Dispersive decay and stationary phase

For the free equation i u_t plus the Laplacian of u equals zero on R to the d, Fourier inversion gives the propagator U(t)=exp(it Laplacian). For nonzero time its convolution kernel is a complex Gaussian with absolute value (4pi times |t|) to the minus d over two. For Schwartz initial data, taking absolute values inside the convolution yields an L¹-to-L∞ estimate. Density and the kernel formula extend the appropriate statement to integrable data, with the usual interpretation of the free evolution.

The estimate is singular at zero time, where the propagator is the identity and no bounded map from all L¹ data to L∞ exists. On the other hand, Plancherel gives exact unitarity on L² for every time because the Fourier multiplier has modulus one. Interpolating these two bounds gives L to the r-prime into L to the r decay with exponent d times one half minus one over r, for r between two and infinity. The domain R to the d and its spreading geometry are essential; a fixed torus has recurrent Fourier modes rather than this global decay law.

Consider an oscillatory integral with a smooth real phase and a smooth compactly supported amplitude. Suppose there is exactly one critical point near the amplitude's support, it lies in the interior region under consideration, and the phase Hessian there is invertible; any other part of the support is nonstationary. For positive large time, stationary phase gives a leading term proportional to t to the minus d over two. Its coefficient uses the amplitude at the critical point, the absolute Hessian determinant, and a phase determined by the Hessian signature.

The proof locally changes coordinates to a nondegenerate quadratic phase and integrates the resulting Gaussian; outside the critical neighborhood, integration by parts gains powers of time because the phase gradient stays nonzero. These are the essential mechanisms, not a complete proof here. Degenerate Hessians, nonsmooth amplitudes, boundaries, and colliding critical points require different theorems. If the amplitude vanishes at the stationary point, the leading coefficient can vanish too, giving faster decay than the generic first term predicts.

In one dimension, initial data exp(−x squared) evolve under free Schrödinger into (1+4it) to the minus one half times exp of minus x squared divided by (1+4it). Its peak magnitude is (1+16t squared) to the minus one quarter. Meanwhile its spatial width increases and its L² norm remains fixed. The lost peak is redistributed over a wider region with changing phase. This exact example separates the qualitative picture from the rough upper bound, which is not sharp at short times.

Dispersive estimates feed nonlinear analysis because they make distributed wave interactions integrable in suitable spacetime norms. A raw pointwise decay estimate can nevertheless be too singular near equal times to integrate directly in a nonlinear Duhamel formula. Strichartz estimates combine decay, unitarity, and time integration more carefully. For other dispersion relations, critical points can be degenerate and the decay exponent changes; a cubic Airy phase in one dimension has a different scaling from the quadratic Schrödinger phase. One should derive the actual phase before borrowing an exponent.