$L^p$ Maximal regularity for vector-valued Schrödinger operators

TL;DR

This work develops $L^p$-maximal regularity theory for vector-valued Schrödinger operators $-\Delta+V$ with matrix-valued potentials $V$ that are symmetric, nonnegative, and have nonpositive off-diagonal entries. The authors combine a vector-valued Fefferman--Phong inequality, $L^1$-maximal estimates via resolvent approximations that preserve positivity, and a vector-valued perturbation framework to obtain $L^p$-maximal bounds for $p$ in a range determined by the reverse Hölder class of the minimal eigenvalue $\lambda_V$, as well as generation and sectoriality results for the associated $L^p$-realizations. The theory extends scalar results to the vector-valued setting, providing domain characterizations and analytic semigroup generation under unbounded matrix coefficients, and includes concrete examples with singular or polynomial-growth potentials. Overall, the results yield well-posedness and regularity for vector-valued elliptic problems with unbounded coefficients and have implications for coupled systems in PDE, spectral theory, and semigroup analysis.

Abstract

In this paper we consider the vector-valued Schrödinger operator $-Δ+ V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in $L^1(\mathbb{R}^d,\mathbb{R}^m).$ Assuming further that the minimal eigenvalue of $V$ belongs to some reverse Hölder class of order $q\in(1,\infty)\cup\{\infty\}$, we obtain maximal inequality in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $p$ in between $1$ and some $q$.

Theorems & Definitions (41)

• Proposition 3.1
• proof
• Remark 4.1
• Lemma 4.2
• proof
• Remark 4.3
• Lemma 5.1
• proof
• Lemma 5.2
• proof