Wave Packet Decomposition

• Wave packet decomposition is a technique that expresses functions as sums of spatially and frequency-localized wave packets, providing detailed phase space resolution.
• It employs frameworks like Gabor frames, curvelets, and Littlewood–Paley packets to capture propagation, regularity, and interaction properties in dispersive and quantum systems.
• The method underpins rigorous estimates in PDE analysis and supports practical applications in quantum dynamics, numerical algorithms, and signal processing.

Wave packet decomposition is a foundational methodology in the analysis of quantum, dispersive, and time-frequency phenomena, providing a precise framework to capture the localized behavior of solutions in phase space. The decomposition systematically expresses a function or solution as a superposition of elementary "wave packets," each sharply localized in both spatial and frequency domains, enabling sharp control of propagation, regularity, and interaction properties in both linear and nonlinear contexts.

1. Formalism of Wave Packet Decomposition

Wave packet decomposition constructs a phase-space-resolved representation of functions or solutions, typically in $L^2(\mathbb{R}^n)$ or more general distributional settings. The core object is a family of atoms $\{\psi_\theta\}_{\theta\in \Theta}$, each microlocalized near a point $(x_\theta,\xi_\theta)$ in phase space $\mathbb{R}^n_x\times\mathbb{R}^n_\xi$.

The analysis operator

$$Wf = \bigl(\langle f, \psi_\theta\rangle\bigr)_{\theta\in\Theta}$$

maps $f$ to its wave packet coefficients. Under suitable normalization, $\{\psi_\theta\}$ forms a tight frame or even an (almost) orthonormal basis, so that

$$f = \sum_{\theta\in\Theta} \langle f, \psi_\theta\rangle\,\psi_\theta$$

with norm identity or equivalence in $L^2$. This decomposition is typically constructed via Gabor frames (Gaussian packets on grids) for modulation spaces, frequency–scale–direction systems (curvelets/wave atoms) for oscillatory PDE, or Littlewood–Paley packets for dyadic frequency analysis (Portal, 3 Feb 2026).

2. Construction for Hamiltonian Flows and Dispersive PDE

Wave packet decompositions for dispersive equations focus on building explicit packets that track both position and momentum under evolution.

Given a phase-space grid $$\Lambda_R = R^{1/2}\mathbb{Z}^d \times R^{-1/2}\mathbb{Z}^d$$ at scale $R\gg 1$, the basis packets are coherent states

$$\phi_{x_0,\xi_0}(y) = C_R \exp\left[i\,\xi_0\cdot(y-x_0) - \frac{1}{2R}|y-x_0|^2 \right],$$

with spatial width $\sim R^{1/2}$ and frequency width $\sim R^{-1/2}$ (Schippa et al., 28 Sep 2025). For evolution under a (possibly rough) Hamiltonian flow, each packet follows the associated bicharacteristic, with the decomposition remaining almost orthogonal and localized.

Specialized constructions employ phase-space tilings and smooth bump functions that yield orthonormal families in $L^2$. For instance, in the study of Schrödinger equations with rough potentials, packets are tube-localized and adapted to the free flow, yielding explicit $L^2$ expansion and tube-wise evolution (Denisov, 2024).

3. Wave Packet Parametrices and Propagation Estimates

Propagation of wave packets under evolution operators—such as $e^{it\Delta}$ or Fourier integral operators—enables fine-grained analysis of dispersive and regularization properties. Representative results include:

• For Schrödinger with rough potentials, each wave packet remains tube-localized—with negligible leakage out of the tube—for times up to $O(T)$, and perturbative effects from the potential can be quantified via precise overlap estimates in phase space (Denisov, 2024).
• For semigroups generated by quadratic differential operators, Gabor wave packet decomposition gives pointwise kernel bounds:

$$|\langle e^{t q^w} \pi(z)g, \pi(w)g\rangle| \leq C_N e^{-\mu t}(1+\|w-z\|)^{-2N}$$

capturing both sharp exponential decay and off-diagonal localization, and yielding function space bounds on modulation spaces $M^p$ (Trapasso, 2024).

These decompositions underpin sharp bilinear and multilinear estimates, including those for rough Hamiltonian flows, by enabling fine spatial, frequency, and time–frequency segmentation and transversality counting (Schippa et al., 28 Sep 2025).

