Oscillatory integral

Definition

An oscillatory integral with large parameter $\lambda$ has the form

I(\lambda)=\int_{\mathbb R^n}e^{i\lambda\phi(x)}a(x)\,dx,

where the real-valued function $\phi$ is the phase and $a$ is the amplitude. The term may also denote an operator obtained by allowing the phase and amplitude to depend on input and output variables.

Cancellation

Away from critical points of $\phi$ , integration by parts gives decay faster than the absolute-value estimate. Near nondegenerate critical points, stationary phase gives an asymptotic expansion whose leading order is $\lambda^{-n/2}$ .

Fourier-like relations

Fourier transformation is the model oscillatory integral. In quantum chaos, incoming and outgoing boundary data are related by oscillatory integral operators, so simultaneous localization becomes a fractal uncertainty question.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, 2003. DOI record.