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Description
Quantum mechanics and eigenfunction concentration in phase space
In quantum mechanics, systems are represented by self-adjoint operators on $L^2(\mathbb{R}^d)$, and the time evolution can be written in terms of the eigenvalues and eigenfunctions thanks to the spectral theorem. In the specific framework of semiclassical analysis, the data depend on a small parameter $h$, which mathematically means that we study the semiclassical limit $h\rightarrow 0$. In this asymptotic, the eigenfunctions concentrate on subsets of $\mathbb{R}^d$, which correspond to regions where the system could be if it was classical (non-quantum). However, understanding the rate at which they concentrate is more challenging.
In a first part I will introduce the mathematical formulation of quantum mechanics, and the notion of quantisation. In a second part, we will see how the eigenfunctions behave in the semiclassical limit with an explicit example, and with some references. In particular, we will see that the Fourier transforms of the eigenfunctions also concentrate, and that a specific tool, called the FBI transform, enables to describe both concentration simultaneously. Finally, I will present a different class of operators for which the concentration can be described even on compact manifolds.
