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 Schrödinger 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:
| # | 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 |
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| 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.
Grading. 100% homework. The bottom assignment will be dropped.