Ivrii's generic spectral asymptotics conjecture for even-dimensional magnetic Schrödinger operators

Let $d$ be even, let $g^{jk}$ be the metric coefficients, let $(V_j)$ be the magnetic vector potential with magnetic intensity matrix $(F_{jk})$, and let $V$ be the electric potential. Write ${\mathcal E}^{\rm {MW}}$ for the magnetic Weyl principal term, and let ${\mathfrak A}_g$ and ${\mathfrak B}_g$ be the nowhere dense closed exceptional sets of magnetic potentials and pairs of magnetic and electric potentials, respectively. Assume

$$\mu\leq c h^{-1}.$$

Ivrii's conjecture. The main part of the spectral asymptotics is given by ${\mathcal E}^{\rm {MW}}$, while: (i) for fixed $(g^{jk})$ and general $(V_j),V$, the remainder is $O(\mu h^{1-d})$; (ii) for fixed $(g^{jk})$ and generic $(V_j)$, meaning $(V_1,\dots,V_d)\notin{\mathfrak A}_g$, and general $V$, the remainder is

$$O(\mu^{-1}h^{1-d}+\mu^{d/2}h^{-d/2})$$

when $(F_{jk})$ has full rank everywhere, and

$$O(\mu^{-1/2}h^{1-d}+\mu^{d/2}h^{-d/2})$$

otherwise; (iii) for fixed $(g^{jk})$ and generic $(V_j,V)$, meaning $(V_1,\dots,V_d;V)\notin{\mathfrak B}_g$, the remainder is

$$O(\mu^{-1}h^{1-d})$$

when $(F_{jk})$ has full rank everywhere, and

$$O(\mu^{-1/2}h^{1-d})$$

otherwise. This predicts sharp remainder estimates for the magnetic Weyl asymptotics in the generic even-dimensional setting, distinguishing general potentials from potentials outside the exceptional sets and full-rank magnetic fields from degenerate ones.