Distinguish dispersive phases from diffusion

Try

Heat suppresses high frequencies. A dispersive equation can instead rotate their phases at different rates while preserving total L² size. Consider the free one-dimensional Schrödinger equation i uₜ=−uₓₓ, where u is complex-valued and i²=−1. This is a normalized mathematical equation; physical constants have been scaled out.

Substitute a plane wave e^(i(kx−ωt)). Such a wave is a useful exact mode but is not square integrable on the whole line, so whole-line finite-energy claims must be applied to packets rather than to this mode alone.

Predict

Will the amplitude of a plane wave decay? Calculate its modulus, then compare the frequencies for k=1 and k=2. Predict whether a packet containing both can keep a rigid shape indefinitely.

Experiment

Substitution gives ω=k². Fourier transformation yields û(ξ,t)=e^(−iξ²t)f̂(ξ).

Wave number k	Frequency ω	Phase velocity ω/k for nonzero k	Group velocity ω′
1	1	1	2
2	4	2	4
−1	1	−1	−2

The multiplier has modulus 1 at every frequency. A localized spectrum near k travels approximately at group velocity 2k, but different frequencies accumulate different phases and the packet spreads. This approximation has a bandwidth and time range; it is not an exact rigid translation of arbitrary data.

Understand

Plancherel gives ∥u(t)∥₂=∥f∥₂ because the multiplier preserves every frequency magnitude. For smooth rapidly decaying solutions, the same fact follows in physical space. Differentiate ∫|u|² to obtain 2Re∫uₜ conjugate(u). Since uₜ=i uₓₓ, integration by parts gives 2Re(−i∫|uₓ|²)=0. The real part and complex conjugation are essential; treating u as an ordinary real heat field would give the wrong sign and interpretation.

For positive time, the formal kernel is (4πit)^(−1/2)exp(ix²/(4t)), with a consistent square-root branch. Its magnitude is constant in x, so it is not absolutely integrable. It is interpreted through oscillatory regularization or tempered distributions. For sufficiently integrable data the resulting solution can exhibit pointwise decay estimates, even though its L² norm is conserved. Spreading can lower local amplitudes without destroying total squared norm.

There is no heat-style positivity or maximum principle: complex phase cancellation replaces positive averaging. Nor does this equation share the finite propagation cone of the classical wave equation. These distinctions arise from the multipliers and kernels, not merely from calling all three equations “evolution equations.”

The general theory of dispersive estimates, nonlinear Schrödinger equations and quantum measurement is outside this lesson. The proved core is the free multiplier, dispersion relation and norm conservation under stated hypotheses.

Apply

Compare the same smooth initial packet under translation, heat and free Schrödinger evolution in frequency space. Identify which changes magnitude, which changes phase linearly in frequency and which changes phase quadratically. Explain why a smaller peak after evolution does not by itself establish dissipative loss.

Continue

Weak elliptic formulations use a different kind of abstraction: an integrated equation and coercive energy supply existence even when classical second derivatives are unavailable.