Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data

Abstract

We develop a holomorphic functional calculus for first-order operators $DB$ to solve boundary value problems for Schrödinger equations $-\mathrm{div}\, A \nabla u + a V u = 0$ in the upper half-space $\mathbb{R}^{n+1}_+$ with $n\in\mathbb{N}$. This relies on quadratic estimates for $DB$, which are proved for coefficients $A,a,V$ that are independent of the transversal direction to the boundary, and comprised of a complex-elliptic pair $(A,a)$ that are bounded and measurable, and a singular potential $V$ in either $L^{n/2}(\mathbb{R}^n)$ or the reverse Hölder class $B^{q}(\mathbb{R}^n)$ with $q\geq\max\{\tfrac{n}{2},2\}$. In the latter case, square function bounds are also shown to be equivalent to non-tangential maximal function bounds. This allows us to prove that the (Dirichlet) Regularity and Neumann boundary value problems with $L^2(\mathbb{R}^n)$-data are well-posed if and only if certain boundary trace operators defined by the functional calculus are isomorphisms. We prove this property when the principal coefficient matrix $A$ has either a Hermitian or block structure. More generally, the set of all complex coefficients for which the boundary value problems are well-posed is shown to be open.

Theorems & Definitions (109)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3: Fefferman--Phong
• proof