Semiclassical measures and Wigner transforms

For an L²-bounded family u_h and compact smooth symbols a, the expectations ⟨a^w u_h,u_h⟩ are bounded. Along a sequence h_j tending to zero, a diagonal compactness argument selects a subsequence on which they converge for all such tests. Sharp Gårding positivity shows that the resulting functional is positive when a is nonnegative, and hence is represented by a positive Radon measure on finite phase space. The measure can depend on the subsequence; this theorem does not choose a unique classical state.

For normalized states its total mass is at most one when testing only compact phase-space symbols. To ensure mass exactly one, impose position and semiclassical-frequency tightness: the L² masses outside large position balls and outside large momentum balls after F_h both tend uniformly to zero in the limiting sense. A common compact energy localization for an elliptic confining Hamiltonian supplies such control. On noncompact spaces, merely writing ||u_h||₂=1 is insufficient because every compact test can miss an escaping family.

For Schwartz states, the displayed Wigner formula satisfies the exact identity that integrating a against W_h equals the Weyl expectation. Its position marginal is |u_h(x)|² and its momentum marginal is |F_hu_h(ξ)|². The distribution can take negative values because off-diagonal interference enters before the limit. Thus sampling it as an ordinary nonnegative phase-space probability is generally invalid. The positive limiting measure relies on a limiting argument, not on pointwise positivity of every Wigner function.

For a single coherent Gaussian centered at (x₀,ξ₀), the Wigner function happens to be a positive Gaussian on phase space. Its widths shrink as sqrt(h) in both conjugate directions, giving a delta measure at that center. This special positivity should not be generalized to arbitrary superpositions. Cross terms can oscillate, and they may disappear weakly even though they remain visible at finite h. A measure records the leading quadratic mass distribution, not a complete phase-sensitive description of the original state.

The support of a semiclassical measure lies in the family's semiclassical wavefront set, but equality need not hold. A component of polynomially small amplitude may survive every O(h^∞) test and still contribute vanishing quadratic mass. For instance, combine two separated coherent packets with amplitudes approaching one and sqrt(h), then normalize the sum. Both centers can remain in the wavefront set, while the limiting measure keeps only the first. Different notions of negligibility therefore lead to different supports.

For a fixed normalized envelope χ and a smooth real phase S, the WKB family χ exp(iS/h) has measure |χ(x)|² dx concentrated on ξ=S′(x). This gives a spatially distributed measure on a momentum graph rather than a point mass. Tightness must still be checked when envelopes or centers move with h. A translation to position one over h loses all finite limiting mass, just as a frequency moving to one over h does in the preceding lesson.