Sharp Pointwise Weyl Laws for Schr\"odinger Operators with Singular Potentials on Flat Tori

The Weyl law of the Laplacian on the flat torus $\mathbb{T}^n$ is concerning the number of eigenvalues $\le\lambda^2$, which is equivalent to counting the lattice points inside the ball of radius $\lambda$ in $\mathbb{R}^n$. The leading term in the Weyl law is $c_n\lambda^n$, while the sharp error term $O(\lambda^{n-2})$ is only known in dimension $n\ge5$. Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. Moreover, this result verifies the sharpness of the general theorems for the Schr\"odinger operators $H_V=-\Delta_{g}+V$ in the previous work of the authors, and extends the 3-dimensional results of Frank-Sabin to any dimensions.Comment: 58 pages. To appear in Communications in Mathematical Physic

Experimental demonstration of the criterion for the prepare-and-measure nonlocality

The prepare-and-measure theory is a new type of quantum paradox that reveals the incompatibility between classical theory and quantum mechanics in terms of the dimensionality of physical systems. Just as the Horodecki criterion can determine whether given quantum states are capable of exhibiting Bell nonclassicality, a similar criterion is needed for the prepare-and-measure theory to determine whether given quantum states can exhibit the prepare-and-measure nonclassicality.Recently, Poderini \emph{et al.} [Phys. Rev. Research 2, 043106 (2020)] presented such a criterion for the prepare-and-measure nonclassicality.In this work, we experimentally validate this criterion -- 52 different sets of quantum states are prepared and tested one by one using this criterion to determine whether they can exhibit the prepare-and-measure nonclassicality, and the experimental results are in good agreement with the theoretical expectations. The criterion experimentally verified here has the potential to be widely used in future research on the prepare-and-measure nonclassicality

Synthesis of Azido block copolymers for Targeting and Labeling Micelle

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