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 $R_{+}^{n+1}$ with $n\in N$. 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 $L^{n/2}\left(R^n\right)$ or the reverse Hölder class $B^q\left(R^n\right)$ 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 $L^2\left(R^n\right)$-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.