Spectra, resolvents and scattering

Start with the worked comparisons and five proof routes. The lessons connect spectral measurements, escape, operator domains, reconstruction and long-range limits. Each lesson begins with a working question and includes five exercises with complete solutions.

Written by GPT-6.1 Sol (OpenAI) and GPT-6 Astra (OpenAI). Self-checked by the writing AI. Human mathematical sources are credited in the lessons.

What can a spectral measurement tell us?

  1. Resolvents, domains and spectral density
  2. Wave evolution and cotangent flow
  3. Local spectral density and the subprincipal correction
  4. Return times and spectral counting
  5. Reflection and the Dirichlet boundary coefficient

How does escape select a real-energy solution?

  1. Wave operators and modified phases
  2. Endpoint spaces and flat energy shells
  3. Fourier traces on curved energy surfaces
  4. Mild weights and frequency localization
  5. Division and radiation at regular energies
  6. Polynomial translations and regular energies
  7. Global polynomial resolvent estimates
  8. Global radiation and flux
  9. Hamilton trajectories under a long-range force
  10. Smooth long-range phases from Hamilton trajectories
  11. Modified waves and the direction of escape

Which perturbations define the operator we need?

  1. Short-range compactness and local tests
  2. Self-adjoint short-range operators
  3. Wave operators for differential perturbations
  4. Compact perturbations in weighted Hilbert spaces

Can the observations reconstruct every state?

  1. Limiting absorption and point spectrum
  2. Quadratic weights and uniqueness at infinity
  3. Distorted Fourier transforms and spectral density
  4. Asymptotic completeness for short-range operators
  5. Scattering matrices at regular energies
  6. Polynomial localizations and rough coefficients
  7. One-dimensional scattering and phase shifts
  8. Compressed spectral measures and symbol distributions
  9. Positive real powers and spectral rescaling
  10. Generalized rays and the Dirichlet Weyl law
  11. Arithmetic spectral clusters and their distributions
  12. Averaging a perturbation around closed trajectories

Which estimates survive a long-range limit?

  1. Regularizing long-range coefficients
  2. Admissible differential perturbations
  3. The Sobolev domain of an elliptic operator
  4. Weighted Sobolev spaces and rough elliptic estimates
  5. The resolvent away from the energy surface
  6. A resolvent estimate at noncritical frequencies
  7. Combining the long-range resolvent estimates
  8. Radiation for limits of long-range resolvents
  9. Outgoing flux and vanishing shell mass
  10. Weighted endpoint estimates and polynomial decay
  11. Limiting absorption for long-range differential perturbations
  12. Escaping Lagrangians on regular energy surfaces
  13. Generating functions and the end of a localized force
  14. Energy-shell factors and outgoing equations
  15. Frequency cutoffs and compact scattering remainders
  16. Commuting coordinates for long-range evolution
  17. Transverse moments and outgoing amplitudes
  18. Truncated operators and stable scattering amplitudes
  19. Spectral transforms and completeness of modified waves

Original text and figures are CC0, with component exceptions: the analytic preparation supplement retains Demailly’s custom OpenContent grant; the coordinate-integration supplement follows Jiří Lebl under CC BY-SA 4.0; integrated Stacks project excerpts retain GNU FDL 1.2. KaTeX assets retain the MIT licence. Course metadata.