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

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=−(∇−ia)+V with a singular or irregular magnetic field B on R, n≥3. We do this by constructing a new landscape function for L, and proving its corresponding uncertainty principle, under certain directionality assumptions on B, but with no assumption on ∇B. These results arise as applications of our study of the Filoche-Mayboroda landscape function u, a solution to the equation Lu=−divA∇u+Vu=1, on unbounded Lipschitz domains in R, n≥1, and 0≤V∈L , under a mild decay condition on the Green's function. For L, 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. Furthermore, when n≥3 and V satisfies a scale-invariant Kato condition and a weak doubling property, we show that 1/u is pointwise equivalent to the Fefferman-Phong-Shen maximal function m(⋅,V) (also known as Shen's critical radius function); in particular this gives a setting where the Agmon distance with weight 1/u is not too degenerate. Finally, we extend results from the literature for L regarding exponential decay of the fundamental solution and eigenfunctions, to the situation of irregular magnetic fields with directionality assumptions.