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

Holomorphic functional calculus Kato square root estimatesWe develop a holomorphic functional calculus for firstorder operators DB to solve boundary value problems for Schrdinger equationsdiv Au +aV u = 0 in the upper halfspace R n+1 + with n 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 (R n ) or the reverse Hlder class B q (R n ) with q max{ 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 (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.