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Register a new userEigenvalue approximation of sums of Hermitian matrices from eigenvector localization/delocalization
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We propose a technique for calculating and understanding the eigenvalue distribution of sums of random matrices from the known distribution of the summands. The exact problem is formidably hard. One extreme approximation to the true density amounts to classical probability, in which the matrices are assumed to commute; the other extreme is related to free probability, in which the eigenvectors are assumed to be in generic positions and sufficiently large. In practice, free probability theory can give a good approximation of the density. We develop a technique based on eigenvector localization/delocalization that works very well for important problems of interest where free probability is not sufficient, but certain uniformity properties apply. The localization/delocalization property appears in a convex combination parameter that notably, is independent of any eigenvalue properties and yields accurate eigenvalue density approximations. We demonstrate this technique on a number of examples as well as discuss a more general technique when the uniformity properties fail to apply.
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The Nearest Neighbour Spacing (NNS) distribution can be computed for generalized symmetric 2x2 matrices having different variances in the diagonal and in the off-diagonal elements. Tuning the relative value of the variances we show that the distributions of the level spacings exhibit a crossover from clustering to repulsion as in GOE. The analysis is extended to 3x3 matrices where distributions of NNS as well as Ratio of Nearest Neighbour Spacing (RNNS) show similar crossovers. We show that it is possible to calculate NNS distributions for Hermitian matrices (N=2, 3) where also crossovers take place between clustering and repulsion as in GUE. For large symmetric and Hermitian matrices we use interpolation between clustered and repulsive regimes and identify phase diagrams with respect to the variances.
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Eigenvector continuation has recently attracted a lot attention in nuclear structure and reactions as a variational resummation tool for many-body expansions. While previous applications focused on ground-state energies, excited states can be accessed on equal footing. This work is dedicated to a detailed understanding of the emergence of excited states from the eigenvector continuation approach. For numerical applications the one-dimensional quartic anharmonic oscillator is investigated, which represents a strongly non-perturbative quantum system where the use of standard perturbation techniques break down. We discuss how different choices for the construction of the EC manifold affect the quality of the EC resummation and investigate in detail the results from EC for excited states compared to results from a full diagonalization as a function of the basis space size.
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