Approximation of Feynman path integrals with non-smooth potentials

We study the convergence in $L^2$ of the time slicing approximation of Feynman path integrals under low regularity assumptions on the potential. Inspired by the custom in Physics and Chemistry, the approximate propagators considered here arise from a series expansion of the action. The results are ultimately based on function spaces, tools and strategies which are typical of Harmonic and Time-frequency analysis.Comment: 18 page

Exploration of the Transference of Cognitive Skills Gained from a Movement-Based Program Incorporating Modified Dance to Occupational Performance for Individuals with Parkinson’s Disease

A movement-based program incorporating modified dance was used with individuals with Parkinson’s disease (PD) to assess changes in cognition as it relates to occupational performance. This qualitative research study provided 1-hour dance sessions 3 days per week over the course of 16 weeks with 6 participants who have mild-moderate PD. Various dance styles were used including salsa, tango, waltz, line dancing, and others. Cognition and occupational performance were assessed using the Montreal Cognitive Assessment (MOCA), the Canadian Occupational Performance Measure (COPM), and surveys. Findings indicated improvements in perceived occupational performance via the COPM and surveys, while MoCA results indicated improvements in the areas of visuospatial/executive function, attention, and memory recall. Further research is warranted to address limitations of this research study and explore the possible benefits of dance as a movement-based intervention to address cognitive deficits within the Parkinson’s population.https://soar.usa.edu/otdcapstones-spring2022/1025/thumbnail.jp

Time-frequency analysis of the Dirac equation

The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for the nonlinear Dirac equation with a general nonlinearity, including power-type terms and the Thirring model. For this study we adopt the unifying framework of vector-valued time-frequency analysis as developed by Wahlberg; most of the preliminary results are stated under general assumptions and hence they may be of independent interest.Comment: 26 page

On the convergence of a novel time-slicing approximation scheme for Feynman path integrals

In this note we study the properties of a sequence of approximate propagators for the Schrodinger equation, in the spirit of Feynman's path integrals. Precisely, we consider Hamiltonian operators arising as the Weyl quantization of a quadratic form in phase space, plus a bounded potential perturbation in the form of a pseudodifferential operator with a rough symbol. The corresponding Schrodinger propagator belongs to the class of generalized metaplectic operators, a fact that naturally motivates the introduction of a manageable time-slicing approximation scheme consisting of operators of the same type. By leveraging on this design and techniques of wave packet analysis we are able to prove several convergence results with precise rates in terms of the mesh size of the time slicing subdivision, even stronger then those that can be achieved under the same assumptions using the standard Trotter approximation scheme. In particular, we prove convergence in the norm operator topology in L-2, as well as pointwise convergence of the corresponding integral kernels for non-exceptional times

The role of G-CSF in the treatment of advanced tumors

Commentary to:Peritumoral administration of granulocyte colony-stimulating factor induces an apoptotic response on a murine mammary adenocarcinomaJulieta Marino, Veronica A. Furmento, Elsa Zotta, Leonor P. Rogui

Phase space analysis of spectral multipliers for the twisted Laplacian

We prove boundedness results on modulation and Wiener amalgam spaces concerning some spectral multipliers for the twisted Laplacian. Techniques of pseudo-differential calculus are inhibited due to the lack of global ellipticity of the special Hermite operator, therefore a phase space approach must rely on different pathways. In particular, we exploit the metaplectic equivalence relating the twisted Laplacian with a partial harmonic oscillator, leading to a general transference principle for spectral multipliers. We focus on a wide class of oscillating multipliers, including fractional powers of the twisted Laplacian and the corresponding dispersive flows of Schr\"odinger and wave type. On the other hand, elaborating on the twisted convolution structure of the eigenprojections and its connection with the Weyl product of symbols, we obtain a complete picture of the boundedness of the heat flow for the twisted Laplacian. Results of the same kind are established for fractional heat flows via subordination.Comment: 30 page