4. Wave Packet Decomposition and Function Spaces

By leveraging the mapping

$$f \mapsto Wf(x,\xi) = \langle f, \psi_{(x,\xi)}\rangle$$

function spaces central to harmonic and microlocal analysis can be constructed as "retracts" of vector-valued function spaces over phase space.

• Modulation spaces $M^{p,q}_s$: Built from Gabor frames, normed by mixed sequence norms in $(x,\xi)$ (Portal, 3 Feb 2026).
• Besov and Triebel–Lizorkin spaces: Via Littlewood–Paley wave packet projections, suitable for Calderón–Zygmund and pseudo-differential calculus.
• Coorbit/curvelet spaces: Structured to capture anisotropic scaling and wave propagation.

The choice of wave packet system adapts to linear symbol geometry, dispersion relation, and operator class, with function space boundedness of evolution operators (e.g., $e^{it\Delta}$, Fourier integral operators, semigroups) following as a direct consequence (Portal, 3 Feb 2026, Trapasso, 2024).

5. Quantum Dynamical Applications: Splitting, Fractional Revivals, and Entanglement

In quantum systems, wave packet decomposition elucidates phenomena such as perfect splitting, fractional revivals, and mode entanglement.

• On a one-dimensional mirror-symmetric lattice, by engineering the spectrum such that

$e^{-i E_k t^*/\hbar} = (1/\sqrt{2})[1 + i(-1)^k]$

an initial delta-like state undergoes perfect splitting at time $t^*$ into two components at mirror sites, and full revival at $2t^*$, analogous to a 50:50 beam splitter (Banchi et al., 2015).

• In multi-particle/cavity QED, the two-photon joint spectral amplitude (2PJS) can be Schmidt-decomposed,

$$\Phi^{\rm out}_{\omega,\omega'} = \sum_n \sqrt{\lambda_n}\,\phi_n(\omega)\,\phi_n(\omega')$$

with entanglement entropy $S_{\rm vN} = -\sum_n \lambda_n \log_2 \lambda_n$, capturing induced mode correlations by nonlinear scattering (Stolyarov, 2018).

• In quantum billiards, an initial Gaussian decomposes into cubical eigenstates, with time evolution manifesting collapse, classical periodic motion, and fractional/full revivals, fully determined by spectral decomposition and the resulting phase relations (Kaur et al., 2015).
• Metaplectic-extended WKB schemes provide semiclassical uniformization of the spread and decomposition of coherent packets to Ehrenfest times, showing decomposition into Lagrangian states associated with unstable manifolds (Schubert et al., 2011).

6. Numerical and Signal Processing Frameworks

Wave packet decomposition underpins numerical algorithms and signal analysis tools. A prominent example is wavelet packet (WP) decomposition, which refines both approximation and detail branches at every level, and is efficiently used for feature extraction in time–frequency representations.

In gravitational wave detection, WP decomposition (e.g., with six levels and Haar filters) yields a time–frequency tiling used as CNN input, where the preprocessing both separates signal from noise and drastically increases detection speed relative to matched filtering (Lin et al., 2019).

7. Applications, Strategy Selection, and Functional Adaptivity

The selection of a wave packet system is dictated by the analysis goals:

• For constant-coefficient dispersive equations (e.g., Schrödinger, wave), Gabor or curvelet systems tuned to the symbol’s geometry are optimal (Portal, 3 Feb 2026).
• For variable-coefficient or potential-perturbed flows, phase-space adaptive packets (FBI transforms, tube packets) are required for sharp analysis of localization, interaction, and control of resonance effects (Denisov, 2024, Schippa et al., 28 Sep 2025).
• For operator-induced function spaces, the atomic decomposition can be adapted via operator-dependent projections (e.g., for Schrödinger operators with potential, construct packets localized at scales defined by the potential profile) (Portal, 3 Feb 2026).

This framework provides a unifying narrative for both abstract harmonic analysis spaces and the propagation/interaction analysis in evolution equations, quantum systems, and signal processing applications. It enables both rigorous mathematical results—such as sharp propagation, boundedness, and bilinear estimates—and concrete algorithmic realizations tailored to underlying operator structure and physical context.