Quadratic estimates and functional calculi for inhomogeneous first-order operators and applications to boundary value problems for Schrödinger equations

Turner, Andrew James (2021). Quadratic estimates and functional calculi for inhomogeneous first-order operators and applications to boundary value problems for Schrödinger equations. University of Birmingham. Ph.D.

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Abstract

We develop a holomorphic functional calculus for first-order operators DB to solve boundary value problems for Schrödinger equations −div A∇u + aVu = 0 in the upper half-space $ℝ^{n+1}_+$ when n ≥ 3. 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 the reverse Hölder class $B^{\frac{n}{2}} (ℝ^n)$ . The 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 coefficient matrices A and a are either a Hermitian or block structure. More generally, the set of all complex-elliptic A for which the boundary value problems are well-posed is shown to be open in $L^∞$ . We also prove these solutions coincide with those generated from the Lax–Milgram Theorem. Furthermore, we extend this theory to prove quadratic estimates for the magnetic Schrödinger operator (∇ + ib)∗A(∇ + ib) when the magnetic field curl (b) is in the reverse Hölder class $B^{\frac{n}{2}} (ℝ^n)$ .