Spectral Theory and Applications to Schrödinger operators
(run as Topics in Material Science)

Math 7384
Louisiana State University
Spring Semester, 2024

Prof. Stephen Shipman

Place: Lockett 232
Time: 2:30-3:20 Monday, Wednesday, Friday

Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@lsu.edu
Office Hours: Monday 1:00-2:20, Wednesday and Friday 3:30-5:00, and by appointment.

Assistant: Jorge Villalobos

Office: Room 377 of Lockett Hall
Email: jvill38@lsu.edu
Office Hours: by appointment.

Course Description

This course concentrates on the spectral theory of Schrödinger operators with a view toward modern research in the field. The huge literature in this field was spawned by non-relativistic quantum mechanics and has led to rich advances in pure spectral theory and applications. The course first develops abstract spectral theory of self-adjoint operators in Hilbert space with some emphasis on classical ideas of harmonic analysis, namely spectral resolutions induced by symmetry groups. Then we introduce continuous and discrete Schrödinger operators with electric and magnetic potentials and some of the standard theorems. We treat periodic, quasi-periodic, and ergodic operators, in decreasing detail. The treatment of periodic operators will emphasize the connections to commutative algebra centering around the Fermi and Bloch algebraic or analytic varieties. The course will conclude with analysis in physical, momentum (dual), configuration, and reciprocal space.

• Introduction: Finite-dimensional spectral theory and examples in infinite dimensions
  
  • Review of spectral theory in finite dimension
  • Self-adjointness with respect to an indefinite inner product
  • Multiplication operators, bounded and unbounded
  • Examples from Fourier analysis
• The spectral theorem for self-adjoint operators in Hilbert space
  • Unbounded operators
  • Some prerequisites from analysis
  • The spectral theorem
  • Stone's resolvent formula
  • Stone's theorem on unitary groups
• Prerequisites from operator theory
  
  • Compact operators in Hilbert space
  • Trace-class operators
  • Hilbert-Schmidt operators
  • Integral kernels of operators
• Schrödinger operators
  
  • Continuous and discrete operators
  • Schnol's theorem for bounded potentials
  • Some standard theorems
  • Time dynamics
• Periodic operators
  
  • Spectral resolution by discrete symmetry group
  • The Bloch and Fermi varieties in momentum and energy space
  • Reducibility of the Fermi variety
  • Defect states and embedded eigenvalues
  • Application to multi-layer graphene
  • Nondegeneracy of spectral band edges
• Ergodic and quasi-periodic operators
  
  • General theory
  • A general theorem on almost sure spectrum
• Physical, momentum (dual), configuration, and reciprocal space
  
  • Definitions of configuration and reciprocal space
  • Applications to twisted bilayer graphene structures

Prerequisite

Assignments and Evaluation

I will give you problem sets throughout the semester. I would like you to do as many as you are able to. However the grade in the course will be based on just a few problems (maybe one to three) per set. There will no mid-term or final exams.

This is my general policy on academic integrity.

Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone.

Lecture Notes

Problem Sets

Grading scale (required by LSU)

A+: at least 95%	A: at least 90%	A-: at least 88%
B+: at least 85%	B: at least 80%	B-: at least 78%
C+: at least 75%	C: at least 70%	C-: at least 68%
D+: at least 65%	D: at least 60%	D-: at least 50%
F: less than 50%

Bibliography

1. Excerpts on compact operators:
   Naylor and Sell
   Kress
   Reed and Simon
   Riesz and Sz.-Nagy
   Akhiezer and Glazman, Chapter V; see the link below.
2. Michael Reed and Barry Simon, Methods of Modern Mathematics: Vol. I Functional Analysis, Vol. II Fourier Analysis and Self-Adjointness, Vol. III Scattering Theory, Vol. IV Analysis of Operators, excerpts from Vol. I (spectral theorem), Vol. II (self-adjoint extensions), Academic Press, 1980.
3. Jean-Pierre Aubin, Applied Functional Analysis, Wiley, 2000.
4. Hans L. Cycon and Richard G. Froese and Werner Kirsch and Barry Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Springer-Verlag, 1987.
5. Werner Kirsch, An Invitation to Random Schrödinger operators, Notes, 2007.
6. I. Gohberg and P. Lancaster and L. Rodman, Indefinite Linear Alegebra and Applications, Birkhäuser, 2005.
7. Ngoc Do and Peter Kuchment and Frank Sottile Generic properties of dispersion relations for discrete periodic operators, J Math Phys 61 (2020).
8. Gerald B. Folland, Introduction to Partial Differential Equations, Chapter 0, Second Edition, Princeton University Press, 1995.
9. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, 1993, replication of the edition of F. Ungar Publishing, NY, 1961, 1963.