Hamiltonians Generated by Parseval Frames

It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a ${\mathcal{D}}$ D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of $E$ E -connection for observables. Several examples are discussed.

Data-Driven Redundant Transform Based on Parseval Frames

The sparsity of images in a certain transform domain or dictionary has been exploited in many image processing applications. Both classic transforms and sparsifying transforms reconstruct images by a linear combination of a small basis of the transform. Both kinds of transform are non-redundant. However, natural images admit complicated textures and structures, which can hardly be sparsely represented by square transforms. To solve this issue, we propose a data-driven redundant transform based on Parseval frames (DRTPF) by applying the frame and its dual frame as the backward and forward transform operators, respectively. Benefitting from this pairwise use of frames, the proposed model combines a synthesis sparse system and an analysis sparse system. By enforcing the frame pair to be Parseval frames, the singular values and condition number of the learnt redundant frames, which are efficient values for measuring the quality of the learnt sparsifying transforms, are forced to achieve an optimal state. We formulate a transform pair (i.e., frame pair) learning model and a two-phase iterative algorithm, analyze the robustness of the proposed DRTPF and the convergence of the corresponding algorithm, and demonstrate the effectiveness of our proposed DRTPF by analyzing its robustness against noise and sparsification errors. Extensive experimental results on image denoising show that our proposed model achieves superior denoising performance, in terms of subjective and objective quality, compared to traditional sparse models.