Gross Transform: Theory & Applications

• Gross Transform is a transformation principle that bridges nonlinear PDEs, spectral theory, and arithmetic geometry through variable changes and invariant transfers.
• It enables mapping between families of exactly solvable models, such as GP equations with varied coefficients, and facilitates hydrodynamic reformulations like the Madelung transform.
• The technique underpins applications in quantum fluids, supersolid phase transitions, and numerical stability via advanced Fourier methods and spectral analysis.

A Gross Transform is a transformation principle or technique that arises in several areas of mathematics and mathematical physics where the Gross–Pitaevskii equation or related frameworks occur. It typically denotes a change of variables, a spectral transformation, or a transfer of invariants connecting distinct classes of equations, solutions, or arithmetic phenomena. The exact meaning depends on context, encompassing nonlinear PDEs describing quantum fluids, arithmetic geometry, and spectral theory. In all instances, the Gross Transform facilitates analysis or solution generation by exploiting structural relationships inherent in the target models.

1. Transformations in Gross–Pitaevskii-Type Equations

In the context of exactly solvable Gross–Pitaevskii (GP) type equations, the Gross Transform denotes a parametric transformation that interrelates families of GP-type equations. For a base GP equation in variable $x$ and wave function $\psi(x)$,

$$\psi''(x) - [U_{\text{eff}}(x) + g_1(x)|\psi(x)| + \ldots] \psi(x) = 0,$$

the transform is defined by

$$x = |\xi|^\sigma, \quad \psi(x) = |\xi|^{(\sigma-1)/2} \varphi(\xi)$$

where $\sigma$ is an arbitrary constant. The effective potentials and nonlinear coefficients in the transformed equation for $\varphi(\xi)$ are explicitly related to those in the base equation via scaling relations, e.g.,

$$V_{\text{eff}}(\xi), \; G_\ell(\xi) \text{ are determined by }\sigma, U_{\text{eff}}(x), g_\ell(x).$$

Solving one member of the family yields solutions for all, as shown by direct substitution. This construction allows the generation of families of exactly solvable nonlinear Schrödinger-type equations, including those with variable coefficients and higher-order nonlinearities, such as the quintic or cubic-quintic GP equations. Analytic solutions are mapped between family members via the Gross Transform (Liu et al., 2020).

2. Hydrodynamic Formulation and the Madelung Transform

The analysis and solution of the GP equation benefit from the Madelung transform, which is closely related to the Gross Transform in quantum hydrodynamics. The GP equation

$$i\,\partial_t\psi + \Delta\psi + (1 - |\psi|^2)\psi = 0$$

can be cast into hydrodynamic variables by the polar decomposition

$$\psi(t,x) = \sqrt{p(t,x)}\,\exp(i\varphi(t,x)/\varepsilon)$$

where $p \geq 0$ is the density and $\varphi$ is the phase. Inserting into the GP equation and separating real and imaginary parts leads to the quantum hydrodynamics system: $$\begin{align*} &\partial_t p + \operatorname{div}(p \nabla\varphi) = 0 \quad \text{(continuity)}, \ &\partial_t \nabla\varphi + (\nabla\varphi \cdot \nabla)\nabla\varphi + \nabla p = \varepsilon^2 \nabla \left( \Delta\sqrt{p}/\sqrt{p} \right) \quad \text{(momentum)}. \end{align*}$$ The quantum pressure term $\varepsilon^2 \nabla(\Delta \sqrt{p}/\sqrt{p})$ encodes the dispersive, capillarity-like correction absent in classical Euler equations. The extended formulation using a complex vector $z = v + i w$ further aids energy estimates and well-posedness (Carles et al., 2011).

3. Nonlocal Gross–Pitaevskii Equations and the Gross Transform

A related Gross Transform is deployed in the analysis of nonlocal GP equations, especially those with Hartree-type nonlocal nonlinearities:

$$i \partial_t v - \Delta v = v \cdot (W * (1 - |v|^2)).$$

A transformation of variables $u = v - 1$ leads to analysis around the "ground state" $v = 1$. The nonlocal energy functionals and the convolution operator $W$ are controlled using harmonic analysis, particularly via the decay properties of the Fourier transform $\widehat{W}(\xi)$. The adoption of the I-method and careful energy increment estimates results in global well-posedness for large, rough data, even when the energy is not positive definite (Pecher, 2012).

