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

We develop a holomorphic functional calculus for first-order operators $DB$ to solve boundary value problems for Schr"{o}dinger equations $-$\mathrm{d}\mathrm{i}\mathrm{v}$, A \nabla u + a V u = 0$intheupperhalf-space$$\mathbb{R}$^{n+1}_+$with$n\in$\mathbb{N}$$\mathrm{.}Thisreliesonquadraticestimatesfor$DB$, 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 $L^{n\mathrm{/}2}({\mathbb{R}}^n)$ or the reverse H"{o}lder 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.