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 $-\mathrm{div}A\nabla u+aVu=0$ 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 $q\ge \max \{\frac{n}{2},2\}$. 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.

63 pages, minor updates prior to publication, to be published in Journal of Functional Analysis