4. Spectral Theory: Distorted Fourier Transform Around Vortices

The Gross Transform in spectral theory refers to the diagonalization of the linearized GP operator around a vortex by constructing a distorted Fourier transform. Let $\mathcal{A}$ be the linearized operator. Generalized eigenfunctions $\psi(\xi, r)$ solve

$$\mathcal{A} \psi(\xi, r) = \lambda(\xi) \psi(\xi, r), \qquad \lambda = \xi \sqrt{\xi^2 + 2}.$$

Define the distorted Fourier transform and its inverse as

$$\begin{align*} &\widetilde{\mathcal{F}}(\varphi)(\xi) = \int_0^\infty \psi(\xi, r) \sigma_3 \varphi(r) r dr, \ &\widetilde{\mathcal{F}}^{-1}(\zeta)(r) = \frac{1}{\pi} \int_{-\infty}^{+\infty} \zeta(\xi) \psi(\xi, r) \lambda'(\xi) \operatorname{sgn}(\xi) d\xi. \end{align*}$$

The evolution generated by $\mathcal{A}$ is then

$$e^{it\mathcal{A}} \varphi = \widetilde{\mathcal{F}}^{-1}(e^{it\lambda} \widetilde{\mathcal{F}}(\varphi)).$$

Sharp dispersive estimates are derived using stationary phase in the spectral variable $\xi$, with explicit decay rates and growth phenomena in $L^2$ due to zero-energy resonances (Collot et al., 4 Mar 2025, Luhrmann et al., 10 Mar 2025). The full construction relies on matching power-series solutions near the vortex core and oscillatory (Hankel-function) solutions at infinity.

5. Gross Transform in Arithmetic Geometry

In arithmetic geometry and the theory of modular forms, the Gross Transform encompasses lifting and transformation properties for invariants of quadratic forms. The classical Gross–Zagier formula relates geometric intersection numbers (e.g., the Néron–Tate heights of Heegner points) to derivatives of $L$-functions. Higher-dimensional generalizations link arithmetic intersection pairings of cycles on Shimura varieties to $L$-function derivatives:

$$L'(1/2, T) = 2^{-B_T} \langle z_0, z_0 \rangle_{BB}$$

for arithmetic cycles $z_0$ and automorphic representations $T$. The transform occurs via the relative trace formula, connecting test functions, Hecke correspondences, and distributional identities matching local orbital integrals to local intersection numbers; see formulas such as

$$\omega(y)\cdot 2\cdot \operatorname{Orb}(y, f', 0) = -2\cdot \operatorname{Int}(g)\cdot \log q.$$

This arithmetic Gross Transform synthesizes analytic and geometric data and is central to modern work on the Arithmetic Gan–Gross–Prasad conjecture (Zhang, 27 Feb 2024).

6. Connections to Quantum Fluids, Supersolids, and Phase Transitions

The existence and structure of positive-density ground states for the GP equation are informed by Gross Transform techniques in both variational analysis and Fourier space. For nonlocal potentials whose Fourier transform $\widehat{w}(k)$ takes negative values, the system undergoes a phase transition: at chemical potentials above a critical value $\mu_c$, constant fluid states cease to be ground states, giving way to spatially modulated (supersolid) or vortex-type configurations. The transform here links the spectral properties of the interaction potential to the macroscopic state realized in the quantum fluid, enabling rigorous phase diagrams and stability analysis (Lewin et al., 2023).

7. Computational Methods and Gross Transform Implementation

In numerical analysis of GP equations, the Gross Transform principle often motivates discretization strategies that enhance robustness to irregularities in coefficients or potentials. For example, the extended Fourier pseudospectral method (eFP), while not itself a Gross Transform, implements a variable projection (analogous in spirit to Gross Transform mappings) to achieve optimal convergence rates for rough potentials. Fourier interpolation and projection operators are computed on extended grids to ensure accuracy in the presence of low regularity (Bao et al., 2023).

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The Gross Transform, as documented in multiple research areas, serves as a unifying theme linking transformation techniques in nonlinear PDEs, spectral theory, arithmetic geometry, and computational analysis. It enables the translation of structural, spectral, and invariance properties between models, facilitating both analytic and computational breakthroughs across quantum mechanics, geometry, and number theory.