Spectral asymptotics for the Schrödinger equation with bounded, unstructured potentials

Abstract: High energy spectral asymptotics for Schrödinger operators on compact manifolds have been well studied since the early 1900s and it is now well known that they are intimately related to the structure of periodic geodesics. In this talk, we discuss analogous questions for Schrödinger operators, $-\Delta +V$ on $\mathbb{R}^d$, where $V$ is bounded together with all of its derivatives. Since the geodesic flow on $\mathbb{R}^d$ has no periodic trajectories (or indeed looping trajectories) one might guess that the spectral projector has a full asymptotic expansion. Indeed, for (quasi) periodic $V$ this has been known since the work of Parnovski–Shterenberg in 2016. We show that when $d=1$, full asymptotic expansions continue to hold for any such $V$. When $d=2$, we give a large class of potentials whose spectral projectors have full asymptotics. Nevertheless, in $d\geq 2$, we construct examples where full asymptotics fail. Based on joint work with L. Parnovski and R. Shterenberg.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsoptimization and control

Audience: researchers in the topic

Potomac region PDE seminar

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