Solvability for non-smooth Schr\"{o}dinger equations with singular potentials and square integrable data

TL;DR

This paper develops a functional calculus approach to solve boundary value problems for non-smooth Schrödinger equations with singular potentials, establishing conditions for well-posedness in the upper half-space.

Contribution

It introduces a novel holomorphic functional calculus for first-order operators to handle Schrödinger equations with singular potentials and proves well-posedness criteria for boundary value problems.

Findings

Quadratic estimates for $DB$ are established for complex-elliptic coefficients.

Square function bounds are shown to be equivalent to non-tangential maximal bounds.

Well-posedness of boundary problems is characterized by boundary trace operators being isomorphisms.

Abstract

We develop a holomorphic functional calculus for first-order operators $DB$ to solve boundary value problems for Schr\"{o}dinger 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 H\"{o}lder 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.