Strichartz estimates for a class of Schrödinger equations with a drift

1. Problem Statement and Context

The paper investigates the Cauchy problem for a class of possibly degenerate Schrödinger equations with a real drift term in $\mathbb{R}^n \times [0, \infty)$:
$$\partial_t u - i \text{tr}(Q\nabla^2 u) - \langle Bx, \nabla u \rangle = F(x, t), \quad u(x, 0) = \phi(x)$$
where:

• $Q \in \mathbb{R}^{n \times n}$ is a symmetric, positive semi-definite matrix ($Q = Q^*, Q \ge 0$).
• $B \in \mathbb{R}^{n \times n}$ is a constant drift matrix.
• The equation may be highly degenerate (e.g., $\text{Rank}(Q) = 1$), modeling systems like quantum wires or kinetic equations.
• The presence of the drift term $\langle Bx, \nabla u \rangle$ destroys the standard variational structure and global dilation invariance found in the classical Schrödinger equation ($Q=I, B=0$).

The Core Challenge:
Establishing Strichartz estimates (mixed-norm space-time bounds) for solutions. In the classical case, these estimates rely on the dispersion of the wave packet and the scaling invariance of the Laplacian. In this generalized setting, the degeneracy of $Q$ and the drift $B$ complicate the dispersion mechanism, and the lack of global homogeneity prevents the direct application of standard scaling arguments.

Key Assumption (H):
The authors assume a controllability condition: $\text{Ker}(Q)$ contains no non-trivial invariant subspace of $B^*$. This is equivalent to the strict positivity of the controllability Gramian $Q(t) = \int_0^t e^{sB} Q e^{sB^*} ds$ for all $t > 0$. This ensures the equation is hypoelliptic (in the sense of Hörmander) and admits a well-defined fundamental solution.

2. Methodology

The authors employ a combination of semigroup theory, Fourier analysis, and geometric control theory to derive the estimates.

A. The Schrödinger Group and Fundamental Solution

Using the representation formula derived in previous works (e.g., [17]), the solution operator $T(t)$ is expressed via the fundamental solution involving the volume function $V(t) = \det Q(t)$.
The authors extend the semigroup $T(t)$ defined for $t \ge 0$ to a group $U(t)$ on the entire real line $\mathbb{R}$. This extension is crucial for applying the $T^*T$ argument. The group action incorporates the drift via the transformation $v(x,t) = u(e^{-tB}x, t)$, which removes the drift term but introduces a time-dependent diffusion matrix.

B. The $T^*T$ Approach (Ginibre-Velo)

The core of the proof relies on the Ginibre-Velo $T^*T$ method, originally developed for the classical Schrödinger equation.

1. Operator Definition: Define an operator $T^*: L^2(\mathbb{R}^n) \to L^\infty_t L^2_x$ by $T^*\phi(x,t) = e^{\frac{t}{2}\text{tr} B} U(t)\phi(x)$.
2. Adjoint and Composition: Compute the adjoint $T$ and the composition $T^*T$. The kernel of $T^*T$ involves the dispersive decay of the group $U(t-s)$.
3. Dispersive Inequality: The authors establish a dispersive estimate for the group:
   $$\|U(t)\phi\|_{L^r} \le C |t|^{-\frac{D}{2}(1/2 - 1/r)} e^{-\frac{t}{r}\text{tr} B} \|\phi\|_{L^{r'}}$$
   (Note: The exponential weight depends on the sign of $t$ and $\text{tr} B$).
4. Hardy-Littlewood-Sobolev (HLS): The composition $T^*T$ is shown to be a fractional integration operator (convolution with a kernel behaving like $|t|^{-\alpha}$). The boundedness of this operator from $L^{q'}_t L^{r'}_x$ to $L^q_t L^r_x$ is established using the HLS theorem.

C. The Role of the Volume Function $V(t)$

The decay rate of the solution is governed by the asymptotic behavior of the volume function $V(t) = \det Q(t)$.

• The authors define a local homogeneous dimension $D$ based on the canonical form of the matrices $Q$ and $B$ (specifically the ranks of blocks in the Jordan-like decomposition).
• They analyze the large-time behavior of $V(t)$ to determine whether the decay follows a power law $t^D$ (Hypothesis A) or a mixed behavior involving a lower dimension $D_\infty$ (Hypothesis B).

