Nodal set for the Schrödinger equation under a local growth condition

Abstract

We address the upper bound on the size of the nodal set for a solution $w$ of the Schrödinger equation $\Delta w=W\cdot \nabla w+Vw$ in an open set in ${\mathbb{R}}^n$, where the coefficients belong to certain Sobolev spaces. Assuming a local doubling condition for the solution $w$, we establish an upper bound on the $(n-1)$-dimensional Hausdorff measure of the nodal set, with the bound depending algebraically on the Sobolev norms of $W$ and $V$.

arXiv comment24 pages