Nodal set for the Schr\"odinger equation under a local growth condition

TL;DR

This paper establishes an upper bound on the size of the nodal set for solutions to a Schrödinger equation with certain Sobolev coefficients, under a local doubling condition, advancing understanding of solution behavior in mathematical physics.

Contribution

It provides a new upper bound on the nodal set size for Schrödinger equations with Sobolev coefficients under local doubling assumptions, extending previous results.

Findings

Upper bound on the Hausdorff measure of the nodal set derived

Bound depends algebraically on Sobolev norms of coefficients

Results applicable to solutions satisfying local doubling 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 $V$.