## Math 224a: Mathematical Methods for the Physical Sciences. Fall 2020.

Instructor: Nikhil Srivastava, email: firstname at math.obvious.edu

Lectures: TTh 3:40pm-5:00pm, Zoom.

Office Hr: T 5pm-6pm, Zoom.

Textbooks. Much of the material will be drawn from the following four books. All are available to Cal students for free on Oskicat or Springerlink.
Reed and Simon, Functional Analysis, Vol I
Jonathan Dimock, Quantum mechanics and quantum field theory [electronic resource] : a mathematical primer
Geralrd Teschl, Mathematical Methods in Quantum Mechanics With Applications to SchrÃ¶dinger Operators
Trefethen, Approximation Theory and Approximation Practice I will also draw on several other resources and frequently post lecture notes on this webpage.

Announcements

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.
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

 # Date Topics Readings Notes Remarks 1 Th 8/27 Lebesgue integral, monotone and dominated convergence, completeness of L1 RS I.3 lec1 2 T 9/1 L2, Hilbert spaces, separability, orthonormal bases. RS II.1,II.3 lec2 3 Th 9/2 Weierstrass thm, separability of L2, projections RS II.2-II.3 lec3 4 T 9/8 dual space, adjoint, positivity, convergence in norm RS VI.1-VI.2, VI.4 lec4 5 Th 9/10 square root, range and kernel, polar decomposition RS VI.4 lec5 6 T 9/15 compact operators, spectral thm for compact operators RS VI.5 lec6 7 Th 9/17 trace class and Hilbert-Schmidt operators RS VI.6 lec7 8 T 9/22 Fredholm Alternative, 2nd order BVP lec8 9 Th 9/24 Green's Functions, Regular SL theory lec9 10 T 9/29 resolvent, spectrum RS VI.3 lec10 11 Th 10/1 uniform boundedness, spectral radius, cts functional calculus RS I.4, VI.3, VI.1 lec11 12 T 10/6 measure spaces, spectral measures, spectral theorem (mult. form) RS VII.2 lec12 13 Th 10/8 spectral projections, projection valued measures RS VII.3, see Kowalski sec 3.4. 14 T 10/13 example: the infinite tree, Stieltjes transform, Kesten-McKay law spectrum: Friedman sec 2 stieltjes inversion: wikipedia spectral measure: Avni-Breuer-Simon example 7.1 15 Th 10/15 Fourier transform Teschl 7.1 lec15 16 T 10/20 Gaussian Integral, Uncertainty Principle Teschl 7.1 lec16 higher dimensions 17 Th 10/22 Unbounded, closed, and selfadjoint operators RS VIII.1-2 lec17 18 T 10/27 Unbounded spectral theorem, Kato-Rellich RS VIII.3, Dimock 4.2 lec18 19 Th 10/29 Harmonic oscillator, time evolution Dimock 4.4, RS VIII.4 20 T 11/3 Tempered distributions RS V.3 lec20 21 Th 11/5 Operations on tempered distributions, applications 22 T 11/10 orthogonal polys, 3 term rec, Jacobi coefficients, real rootedness lecs22-26 23 Th 11/12 Gauss quadrature, Favard's theorem, infinite Jacobi matrices Trefethen Ch 19 lecs22-26 see also Simon 24 T 11/17 Chebyshev polyonomials and series, rates of convergence Trefethen 3,7,8 lecs22-26 25 Th 11/19 Chebyshev interpolation, Hermite integral formula Trefethen 4,11 lecs22-26 26 T 11/24 Potential Theory, Lebesgue Constants Trefethen 4,11 lecs22-26 27 T 11/26 Anderson Localization lecs27-28 28 Th 11/28 Anderson Localization, Minami Estimate lecs27-28

Homework. Will be due every two weeks, on Thursdays by email. The subject of the email should be [224a HW # ...]. 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 9/10.
2. HW2, due 10/1.
3. HW3, due 10/15.
4. HW4, due 11/6.
5. HW5, due 11/30.
6. HW6, due 12/16.

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