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Semiclassical resolvent bounds for weakly decaying potentials
Jeffrey Galkowski, Jacob Shapiro
In this note, we prove weighted resolvent estimates for the semiclassical Schrödinger operator , . The potential $…
Existence and asymptotics of nonlinear Helmholtz eigenfunctions
Jesse Gell-Redman, Andrew Hassell, Jacob Shapiro +1
We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (Δ- λ^2) u = N[u], \end{equation*} wher…
Semiclassical Estimates for Scattering on the Real Line
Kiril Datchev, Jacob Shapiro
We prove explicit semiclassical resolvent estimates for an integrable potential on the real line. The proof is a comparatively easy case of the spherical energies method, which has…
Semiclassical resolvent bound for compactly supported potentials
Jacob Shapiro
We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators in dimension . We require the potential belong to $L^\infty(\mathbb{R}…