Parent Hamiltonians of Jastrow wavefunctions

We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to an arbitrary spatial dimension. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.

Trimmed estimators for large dimensional sparse covariance matrices

In this paper, we will propose two new estimators for sparse covariance matrix. Our starting point is to make the estimator of each element of covariance matrix more robust. More precisely, we will trim the observations for each pairwise product of components of population as a first step. Then we form the sample covariance matrices based on the trimmed data. Finally, we apply the thresholding to the derived sample covariance matrices. These two new estimators will be shown to achieve the optimal convergence rate.