Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well

Nonlocal Hamiltonian-type operators, like e.g. fractional and quasirelativistic, seem to be instrumental for a conceptual broadening of current quantum paradigms. However physically relevant properties of related quantum systems have not yet received due (and scientifically undisputable) coverage in the literature. In the present paper we address Schr\"{o}dinger-type eigenvalue problems for $H=T+V$, where a kinetic term $T=T_m$ is a quasirelativistic energy operator $T_m = \sqrt{-\hbar ^2c^2 \Delta + m^2c^4} - mc^2$ of mass $m\in (0,\infty)$ particle. A potential $V$ we assume to refer to the harmonic confinement or finite well of an arbitrary depth. We analyze spectral solutions of the pertinent nonlocal quantum systems with a focus on their $m$-dependence. Extremal mass $m$ regimes for eigenvalues and eigenfunctions of $H$ are investigated: (i) $m\ll 1$ spectral affinity ("closeness") with the Cauchy-eigenvalue problem ($T_m \sim T_0=\hbar c |\nabla |$) and (ii) $m \gg 1$ spectral affinity with the nonrelativistic eigenvalue problem ($T_m \sim -\hbar ^2 \Delta /2m$). To this end we generalize to nonlocal operators an efficient computer-assisted method to solve Schr\"{o}dinger eigenvalue problems, widely used in quantum physics and quantum chemistry. A resultant spectrum-generating algorithm allows to carry out all computations directly in the configuration space of the nonlocal quantum system. This allows for a proper assessment of the spatial nonlocality impact on simulation outcomes. Although the nonlocality of $H$ might seem to stay in conflict with various numerics-enforced cutoffs, this potentially serious obstacle is kept under control and effectively tamed.Comment: 23 pages, 16 figure

Spectral equations for the modular oscillator

Motivated by applications for non-perturbative topological strings in toric Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting modular conjugate (in the sense of Faddeev) Harper type operators, corresponding to a special case of the quantized mirror curve of local $\mathbb{P}^1\times\mathbb{P}^1$ and complex values of Planck's constant. We illustrate our analytical results by numerical calculations.Comment: 23 pages, 9 figures, references added and interpretation of the numerical results of Section 6 correcte

High-Fidelity and Perfect Reconstruction Techniques for Synthesizing Modulation Domain Filtered Images

Biomimetic processing inspired by biological vision systems has long been a goal of the image processing research community, both to further understanding of what it means to perceive and interpret image content and to facilitate advancements in applications ranging from processing large volumes of image data to engineering artificial intelligence systems. In recent years, the AM-FM transform has emerged as a useful tool that enables processing that is intuitive to human observers but would be difficult or impossible to achieve using traditional linear processing methods. The transform makes use of the multicomponent AM-FM image model, which represents imagery in terms of amplitude modulations, representative of local image contrast, and frequency modulations, representative of local spacing and orientation of lines and patterns. The model defines image components using an array of narrowband filterbank channels that is designed to be similar to the spatial frequency channel decomposition that occurs in the human visual system. The AM-FM transform entails the computation of modulation functions for all components of an image and the subsequent exact recovery of the image from those modulation functions. The process of modifying the modulation functions to alter visual information in a predictable way and then recovering the modified image through the AM-FM transform is known as modulation domain filtering. Past work in modulation domain filtering has produced dramatic results, but has faced challenges due to phase wrapping inherent in the transform computations and due to unknown integration constants associated with modified frequency content. The approaches developed to overcome these challenges have led to a loss of both stability and intuitive simplicity within the AM-FM model. In this dissertation, I have made significant advancements in the underlying processes that comprise the AM-FM transform. I have developed a new phase unwrapping method that increases the stability of the AM-FM transform, allowing higher quality modulation domain filtering results. I have designed new reconstruction techniques that allow for successful recovery from modified frequency modulations. These developments have allowed the design of modulation domain filters that, for the first time, do not require any departure from the simple and intuitive nature of the basic AM-FM model. Using the new modulation domain filters, I have produced new and striking results that achieve a variety of image processing tasks which are motivated by biological visual perception. These results represent a significant advancement relative to the state of the art and are a foundation from which future advancements in the field may be attained