Nodal set for the Schr\"odinger 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 Schr\"odinger equation $\Delta w= W\cdot \nabla w+V w$ 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 $V$.