Here are some notes to accompany my May 2024 lectures at Kyoto University on spectral theory. My aim in writing the notes has been to recall the main topics covered, the main definitions, the statements the main theorems, and so on. Occasionally I have added some details that I did not mention in the lectures.
Here is the current list of references cited (or to be cited) in the lectures.
For students registered in the class, there is some information about homework at the bottom of this page.
Nigel Higson, May 2024
Lecture 1: Spectral theory and integral operators.
Eigenvalues and eigenvectors. Eigenvalue problems for differential equations. Integral equations.
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Lecture 2: Hilbert's spectral theorem.
Hilbert space. Hilbert’s spectral theorem for integral equations. First applications.
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Lecture 3: Weyl’s asymptotic law.
The Lorentz problem. Explicit solutions of the Helmholtz problem. Singular values. Positivity of operators. Proof of the asymptotic law.
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Lecture 4: Representations of compact groups.
The idea of representation theory. Basic representation theory for finite groups. Characters. Special unitary groups. The Peter-Weyl theorem. Maximal torus and conjugacy classes.
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Lecture 5: The Weyl character formula.
Maximal torus. Weights and dominant weights. Statement of the character formula. The trivial representation. Proof of the character formula. The dimension formula.
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Lecture 6: Spectral theory for densely defined operators.
Densely-defined operators. Examples. Model operators for spectral theory and measurable fields of Hilbert spaces.
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Lecture 7: Spectral theory according to von Neumann.
Self-adjoint operators, after von Neumann. Resolvent operators. Deficiency indices. The Cayley transform.
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Lecture 8: Sturm-Liouville operators.
Basic definitions. The Wronskian. Green's operators from the Wronskian. The Sturm-Liouville theorem and Weyl's boundary conditions.
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Lecture 9: Sturm-Liouville operators on a half-line.
Singular operators. Limit point and limit circle boundary values. Weyl’s analysis of the continuous spectrum. One-parameter unitary groups and the Friedrichs extension and essentially self-adjoint operators. Kodaira's work on the Schrödinger equation for the hydrogen atom.
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Lecture 10: The prolate spheroidal wave operator.
Time-limited and band-limited signals. The uncertainty principle. An improved uncertainty principle in electrical engineering. The role of the prolate spheroidal wave operator. Self-adjoint extensions of the prolate spheroidal wave operator on the interval and on the line. Discreteness of the spectrum. Weyl asymptotic law for the eigenvalues of the prolate spheroidal wave operator, and asymptotics for the zeros of the Riemann zeta function.
Lecture notes
Information About Homework
If you are registered in the class, then please write a one-page (only) summary of one of the lectures (only). Here is a sample summary (pdf and tex) of a lecture that I’ve taken from another class. Please use LaTeX, and email your summary to me in pdf format on or before 12:00 midday on Wednesday May 29 (this is two days after the final lecture in this series). My email address is higson@psu.edu. Thank you!