Mathematics / research
Semiclassical and microlocal analysis
Resolve oscillatory phenomena through phase-space symbols, stationary phase, WKB states, and spectral approximations.
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Module 01 research
Semiclassical and microlocal analysis
Resolve oscillatory phenomena through phase-space symbols, stationary phase, WKB states, and spectral approximations.
- 01
Semiclassical scaling and classical limits
Introduce a dimensionless small action parameter, fix the Fourier convention, and distinguish high frequency from small amplitude.
30 min - 02
Symbol classes and pseudodifferential operators
Track momentum order and h-uniform derivatives before quantizing a phase-space function or claiming a bounded remainder.
31 min - 03
Weyl quantization and symbol composition
Use symmetric ordering and a declared Poisson bracket to calculate products and commutators with consistent signs.
32 min - 04
Wavefront sets and microlocal regularity
Locate nonnegligible h-dependent oscillations and distinguish semiclassical phase-space localization from a fixed distribution's conic wavefront set.
31 min - 05
Semiclassical measures and Wigner transforms
Take subsequential quadratic-observable limits, retain positivity and tightness hypotheses, and distinguish limiting mass from wavefront support.
33 min - 06
Egorov's theorem and classical transport
Transport observables with a sign-consistent Hamiltonian flow and state the fixed-time scope before considering logarithmically long times.
33 min - 07
WKB states and Bohr–Sommerfeld quantization
Separate eikonal and transport equations, match through simple turning points, and retain the Maslov correction and regular-energy scope.
34 min - 08
Tunneling and Agmon estimates
Use weighted energy identities to control forbidden-region decay and distinguish decay estimates from a full tunneling-splitting asymptotic.
34 min - 09
Propagation of singularities
Follow microlocal nonnegligibility along real Hamilton trajectories and distinguish residual wavefront propagation from weaker measure transport.
33 min - 10
Semiclassical trace formulas and periodic orbits
Separate smooth Weyl traces from time-localized orbit terms and retain transverse nondegeneracy, Maslov phases, and distributional scope.
35 min