**Title:** Refined Weyl law for the perturbed harmonic oscillator

**Speaker:** Moritz Doll

**Speaker Info:** Hannover

**Brief Description:**

**Special Note**:

**Abstract:**

Abstract: We consider the quantum harmonic oscillator $H_0 = 1/2 (\Delta + |x|^2)$. The underlying classical flow is periodic with period $2\pi$. By an explicit calculation one can see that the SchrÃ¶dinger propagator of $H_0$ is the identity (modulo a sign) at $2\pi \mathbb{Z}$ and smoothing otherwise.The first part of the talk will discuss propagation of singularities for potential perturbations of the harmonic oscillator. If the potential is a 1-symbol then the singularities reappear at $t = 2\pi$, but possibly at different locations. In the case of an arbitrary second order isotropic pseudodifferential operator, we obtain a similar description.

In the second part we consider the spectral theory of the perturbed harmonic oscillator. There we will show that for perturbations in a certain class of isotropic 1-symbols we obtain an improved remainder term in the Weyl law. This is based on joint work with Oran Gannot and Jared Wunsch.

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