Applications of the landscape function for Schrödinger operators with singular potentials and irregular magnetic fields

Abstract

We resolve both a conjecture and a problem of Z. Shen from the 90's regarding non-asymptotic bounds on the eigenvalue counting function of the magnetic Schrödinger operator $L_{\mathbf{a},V}=-(\nabla -i\mathbf{a}{)}^2+V$ with a singular or irregular magnetic field $\mathbf{B}$ on ${\mathbb{R}}^n$, $n\ge 3$. We do this by constructing a new landscape function for $L_{\mathbf{a},V}$, and proving its corresponding uncertainty principle, under certain directionality assumptions on $\mathbf{B}$, but with no assumption on $\nabla \mathbf{B}$. These results arise as applications of our study of the Filoche-Mayboroda landscape function $u$, a solution to the equation $L_{V}u=-\mathrm{div}A\nabla u+Vu=1$, on unbounded Lipschitz domains in ${\mathbb{R}}^n$, $n\ge 1$, and $0\le V\in L_{\mathrm{loc}}^1$, under a mild decay condition on the Green's function. For $L_V$, we prove a priori exponential decay of Green's function, eigenfunctions, and Lax-Milgram solutions in an Agmon distance with weight $1/u$, which may degenerate. Similar a priori results hold for $L_{\mathbf{a},V}$. Furthermore, when $n\ge 3$ and $V$ satisfies a scale-invariant Kato condition and a weak doubling property, we show that $1/\sqrt{u}$ is pointwise equivalent to the Fefferman-Phong-Shen maximal function $m(\cdot ,V)$; in particular this gives a strong scale-invariant Harnack inequality for $u$, and a setting where the Agmon distance with weight $1/u$ is not too degenerate. Finally, we extend results from the literature for $L_{\mathbf{a},V}$ regarding exponential decay of the fundamental solution and eigenfunctions, to the situation of irregular magnetic fields with directionality assumptions.

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