Propagation of singularities

Let P_h be a self-adjoint semiclassical pseudodifferential operator with smooth real principal symbol p, in a controlled local symbol calculus. Fix a real energy E and an h-tempered family u_h. Where p−E is nonzero, a microlocal parametrix shows that u_h is residual if (P_h−E)u_h is residual there. Thus away from the source wavefront, WF_h(u_h) can lie only on the characteristic set p=E. This is elliptic regularity and does not yet describe transport along that set.

At real-principal-type points, dp is nonzero on p=E, so H_p is a nonvanishing characteristic vector field. On an open phase-space region where the source is O(h^∞), the wavefront set of u_h is a union of segments of H_p trajectories as long as they stay in that region. Nonnegligibility cannot simply stop in the middle of a source-free segment. Fixed cutoffs, polynomial h bounds and a suitable pseudodifferential calculus are part of the hypothesis, rather than consequences of a ray drawing.

The positive-commutator method compares a localized norm before and after motion along a characteristic. Its proof uses the sign of H_p acting on a carefully selected test symbol, then estimates lower-order and source terms. This is not an assertion that the waveform itself remains unchanged along a trajectory. Amplitudes, phases and transverse spreading can evolve while microlocal nonnegligibility persists. At turning or radial configurations, the chosen local description and hypotheses must be checked rather than replacing the theorem by an unrestricted propagation slogan.

An O(h) residual is not O(h^∞). For an L²-bounded normalized quasimode, a stronger o(h) residual under self-adjointness and suitable localization gives invariance of a semiclassical measure through the limiting commutator identity. That measure-level conclusion does not automatically determine every point of the wavefront set. A polynomially weak component may disappear from the measure and remain in wavefront support. It is therefore necessary to state whether one is transporting a residual regularity estimate, a leading quadratic measure, or only a formal phase.

A smooth spatial cutoff can make an exact oscillatory solution square integrable, but derivatives of that cutoff create a source. Those terms are often only O(h), not residual. If an oscillation disappears at the edge of the cutoff, this does not contradict source-free propagation because the edge is precisely where the cutoff commutator acts. Inside a region where the cutoff equals one, the exact equation and its characteristic interpretation remain valid. Keeping these regions separate avoids a common false counterexample.

For a first-order transport operator with linear momentum symbol, the solution translates exactly. Its support shifts at the group velocity, while its momentum stays fixed. A finite sine-wave lab can illustrate this speed only for its own dispersion relation. The displayed wave sin(2x−t) has speed one half, and should not be silently identified with a different Hamiltonian whose flow speed is one. The operator symbol determines the characteristic velocity, not a generic animation clock.