
16 T 10/29 Fourier transform Dimock 1.1.4 lec16
17 Th 10/31 unbounded operators Dimock 1.2-1.3.3 lec17 RS VIII.1-2 for more detail
18 T 11/5 unbdd spectral theorem, physical applications Dimock 1.3.3, 4.1-4.4 lec18
19 Th 11/7 tempered distributions RS V.3 lec19
20 T 11/12 Fourier transform of a distribution, wave equation lec20 see also Folland Ch 0
21 Th 11/14 Malgrange-Ehrenpreis theorem see Folland PDE Ch 1F
22 T 11/19 orthogonal polynomials, 3 term recurrence, Jacobi coeffs
23 Th 11/21 Gauss quadrature, separation theorem
24 T 11/26 pseudospectral methods lec24 guest lecture by Prof. Wilkening
Th 11/28 thanksgiving
25 T 12/3 Chebyshev polyonomials and series, rates of convergence Trefethen 3,7,8 lec25 see Trefethen Ch 7-8.
- Th 12/5 lecture moved to RRR week
26 T 12/10 Chebyshev interpolation, Hermite integral formula Trefethen 4, 11 makeup lecture for 12/5
27 Th 12/12 Potential Theory, Lebesgue Constants Trefethen 12,13,15 makeup lecture for power outage

Homework. Will be due every two weeks, on Thursday in class. HW assignments will be updated (i.e., problems may be added) until upto a week before they are due. Please write clearly or type your solutions using Latex. Collaboration is allowed but you must list your collaborators in your writeup.

  1. HW1, due 9/12.
  2. HW2, due 9/26.
  3. HW3, due 10/10.
  4. HW4, due 10/31.
  5. HW5, due 11/21.
  6. HW6, due 12/17.

Grading. 100% homework. The bottom assignment will be dropped.