Sharp Semiclassical Spectral Asymptotics for Local Magnetic Schrödinger Operators on \({\mathbb {R}}^d\) Without Full Regularity

We consider operators acting in \(L^2({\mathbb {R}}^d)\) $L^2({\mathbb{R}}^d)$ with \(d\ge 3\) $d\ge 3$ that locally behave as a magnetic Schrödinger operator. For the magnetic Schrödinger operators, we suppose the magnetic potentials are smooth and the electric potential is five times differentiable and the fifth derivatives are Hölder continuous. Under these assumptions, we establish sharp spectral asymptotics for localised counting functions and Riesz means.