3. Key Contributions

A. Definition of Local Homogeneous Dimension ($D$)

The paper introduces a novel definition of the dimension $D$ for degenerate operators with drift:
$$D = p_0 + 3p_1 + \dots + (2r+1)p_r$$
where $p_j$ are the ranks of specific blocks in the canonical form of $B$ relative to $Q$.

• If $Q$ is invertible, $D=n$.
• If $Q$ is degenerate, $D > n$.
  This $D$ dictates the admissible Strichartz pairs $(q, r)$ via the scaling relation:
  $$\frac{2}{q} = D \left( \frac{1}{2} - \frac{1}{r} \right)$$

B. Two Main Theorems (A and B)

The authors prove Strichartz estimates under two distinct hypotheses regarding the growth of $V(t)$:

• Theorem A (Hypothesis A): Assumes $V(t) \ge \gamma t^D$ for all $t > 0$.

  • This applies when the drift has eigenvalues with positive real parts or when $B$ is nilpotent in a specific form (allowing global dilations).
  • Result: Standard Strichartz estimates with exponential weights $e^{\frac{t}{2}\text{tr} B}$. If $\text{tr} B = 0$, the weights vanish, recovering the classical form but with dimension $D$.

• Theorem B (Hypothesis B): Assumes $V(t) \ge \gamma \min(t^D, t^{D_\infty})$ where $D_\infty < D$.

  • This applies to "anomalous" cases where the volume growth is slower at infinity (e.g., when $B$ is skew-symmetric but $Q$ is degenerate).
  • Result: The estimates hold in a sum of spaces ($L^q + L^{q_\infty}$), reflecting the different decay rates at small and large times.

C. Exponential Weights

A significant technical contribution is the rigorous derivation of exponential weights ($e^{\frac{t}{2}\text{tr} B}$) in the estimates. The authors show these arise naturally from the change of variables to conformal self-similar coordinates and the non-unitary nature of the drift group. They demonstrate that these weights are necessary to balance the growth/decay induced by the drift.

D. Classification of Scenarios

The paper provides a comprehensive classification of when Hypothesis A or B holds based on the spectrum of $B$:

1. Generic Case: If $\sigma(B) \not\subset i\mathbb{R}$ or $Q$ is invertible, Hypothesis A holds.
2. Skew-Symmetric Drift: If $B$ is similar to a skew-symmetric matrix and $Q$ is degenerate, Hypothesis B holds with $D_\infty = n$.
3. Anomalous Case: The authors identify a rare scenario where $D_\infty$ can be either greater or less than $D$, requiring a case-by-case analysis of the volume function.

4. Main Results

1. Generalized Strichartz Estimates: The paper establishes that solutions to degenerate Schrödinger equations with drift satisfy Strichartz estimates where the admissible exponents are determined by the local homogeneous dimension $D$ rather than the spatial dimension $n$.
2. Optimality: In cases where global non-isotropic dilations exist (e.g., $B$ is nilpotent in a specific form), the derived exponents are shown to be optimal.
3. Examples:
   • Quantum Wires: The framework applies to 1D nanostructures where electrons are confined, leading to degenerate $Q$.
   • Twisted Laplacians: The results cover Schrödinger equations on Heisenberg-type groups, linking the drift to the sub-Laplacian structure.
   • Kolmogorov Operators: The paper recovers and generalizes estimates for operators like $\partial_t - i\Delta_x - \langle x, \nabla_y \rangle$, showing the dimension $D$ doubles the spatial dimension ($D=2n$).

5. Significance

• Bridging Degeneracy and Dispersion: The work successfully extends the powerful Strichartz machinery to highly degenerate settings where standard scaling fails. It proves that dispersion is still governed by a "local" geometric dimension $D$.
• Unified Framework: It unifies the treatment of various physical models (quantum wires, kinetic theory, sub-Riemannian geometry) under a single algebraic condition (Hypothesis H) and a unified set of estimates.
• New Analytical Tools: The introduction of the volume function $V(t)$ as the primary driver for decay rates, and the detailed analysis of its asymptotic behavior, provides a new toolkit for analyzing hypoelliptic evolution equations.
• Foundation for Nonlinear Problems: By establishing these linear a priori estimates, the paper lays the groundwork for future studies on nonlinear Schrödinger equations with drift and degeneracy, which are relevant in quantum mechanics and fluid dynamics.

In summary, Buseghin and Garofalo have generalized the theory of Strichartz estimates to a broad class of degenerate, drift-driven Schrödinger equations, identifying the correct geometric dimension $D$ that governs dispersion and providing rigorous proofs for the existence of these estimates under various spectral conditions of the drift matrix.